text:
High Wycombe 3°c
Our Facebook feedsBucks Free Press
Our Twitter feeds@BucksFreePress
Best of Bucks
'Is High Wycombe now 'Aldi Town'?' - readers' letters
This is what you have been writing to us about this week.
To send your own letter, email bfpletters@london.newsquest.co.uk.
Please note, any letters sent to the Bucks Free Press office are only being picked up periodically as all staff are still working from home.
LETTER OF THE WEEK: Remembering lost loved ones over festive period
There has been a lot of discussion but, even as our lives are curtailed by another lockdown, I don't think Christmas 2020 will be cancelled.
In fact, we will need the light and love that Christmas brings more than ever after this tumultuous year.
I am writing to invite your readers to join me for The Hospice of St Francis' virtual Light up a Life event on Sunday, December 6 at 4.30pm. Everyone is welcome, regardless of whether you have a hospice connection.
We will be uniting virtually as a community through inspiring music from local young people and comforting words; pausing to light a candle to remember loved ones and our heroic key workers in this short half hour reflection.
If you would like to dedicate a star to a special person to be featured online and on the Hospice's tree of stars, or watch the virtual event, please visit – stfrancis.org.uk/lual or call 01442 869550.
The Hospice of St Francis has continued to be sustained by your ongoing support. For this we thank you.
We hope to see you at Light up a Life but, if you can't be with us, may every blessing be on you and your family this wintertime.
Pru Murray, Hospice of St Francis
'Is High Wycombe now 'Aldi town'?'
So, we are to have a third Aldi supermarket in High Wycombe. Aren't we lucky.
Bicester is known as Tesco Town because of the number of these particular stores in the area. Can we now call High Wycombe Aldi Town? Are we so lacking in our choice of supermarkets that we need yet another one?
Originally we just had Sainsbury's, Tesco and Safeway. Now we have, Sainsburys, Morrisons, Asda, Waitrose, Iceland, two Large Tescos, two Lidl's and soon three Aldis, plus yet another McDonald's adding to the two already in our town.
With this new Aldi opening in Cressex and a new Tesco Express due in Hazlemere, aren't we getting slightly over populated with these supermarkets?
We used to be called a furniture town. Now we seem to be advertising ourselves as a supermarket town.
Michael Curry, Penn
Change negative perception of town
Eight months ago, Buckinghamshire Council took over from Wycombe District Council. In that time, the world has changed due to Covid-19. Shops, gyms, office spaces and leisure venues have been wounded; many may close forever.
There's also been an unprecedented shift in city dwellers seeking to move to the shires in search of a garden, fresh air and more space at home in which to set up a home office.
The latter might help fix the former, if we play our cards right.
High Wycombe town centre has long been good for most shopping needs, more so since Eden arrived in 2008, and HWBIDCo has done a great job to dress and promote the town, especially in the last two years.
Thanks to the council, there are welcome physical improvements currently being made to the road surface in the High Street.
In last week's edition of this paper, we were told that the historic Brunel Train Shed will be renovated at a cost of £4m. We were also told more about the renovation of historic 2/3 High Street, and that the new Centre Square development is tipped for an architectural award.
So why doesn't any of this make a difference to people's perception of the town?
Despite these worthy projects, it's hard to shift negative perceptions – and grab those London exiles to come and live and spend their money here – unless you do something really attention grabbing, and then shout loud from the treetops about it. I believe the answer lies beneath our feet.
Let's give those new London exiles what they want – fresh air, nature and an historic river that's been consigned to a concrete tunnel for too long. We townsfolk want it too.
The High Wycombe Society has long supported the deculverting of the river Wye. The confluence of the river and the roads facilitated trade, milling of paper and flour, furniture and other successful industries. In short, it's the reason that the town exists at all.
Martin Tett and his new council could prove their mettle on this issue. Since the controversial burying of the river in 1967, no council leader has proved capable of bringing it back to the surface.
No town with a waterway would think for one minute of hiding it underground these days; it would be too damaging to the fundamental appeal of the place which is the single biggest reason that people are drawn to a place to live and spend their money on local shops, restaurants and other businesses.
We are all attracted to nature and beauty.
Heather Morley, High Wycombe
'Can council deliver on climate action promise?'
In July, Buckinghamshire Council, under your leadership, committed to taking meaningful action on climate change, and to produce a Strategy and Action Plan by February 2021.
At a council committee meeting on November 12, the results of a survey of Bucks residents were published, showing that 95% were concerned or very concerned about climate change.
Yet at that same meeting, one of your cabinet team announced a surprise delay in the production of the promised strategy, and a planned public consultation on a draft was pulled at the last minute.
No proper explanation was given for the delay, no indication of a revised timetable, and no summary of the emerging thinking.
We recognise that producing and implementing a comprehensive strategy is not a quick or easy task, and we remain ever ready to work constructively with you to help achieve this.
But where is the necessary sense of urgency and purpose? And where is the evidence of actions being taken now to address the climate emergency, actions that do not need and cannot wait for a 'perfect' strategy to arrive?
The situation demands that consideration of climate change impacts has, without delay, to be a key factor in all decision-making across all council functions and all policy areas.
This is not about spending money we haven't got – it is about HOW we spend the money, about investment now for a better and less costly future. And one without runaway climate change.
Please, Cllr. Tett, demonstrate to the 95% that your council is serious about delivering on your climate action promise.
Mike Chadwick, Coordinator, Bucks Climate Action Alliance
PM's Green Industrial Revolution plan welcome
The Prime Minister's ten-point plan for a Green Industrial Revolution is most welcome. It follows on from his speech in October and we have received many hints that it was imminent.
The plan rightly focusses upon the need to harness the talents of British enterprise, to nurture the skills of its workforce and to affirm the drive and innovation of English regions and Scotland and Wales.
However, I am a little concerned at no specific reference to solar PV power. You could read behind the lines and anticipate that plans to 'make homes, schools and hospitals greener' could implicitly mean solar power is utilised.
Further, the 'task force net zero' might well develop detailed plans that are inclusive of solar. Neither is there a mention of offshore wind.
The public recognise the importance of solar power, in recent polling commissioned from the Bright Blue think-tank, stated that 'A majority of the public thinks that we need to use more solar (72%), wind (69%), tidal (64%) and hydro (62%) energy to achieve net zero'.
I would also add that the recent IEA report, 'World Energy Outlook 2020' clearly inferred that solar power has an instrumental role in the future generation of electricity.
Thus, I am a little perplexed as to why the significant potential for developing solar power appears to have dropped off government's radar?
With the long-awaited Energy White Paper, a potential revision of the Industrial Strategy and next year's Cop26 on our shores the UK still has an opportunity to show the scale of its ambition both home and abroad.
Just as the original Industrial Revolution was diverse in its composition, a genuine 'Green Industrial Revolution' will be characterised by all forms of clean energy; be it solar, offshore wind, carbon capture and storage or wave and tidal energy.
Yet their potential needs to be capitalised upon and explicit government support is crucial.
The stakes are too high and the opportunities too pressing for us not to rise to this challenge.
The ten-point plan has much to commend it but it also begs some questions.
Andrew Moore, CEO, UKSOL Ltd, Gerrards Cross
How was school dome built without permission?
I was pleased to see in the November 20 BFP that Buckinghamshire Council has refused planning consent for the large dome at Holmer Green Senior School - but can anyone explain how the situation arose?
So far as I know Buckinghamshire Council own the school and Buckinghamshire Council is the planning authority.
Surely the school is required to ask the council before embarking on such a project. Something does not add up.
Name and address withheld
Help raise money for kids in hospital
Over 12,000 children are admitted to hospital every day in the UK and it has never been a scarier time for them.
With unfamiliar wards, health professionals in additional PPE, only being able to have one parent with them, and no visits from their brothers or sisters during the pandemic, it can be very worrying and lonely.
That is why play and distraction services in hospitals are now more vital than ever.
Having access to toys and entertainment, when recovering or waiting for treatment, or using play as a distraction for procedures can help to alleviate pain, aid recovery and bring joy back into children's lives.
At the children's charity Starlight, thanks to our supporters, we provide hospitals with boxes, filled with toys, games and puzzles to be used on wards.
This year, Starlight's fundraising income has fallen due to many cancelled events, which is why we have started our Time to Play campaign.
Our goal is to raise £300,000 before the end of the year to help 30,000 children get access to play in hospital and limit the negative effects of illness on their childhood.
It's easy to support and our website has fun ideas of how to take part, including our Stream for Starlight fundraiser for computer gamers.
Or if you are able to, please considering making a simple one-off donation. However big or small, it will help Starlight to continue to be there to support children and their families this Christmas.
I'm hoping readers will join together and help support Starlight, to bring back joy into the lives of the 98,000 seriously ill children living in the UK's lives this Christmas.
Visit www.starlight.org.uk/time to help.
Cathy Gilman, CEO, Starlight Children's Foundation
'Empty' climate change promises?
Your report on a planned £4.4m project for a new Aldi and drive-through McDonald's at Cressex Island in High Wycombe (Bucks Free Press, November 20, 2020, pages one and four) illustrates once again the lack of joined-up thinking in Buckinghamshire Council.
How can the council claim, on one hand, to be serious about their commitment to addressing carbon emissions and poor air quality in a promised climate change and Air Quality Strategy, while at the same time be actively facilitating and funding a drive-through restaurant adjacent to one of the most polluted areas of High Wycombe?
Should we not, therefore, assume that their recent announcement of a delay in the strategy actually signals that they have no intention of going beyond rhetoric and empty promises?
Steve Morton, Wycombe Friends of the Earth
EU Commission's staff discipline procedure is better
While Sir Alex Allan resigns and Home Secretary Priti Patel stays in her post, the European Commission's staff disciplinary procedure seems to work better.
In 2019, four lost their jobs. Four got reprimands after initial warnings failed. A retired official's net retirement pension was cut by half for breaking national laws outside of work including illegal possession of arms.
The 2019 report of the Investigation and Disciplinary Office of the Commission (IDOC) has been published.
IDOC covers the European Commission itself. It covers the agencies that work directly for the Commission, such as the Joint Research Centre's scientific testing labs.
It also deals with the European External Action Service, which is like a diplomatic service.
The European Union's Staff Regulations have been published. For example, Commission staff are not allowed to comment publicly on their work while they are employed.
Nor are Commission staff allowed to take instructions from anyone outside of the Commission.
Phil Jones, European Movement UK


lip, chin, cheeks, outer ears, bikini, underarms, back of neck; and large areas like the legs, arms, chest and back.
As opposed to electrolysis, which is slow and painful, laser hair removal offers permanent results much faster. Waxing, shaving, tweezing and creams are painful and may even stimulate hair growth. Laser hair reduction is relatively painless for the vast majority of patients.
The lasers deliver short, powerful pulses of light to target the active growing hair follicle. The laser light is absorbed by the brown-melanin pigment found in the hair shaft in the hair follicles, converted to heat destroying the follicle.
Does it Work for All Hair Color?
hair should be darker than the surrounding skin.
On average, 6-8 treatments are required to achieve optimum results. We offer treatment packages of 6 Sessions spaced about 4-6 weeks apart.
The laser will only be destroying the follicles that are in their "active growth" cycles. Our hair cycles every 4-6 weeks on average, and at any given point in time, contains about 15-20% actively growing follicles. Cycling your treatments catches up with your bodies natural hair cycle to achieve maximum hair reduction in any given area.
Our dynamically cooled laser is powerful enough for the male beard and gentle enough for the bikini line. Most patients experience slight redness and swelling of the skin (similar to sunburn) after a treatment, and the skin returns to normal within 72 hours.
Eliminate the need for waxing, shaving, or chemicals and achieve smooth, beautiful, and long lasting results.


text:
Stormy Daniels Arrested While Performing at Ohio Strip Club … Broke the Community Defense Act
PORN STAR STORMY DANIELS ARRESTED …
According to CBS News, porn actress Stormy Daniels was arrested at an Ohio strip club in Columbus for violating the Ohio law known as the Community Defense Act. This prohibits anyone who isn't a family member to touch a nude or semi-nude dancer. As per reports, Daniels is accused of letting patrons touch her. It certainly makes for one hell of a mugshot. Daniel's attorney, Michael Avenatti, claims it was a set up and politically motivated. Really? Did she or did she not break the law as defined by the State of Ohio and the Community Defense Act? The answer would appear to be yes. The claim that she has done this other places and had not been arrested is a ridiculous argument. That is like telling a police officer he can't arrest you for speeding because you had been speeding all day and had not been arrested. Stormy Daniels has put herself out there. I hardly think this was politically motivated. Had it been it would have more looked like how the Deep state has done after Trump.
Pic – Franklin County Sheriff's Office
Porn actress Stormy Daniels was arrested at an Ohio strip club and is accused of letting patrons touch her in violation of a state law, her attorney said early Thursday. While Daniels was performing Wednesday night at Sirens, a strip club in Columbus, some patrons touched her in a "non-sexual" way, her lawyer, Michael Avenatti, told The Associated Press.
An Ohio law known as the Community Defense Act prohibits anyone who isn't a family member to touch a nude or semi-nude dancer.
Early Thursday, the Franklin County Sheriff's Office released a mugshot of Daniels, whose real name is Stephanie Clifford.
Daniels was in police custody early Thursday morning and was expected to face a misdemeanor charge, Avenatti said.
She was released on bail before 6 a.m. Thursday.
"This was a complete set up," he said. "It's absurd that law enforcement resources are being spent to conduct a sting operation related to customers touching performers in a strip club in a non-sexual manner."
Posted July 12, 2018 by Scared Monkeys
Arrest | no comments
Stormy Daniels Claims CPL Michael Avenatti Sued Trump For Defamation Against Her Wishes
Judge throws out Porn Star Stormy Daniels's Defamation Lawsuit Against President Trump
Stormy Daniels' Attention seeking Attorney Michael Avenatti in Iowa, Says 'I'm exploring a run for the presidency of the United States'
Missing NY Pastor, Reverend Craig Rhodenizer, Found inside Ohio Strip Club
CPL Michael Avenatti Arrested for Felony Domestic Violence

redpajama_set_name: RedPajamaCommonCrawl


As you know, we love grabbing a bargain and there's nothing better than finding a great piece of furniture or fabulous piece of art at a knock down price.
The only problem with picking up pieces at these kinds of places, is that they're usually not in the best of condition. They're always dirty, they're usually scratched or dented and in general are in need of a bit of TLC.
You've probably already seen the little cupboard in our guest bedroom that we found at our local antique/ reclamation shop.
It's been in the bedroom for about a year now, but we have a bit of a confession to make. Except for a quick clean, we haven't really bothered to repair and restore it.
Whilst it looks fine from a distance (and it isn't really in an awful condition), it is a bit scratched and could just do with a general tidy up.
Having purchased more than our fair share of used furniture, we've developed a go to method of refreshing the pieces before they make it into our home and, unless we buy something with the purpose of taking it to pieces, we usually pick things that are in a fairly good condition, which obviously makes our "restoration" work much easier.
For us, the most important step before starting to work on anything, is to give it a really good clean. When we know that we'll be painting the furniture, we always use sugar soap. In this case, as we're keeping the natural wooden finish, we used natural soap for wood (like this one) which is much more gentile to the wood.
Just wipe down your furniture with your cleaner and some water. It should be patted dry straight away to reduce the risk of any water marks. You also shouldn't use microfiber cloths – because of their structure they can leave little scratches in the surface.
It's probably also worth mentioning that sugar soap is really gritty, so can damage delicate surfaces. If you're concerned, try it out on a test area first.
If there are any scratches or dents, these need to be repaired before polishing and treating the surface. Any small dents can be removed by using the same method we used to remove the dents in our floor.
Small scratches can be polished away with a polish that's specific to that type of wood. You shouldn't polish it too often (less than once a year) as the polish will create a film on the surface of the furniture that will become greasy and attract dust.
Alternatively you can use shoe cream in a matching colour or, when treating light furniture, even use Vaseline.
After carrying out any repairs, it's time to refresh the furniture. Old wooden furniture that hasn't been taken care of can be quite dry and really thirsty. How to treat the furniture will depend on the type of wood and how it's been treated in the past (varnished, oiled, waxed, etc.).
Because of the quick and refreshed look that polishes can achieve, they are a popular choice, but are something you should be wary of using. You should also avoid polishes with silicon oils or other mineral oils, as these penetrate the wood and can't be removed, therefore making any future surface restorations impossible. You should also only use products that are specifically designed for your type of wood. Otherwise, if you're working on a proper antique, you could do more harm than good!
We decided to use a wood oil to treat the cupboard. Oil is great, as it penetrates the wood and permanently protects the surface from dirt and moisture. Oil is also great, as it doesn't close the pores of the wood which means that it can still breathe.
Considering the little cupboard (it's actually a record cupboard) didn't look too bad to start with, we were really surprised at what a massive difference a bit of cleaning and polishing made.
It's hard to capture how big difference it actually is. The colour of the wood has changed slightly and is much more vibrant than before. The natural texture and grain of the wood is also much more visible now.
Do you love to shop at flea markets and antique fairs, too? Have you recently restored a piece of furniture?
Good timing! I bought a bed at an auction for £4 a couple of months ago and this fool of Took managed to drop the headboard on Sunday and made a big chip in the wood (I think it's polished oak). I had a look at those hard wax fillers on the internet but I'm a bit dubious. Don't suppose you've tried it before? Any tips?
Love that record cabinet btw!
A bed for £4?! Lucky you! Depending on where and how big the chip is, I'd try to remove it using the method we used to remove the dents in our wooden floor (http://littlehouseonthecorner.com/how-to-remove-dents-in-wood/) It worked really well for us and I'd just repeat the process until the grain of the wood has risen again. If it's too deep, the wax filler sticks aren't actually a bad choice (I'd use the ones where there are multiple colours in the pack so that you can mix them to match your wood exactly – Fiddes does something called Furniture Care Pack).
Good luck with whatever method you go for!
Great find! It is such a cute little cupboard, and it looks so much better now that it had received the treatment it deserves. I used this method recently to refresh my kitchen cabinets and have them looking like new.
Thanks Jessica! We're really happy with how it turned out. Your kitchen looks great, too! It's amazing what a bit of cleaning and polishing can do!
I've recently finished upgrading a grubby old writing bureau and before that a teak sideboard for my dining room. The photographs of both are on my instagram page @moogiesgirl if you fancy a neb.
I used spray paint on both for a flawless finish but would only recommend using spray if you have a garage or shed you can spray in.
Next up a blanket box (which I've had for years and never got round to sorting) for our bedroom then maybe a wardrobe and drawers for the spare bedroom if I can find any I like the shape of for a reasonable price.
Sounds like you have your work cut out, Fin! We always jump at the chance to buy something we like at our local antique place. We may not need it now, but you never know when something fab is going to come along again!
Thanks! We're not experts, but it turned out pretty well. Not sure I'd recommend it for propper antiques, but it definitely worked well for a second hand bargain.
Love your corner house. your local antique shop looks amazing. Please could I have their name / address?
Thanks Sylwia! We mainly visit two antique/second hand shops. There's one in Levenshulme called The Antiques Village and there's another in Failsworth called Old Mill Antiques. Both are of course hit and miss as their stock changes a lot, but it's always worth a try.
We have a kitchen /diner and my oak dining table has gone very sticky. It doesn't matter what I use I can't get rid of the stickiness. I was thinking of using sugar soap to try to remove this sticky coat. Can you please offer any advice.
Many thanks and love the furniture you've restored.
We always use sugar soap to clean any furniture that we're planning on painting. It really removes all dirt, but you have to be careful as it's really strong and can damage the surface. You will probably have to refinish the surface of the table though. For us, sugar soaps works best on varnished or oiled surfaces. I'd suggest trying a small test area somewhere where you can't see it like the underside of the table or a leg first – just to be on the safe side!
No problem, always happy to help if we can! Just remember to be careful – sugar soap is really abrasive.
I recently bought a Rococo bed. It has some damage from someones previous attempts to 'restore' it. Sadly, the heavily carved footboard has a stain of some sort brushed over it. Can see fine brush streaks that are not in line with the grain. I don't know what the application is but I'd like to remove it then re-oil the footboard. I'm hoping you can offer me some advise on how to go about it? Thanks in advance!
My bed frame is walnut. Most of it is as original regarding it finish. Just that outside face of the foot board has been messed with. I think they've already stripped it then stained it… its just in all directions. So far I've tried using 4 0 wirewool n linseed oil. It seems to be rubbing out the random streaks but it's also lacking the 'redness' of the rest of the bed. Beautiful golden colour walnut (bookend veneer) Maybe I shold do the whole bed in this method (slow n laborious I know). I understand this will also de-value the antique n I'm kind of reluctant but I don't know how to restore the reddish colour.
It only cost £163. Very pleased.
It's always difficult to know what the right thing is to do when it comes to more valuable pieces (even if you managed to grab a bargain). It sounds like you could do with some professional advice – maybe it's worth taking some pictures to a local joinery or antique place to see what they say would be best to do?
Thanks. I don't know why I haven't asked my son! He's a bench joiner/cabinet maker.lol. I'll ask him.
I just hope that the scratch cover polishes will cover the worst of the scratches in my furniture and I appreciate the tip of not doing it more than once a year. However, I do have some fairly big scratches on one of my tables. I'm thinking it will be best to just leave that repair job to the pros. I imagine I should still clean and fix as many of the problems I can on my own, though. It should make the pros job easier.
I'd always leave any bigger problems to the pros, Jesse. If you're thinking of taking the table to a professional I wouldn't bother attempting any repairs yourself. It's probably not worth the effort and it I'd be surprised if it made any difference to the pros if they repaired more than one scratch.


text:
Robert Towsend, Ph.D., and the Health of the Humanities
On Wednesday, January 18, we were joined by Dr. Robert Townsend, B.A. 1988, M.A. 1990, as part of the Lunch and Learn Series.
Upskilling Workshop: Graphic Design 101 with Canva
On Thursday, January 12, we were joined by Alexandra Wertz, B.A. 2022, for an introductory workshop on graphic design using Canva.
Garvey Hall officially welcomes the CatholicU community
December 5 was a red-letter day for Catholic University, when the Cardinal community celebrated the opening of Garvey Hall, the long-awaited new campus dining commons. It was also the day that Garvey Hall opened to hungry students as the University's main dining space.
The Metro School Receives Largest Gift to Date of $3.65 Million
This gift, made by the Sheehy family, will provide support for adult learners and is the first endowed scholarship of the Metropolitan School of Professional Services.
How the Feast of the Holy Family Encourages Us to Reflect on Our Relationship to God and Each Other
Today, the Church celebrates the feast day of Holy Family. The family is the fundamental unit of human society; it is the soil in which each of us has roots. In some way we always have family as our goal. God has hallowed the family and made a family the privileged place of His revelation of himself to us in love.
Upskilling Workshop: Excel Bootcamp with Kevin Rensch, B.A. 2014, M.A. 2016
On December 3, we were joined by Kevin Rensch, B.A. 2014, M.A. 2016, for the Upskilling Workshop: Excel Bootcamp. Rensch, took participants through some of Excel's most useful features that make one's work more efficient.
Kevin Walsh Installed as the Inaugural Knights of Columbus Endowed Professor of Law and the Catholic Tradition
On November 8, Kevin Walsh was installed as the inaugural Knights of Columbus Endowed Professor of Law and the Catholic Tradition.
Metro School Earns the Campaign Cup
This spring, the Metropolitan School of Professional Studies earned the Campaign Cup, which is presented to individual schools that surpass their overall fundraising goals for Light the Way: The Campaign for Catholic University.
The Catholic University Community Raises Over $94,000 on Giving Tuesday 2022
The Cardinal community came together for yet another year to help usher in the next generation of Cardinals. Collectively, Catholic University alumni, faculty, staff, parents, friends, and students raised more than $94,000 for student scholarships, surpassing last year's total of more than $75,000. The community's
CatholicU Nets Best Giving Challenge Result Ever
This year marked the most successful Athletics Giving Challenge in University history, with a total of $372,440 dollars raised through 3,721 gifts from a record-setting 2,905 donors. Last year, $313,614 was raised during the Athletics Giving Challenge.
Feast Day of Christ the King celebrates the reign of God and new horizons for humanity
The Solemnity of Our Lord Jesus Christ, King of the Universe, celebrated the last Sunday of the liturgical year, is one of the most recent additions to the calendar of liturgical commemorations in the Church. It asks for believers to rise above the fleeting and limited allegiances that we human beings establish or try to establish among ourselves.
Peter Varga B.A. 2019, on human flourishing through self-transcendence
On November 9, we were joined by Peter Varga, B.A. 2019, for a discussion on his research on aesthetics and self-transcendence in psychology and how those topics relate to well-being.
Kevin Kambo, M.A. 2014, Ph.D. 2018, on philosophy and your career
On October 25, Kevin Kambo, M.A. 2014, Ph.D. 2018, joined us for a discussion on his career and his current research in ethics, moral psychology, and Plato as part of the CUOnline Lunch and Learn series.
Federal grant funds CatholicU program to support DC veterans
Recently, Catholic University was awarded $1.4 million in federal grant funding to launch a program that will help U.S. military veterans prepare for and achieve higher education.
Alumna's childhood, college dreams came true
Mary Bruton Sandifer, B.A. 1975, lives in the Bordeaux region of France, where she and her husband and children run their family winery, Domaine La Tourbeille. She also recently published her first novel, Jeanne: Seeds of Infinity, a multi-generational, coming-of-age story about a young French girl who is introduced to the powers of nature.
Memories, Laughter, and Cardinal Pride: Cardinal Weekend 2022
From October 21-23, nearly 1,500 Catholic University alumni returned to campus to reunite during the annual Cardinal Weekend celebration.
Fr. Peter Galadza installed as first-ever Basil H. Losten Endowed Professor and Chair in Ukrainian Church Studies
On Wednesday, September 28, Catholic University officially endowed Fr. Peter Galadza as the first Bishop Basil H. Losten Endowed Professor and Chair in Ukrainian Church Studies.
Endowed Chairs Light the Way for Our Faculty
When the University launched Light the Way: The Campaign for Catholic University in 2019, one of the primary goals was to sustain the excellence of our faculty through an increased commitment to scholarship, research, training, and engagement.
CatholicU Community Raised $4.88 Million to Fund Impact Scholarships for the Class of 2026
Last winter, the Catholic University Board of Trustees set a bold challenge: to help the University raise an additional $4 million in order to fund 200 gap-filling impact scholarships for incoming first-year students.
Honoring the teachings of Saint John Paul II
Today, the Church celebrates the feast day of Saint John Paul II. He has left the Church with a rich heritage, from the theological wisdom in his extensive writings to his work for the human person against the dehumanizing forces of communism and the culture of death. Through this heritage, John Paul is still present with the Church as a spiritual father, walking with us through our many trials and strengthening us to respond with faith, hope, and love.


As a presidential candidate, Donald J. Trump vowed to end citizenship by birth (jus soli) for the children of undocumented immigrants. On the 150th anniversary of the 14th Amendment's ratification and emboldened by the imminent shift in the Supreme Court's balance, the administration has found a propitious time to begin its attack on the amendment's birthright citizenship clause, a pillar of the U.S. ideal of equality before the law.
In a recent op-ed in The Washington Post, Michael Anton, a former Trump administration speechwriter, resurrected the claim that this clause can be reinterpreted by the president or Congress to exclude the children of undocumented immigrants. In fact, it would require a constitutional amendment to deny citizenship to the U.S.-born children of any politically disfavored group. Moreover, it would represent a disastrous policy shift for a nation whose success has largely turned on its unprecedented ability to incorporate immigrants.
The 14th Amendment has upheld the promise that the United States would never again become a two-tiered society.
For 150 years, the 14th Amendment has upheld the promise that the United States would never again become a two-tiered society, comprised of U.S. citizens on the one hand and a hereditary underclass of "illegal" persons who have broken no laws on the other. The birthright citizenship clause also underlies the nation's historic genius at attracting and integrating immigrants, which should be an overriding priority in a nation with 43 million foreign-born persons. As legal scholar Margaret Stock has argued, its repeal would also require the establishment of a costly new government bureaucracy charged with making often complex citizenship determinations for each of the four million babies born in the United States each year.
Fortunately, it is not within the president's discretion to set the meaning of this clause. The 14th Amendment extends citizenship to "all persons" born in the United States and subject to its jurisdiction. The children of undocumented residents constitute "persons" under the Constitution and are subject to U.S. jurisdiction, meaning subject to its laws, power and sovereign authority. The "subject to the jurisdiction" language was meant to exclude the children of diplomats, enemy soldiers and Native Americans who were subject to the laws of their sovereign tribes. It would come as a great surprise to law enforcement agencies and to undocumented immigrants to learn that their children were "subject to the jurisdiction" of the United States but not required to obey its laws. By this faulty interpretation, U.S.-born non-citizens would presumably not be subject to deportation.
Repeal of birthright citizenship would require the establishment of a costly new government bureaucracy charged with making often complex citizenship determinations.
In fact, partisans in the congressional debate on the passage of the 14th Amendment assumed that this clause would extend citizenship to the children of the foreign-born. The debate focused on whether the amendment should provide citizenship to them, not whether it would as written.
Indeed, in its 1898 decision in United States v. Wong Kim Ark, the Supreme Court concluded that "[e]very citizen or subject of another country, while domiciled here, is within the allegiance and protection, and consequently subject to the jurisdiction, of the United States." The case involved the U.S.-born child of lawful Chinese immigrants who nonetheless could not become U.S. citizens under the Chinese Exclusion Act.
The birthright citizenship clause overturned the infamous Dred Scott decision that held that persons of African descent could never become U.S. citizens. Proponents of the 14th Amendment sought to prevent the political branches of the U.S. government from ever again denying citizenship to disfavored groups. The historian Martha S. Jones recently explained to The New York Times that the clause created "a route to citizenship that could not be denied by virtue of race, by virtue of descent, religion, political party, health, wealth." The administration wishes to undo that remarkable accomplishment.
Some Republicans view birthright citizenship as a tool newly wielded to diminish their political clout. Nativists see it reductively as an immigration loophole. Hate groups fear that it is accelerating the "browning" of the United States. Others view it as an affront to the rule of law and a way to game the immigration system. Yet the undocumented—who pick our crops, build our buildings, serve our food and care for our children and elderly—do not seek to take advantage of U.S. citizens. They seek better lives for their children. As such, they pay homage to the ideals of equality and opportunity that the 14th Amendment embodies.
The solutions to illegal migration are to address the conditions that compel people to leave their homelands, to align the badly outdated U.S. immigration system with the nation's interests and values and to enforce the law, which the federal government has abundant resources to do. It is not to topple a pillar of our constitutional democracy through executive fiat or congressional resolution.
"The birthright citizenship clause overturned the infamous Dred Scott decision that held that persons of African descent could never become U.S. citizens."
I find in this article no justification for calling birthright citizenship "a pillar of our constitutional democracy" or even for judging it a good law at this time. But, my mind is open. Please, address the merits of this issue!
Essential. Hardly. Birthright makes no sense unless anybody can claim citizenship. Why is passing through to have a baby a right to citizenship?
Would the author be for open borders but no social help for anyone who is not documented? That would eliminate most of the illegal problem and the country could then focus on legal immigration and helping others countries develop a free economy where their citizens could thrive. Interesting how none of America, the magazine, authors care about others around the world.
Donald Kerwin is the executive director of the Center for Migration Studies, a Catholic think tank and educational institute devoted to the study of international migration.


text:
\section{Introduction}
\label{sec:intro}

Kohn-Sham density functional theory
(DFT)~\cite{HohenbergKohn1964,KohnSham1965} is the most widely used
electronic structure theory for molecules and systems in condensed
phase. The Kohn-Sham orbitals (a.k.a. Kohn-Sham wavefunctions) are
eigenfunctions of the Kohn-Sham Hamiltonian and are generally
delocalized, \textit{i.e.}{}~each orbital has significant magnitude across the
entire computational domain. Consequently, the information about atomic
structure and chemical bonding, which is often localized in real space,
may be difficult to interpret directly from Kohn-Sham orbitals.
The connection between the delocalized orbitals, and localized ones can
be established
through a \textit{localization} procedure, which has been
realized by various numerical methods in the
literature~\cite{FosterBoys1960,MarzariVanderbilt1997,WannierReview,Gygi2009,SCDM,ELiLu2010,OzolinsLaiCaflischEtAl2013,AquilantePedersenMerasEtAl2006}.
The common goal of these methods is to find a set of
orbitals that are localized in real space and span the Kohn-Sham
invariant subspace, defined as the subspace as that spanned by the Kohn-Sham orbitals. \rr{For simplicity, we restrict our discussion to isolated systems and omit the discussion of Brillouin zone sampling in this manuscript.}

Mathematically, the problem of finding a localized representation of the
Kohn-Sham invariant subspace can be formulated as follows. Assume
the collection of Kohn-Sham orbitals are discretized in the real space representation as a tall
and skinny matrix $\Psi\in \mathbb{C}^{N\times n_{e}}$ with orthonormal columns and where
$N\gg n_{e}$. We seek to compute
a unitary transformation $Q\in\mathbb{C}^{n_{e}\times n_{e}}$ such that
the columns of $\Phi=\Psi Q$ are
\textit{localized}, \textit{i.e.}{}~each column of $\Phi$ becomes a sparse vector with spatially localized support
after truncating entries with relative magnitude smaller than a prescribed threshold. Here, $N$ is the number of grid
points in the discrete real space representation of each Kohn-Sham
orbital, and $n_{e}$ is the number of orbitals. In the absence of spin
degeneracy,  $n_{e}$ is also the number of electrons in the system.

For a general matrix $\Psi$ it may not be possible to construct such a
$Q$ and obtain $\Phi$ with the desired structure. However, when
$\Psi$ represents a collection of  Kohn-Sham orbitals of an insulating
system, such localized orbitals \rr{generally} exist. \rr{An important exception are topological insulators with non-vanishing Chern numbers \cite{hasan2010colloquium,Brouder2007} \textemdash here we restrict our discussion to systems where localized functions are known to exist.} Their construction can be
justified physically by the ``nearsightedness'' principle for electronic
matter with a finite HOMO-LUMO gap~\cite{Kohn1996,ProdanKohn2005}. The
nearsightedness principle can be more rigorously stated as the single
particle density matrix being exponentially localized along the
off-diagonal direction in its real space
representation~\cite{BenziBoitoRazouk2013,Kohn1996,Blount,Cloizeaux1964a,Cloizeaux1964b,
Nenciu,LinLu2015}.

The recently developed selected columns of the density matrix (SCDM)
procedure \cite{SCDM} provides a simple, accurate, and robust way of
constructing localized orbitals. Unlike many existing
methods~\cite{FosterBoys1960,MarzariVanderbilt1997,ELiLu2010,OzolinsLaiCaflischEtAl2013,MustafaCohCohenEtAl2015}, the SCDM method requires no initial
guess and does not involve a non-convex optimization procedure. \rr{The core ideas behind it also readily extend to systems with Brillouin zone sampling \cite{SCDMk}.} The SCDM procedure constructs localized orbitals directly from a
column selection procedure implicitly applied to the density matrix.
Hence, the locality of the basis is a direct consequence of the locality of the
density matrix. The SCDM method can be efficiently performed via a
single column pivoted QR (QRCP) factorization. Since efficient
implementation of the QRCP is available both in serial and
parallel computational environments through the LAPACK \cite{lapack} and
ScaLAPACK~\cite{Scalapack} libraries, respectively, the SCDM method can be
readily adopted by electronic structure software packages.

From a numerical perspective, the computational cost of a QRCP factorization scales as
$\mathcal{O}(N n_{e}^{2})$. The basic form of QRCP \cite{GVL} is not able to take full
advantage of level 3 BLAS operations. Hence for matrices of the same
size, the QRCP factorization can still be relatively expensive compared
to level 3 BLAS operations such as general matrix-matrix
multiplication (GEMM). The computational cost of the single QRCP is not necessarily an issue when the SCDM procedure is used as a post-processing tool, but is a
potential concern when the SCDM procedure needs to be performed repeatedly.  This, for instance, could occur in geometry optimization and molecular dynamics calculations with hybrid exchange-correlation
functionals~\cite{WuSelloniCar2009,GygiDuchemin2012}, where a localized
representation of the Kohn-Sham invariant subspace needs to be
constructed in each step to reduce the large computational cost
associated with the Fock exchange operator. In fact, \cite{SCDM} demonstrates how our existing SCDM algorithm may be used to accelerate Hartree-Fock exchange computations. Therefore, here we focus on accelerating the SCDM computation itself.

Practically, any QRCP algorithm may be used within the SCDM procedure.
In the serial setting, this includes recently developed methods based on
using random projections to accelerate and block up the column selection
procedure \cite{Martinsson,Gu}. In the massively parallel setting one
may alternatively use the recently developed communication avoiding
rank-revealing QR algorithm \cite{DemmelRRQR}.

In this paper, we demonstrate that the computational cost of the SCDM
procedure can be greatly reduced, to the extent that the column
selection procedure is no longer the dominating factor in the SCDM
calculation.
This is based on the observation that SCDM does not really require the
Q and R factors from the QRCP. In fact, only the pivots from
the QRCP are needed.
More specifically, we develop a two-stage column selection procedure that
approximates the behavior of the existing SCDM procedure at a much lower
computational cost. Asymptotically, the computational cost is only dominated
by two matrix-matrix multiplications of the form $\Psi Q$ to
construct the localized orbitals, at least one of which is needed in any localization
procedure starting from the input $\Psi$ matrix. Notably, the only adjustable parameters we introduce are an oversampling factor and truncation threshold. Both of which may be picked without knowledge of the problems physical structure.

The approximate column selection procedure consists of two stages. First,
we use a randomized procedure to select a set of candidate columns
that may be used in an SCDM style localization procedure. The number of
candidate columns is only $\mathcal{O}(n_{e} \log n_{e})$ and is much smaller than
$N$. We may use these candidate columns to quickly construct a basis for
the subspace that is reasonably localized. In some cases, this fast
randomized procedure may provide sufficiently localized columns.
Otherwise, we
propose a subsequent refinement procedure to improve the quality of the
localized orbitals.  This is achieved by using a series of QRCP
factorizations for matrices of smaller sizes that may be performed concurrently.
Numerical results for physical systems obtained from the Quantum
ESPRESSO~\cite{QE} package indicate that the two-stage procedure yields
results that are nearly as good as using the columns selected by a full
QRCP based SCDM procedure. For large systems, the computational time is
reduced by more than one order of magnitude.

The remainder of this paper is organized as follows. In Section
\ref{sec:prelim} we present both a brief introduction to Kohn-Sham DFT
and a summary of the existing SCDM algorithm. Section \ref{sec:algo}
discusses the new two stage algorithm we propose, and details both the
randomized approximate localization stage and the refinement of the
column selection. Finally, Section \ref{sec:numer} demonstrates the
effectiveness of the algorithm for various molecules.

\section{Preliminaries}
\label{sec:prelim}

For completeness we first provide a brief introduction to Kohn-Sham density
functional theory, and the SCDM procedure for finding a localized
basis for the Kohn-Sham subspace.

\subsection{Kohn-Sham density functional theory}

For a given atomic configuration with $M$ atoms at locations $\{R_{I}\}_{I=1}^{M}$,
KSDFT solves the nonlinear eigenvalue problem
\begin{equation}
  \begin{split}
    &\hat{H}[\hat{\rho};\{R_{I}\}]\hat{\psi}_{i} = \varepsilon_{i} \hat{\psi}_{i},\\
    &\hat{\rho}(\bvec{r}) = \sum_{i=1}^{n_e} \abs{\hat{\psi}_{i}(\bvec{r})}^2, \quad \int
    \hat{\psi}^{*}_{i}(\bvec{r}) \hat{\psi}_{j}(\bvec{r}) \,\mathrm{d} \bvec{r} = \delta_{ij}.
  \end{split}
  \label{eqn:KS}
\end{equation}
For simplicity we omit the spin degeneracy.
The number of electrons is $n_{e}$ and the eigenvalues
$\{\varepsilon_{i}\}$ are ordered non-decreasingly.  The lowest
$n_e$ eigenvalues $\{\varepsilon_{i}\}_{i=1}^{n_e}$ are called the occupied state energies, and
$\{\varepsilon_{i}\}_{j>n_e}$ are called the unoccupied state
energies.  We assume
$\varepsilon_{g}:=\varepsilon_{n_e+1}-\varepsilon_{n_e}>0$. Here
$\varepsilon_{n_e}$ is often called the highest occupied molecular
orbital (HOMO), $\varepsilon_{n_e+1}$ the lowest unoccupied
molecular orbital (LUMO), and hence
$\varepsilon_{g}$ the HOMO-LUMO gap.  \rr{For extended systems, if $\varepsilon_{g}$ is uniformly bounded away from zero as the system size increases,}
the quantum system is an insulating system~\cite{Martin2004}.
The eigenfunctions $\{\hat{\psi}_{i}\}_{i=1}^{n_e}$ define the
electron density $\hat{\rho}(\bvec{r})$, which in turn defines the Kohn-Sham
Hamiltonian
\begin{equation}
  \hat{H}[\hat{\rho};\{R_{I}\}] = -\frac12 \Delta +
  \hat{V}_{c}[\hat{\rho}] + \hat{V}_{\mathrm{xc}}[\hat{\rho}] + V_{\mathrm{ion}}[\{R_{I}\}].
\label{eqn:ksdft}
\end{equation}
Here $\Delta$ is the Laplacian operator for the kinetic
energy of electrons,
\begin{equation*}
  \hat{V}_{c}[\hat{\rho}](\bvec{r}) \equiv
  \int \frac{\hat\rho(\bvec{r}')}{\abs{\bvec{r}-\bvec{r}'}} \,\mathrm{d} \bvec{r}'
  \label{}
\end{equation*}
is the Coulomb potential, and $\hat{V}_{c}$ depends linearly with
respect to the electron density $\hat{\rho}$.  $\hat{V}_{\mathrm{xc}}[\hat{\rho}]$
depends nonlinearly with respect to $\hat{\rho}$, and characterizes the
many body exchange and correlation effect. $V_{\mathrm{ion}}[\{R_{I}\}]$ is an
external potential depending explicitly on the ionic positions, and
describes the electron-ion interaction
potential and is independent of $\hat{\rho}$.  Because the eigenvalue
problem (\ref{eqn:KS}) is nonlinear, it is often solved iteratively by a
class of algorithms called self-consistent field iterations
(SCF)~\cite{Martin2004}, until~\eqref{eqn:KS} reaches
self-consistency.

In a finite dimensional discretization of Eq.~\eqref{eqn:ksdft}, let $N$
be the number of degrees of freedom. Using a large basis set such as the
plane-wave basis set, we have $N=c n_{e}$ and $c$ is a large constant
that is often $10^{2}\sim 10^{4}$. Due to this large constant, we
explicitly distinguish $N$ and $n_{e}$ in the complexity analysis below.
For large
scale systems, the cost for storing the Kohn-Sham orbitals is $\mathcal{O}(N
n_e)$, and the cost for computing them is generally $\mathcal{O}(N n_e^2)$
and scales cubically with respect to $n_e$.  In modern KSDFT
calculations the Hartree-Fock exact exchange term is also often taken
into account in the form of hybrid
functionals~\cite{PerdewErnzerhofBurke1996,Becke1993}.  The
computational cost for this step not only scales as $\mathcal{O}(Nn_e^{2})$ but
also has a large pre-constant.

When the self-consistent solution of the Kohn-Sham equation is obtained,
the existence of finite HOMO-LUMO gap has important implications on the
collective behavior of the occupied Kohn-Sham orbitals
$\{\hat{\psi}_{i}\}_{i=1}^{n_e}$. Since any non-degenerate linear
transformation of the set of Kohn-Sham orbitals yields exactly the same
physical properties of a system, the physically relevant
quantity is the subspace spanned by the Kohn-Sham orbitals
$\{\hat{\psi}_{i}\}_{i=1}^{n_e}$. Various
efforts~\cite{FosterBoys1960,MarzariVanderbilt1997,WannierReview,Gygi2009,ELiLu2010,OzolinsLaiCaflischEtAl2013}
have been made to utilize this fact and to find a set of localized
orbitals that form a compressed representation of a Kohn-Sham subspace.
In other words, we find a set of functions
$\{\hat{\varphi}_{i}\}_{i=1}^{n_e}$ whose span is the same as the
span of $\{\hat{\psi}_{i}\}_{i=1}^{n_e}$.
Compared to each Kohn-Sham orbital $\hat{\psi}_{i}$ which is delocalized in
the real space, each compressed orbital $\hat{\phi}_{i}$ is often localized
around an atom or a chemical bond. Hence working with $\hat{\phi}_{i}$'s
can reduce both the storage and the computational cost.

Assume we have access to $\hat{\psi}_{j}(\bvec{r})$'s evaluated at a set of
discrete grid
points $\{\bvec{r}_{i}\}_{i=1}^{N}$.  Let $\{\omega_{i}\}_{i=1}^{N}$ be a set
of positive integration weights associated with the grid points
$\{\bvec{r}_{i}\}_{i=1}^{N}$, then the discrete orthonormality condition
is given by
\begin{equation}
  \sum_{i=1}^{N} \hat{\psi}_{j}(\bvec{r}_{i}) \hat{\psi}_{j'}(\bvec{r}_{i}) \omega_{i} =
  \delta_{jj'}.
  \label{eqn:orthonormal_discrete}
\end{equation}
Let $\hat{\psi}_{j}=[\hat{\psi}_{j}(\bvec{r}_{1}), \hat{\psi}_{j}(\bvec{r}_{2}), \ldots,
\hat{\psi}_{j}(\bvec{r}_{N})]^{T}$ be a column vector, and $\hat{\Psi}=[\hat{\psi}_{1},  \ldots,  \hat{\psi}_{n_e}]$
be a matrix of size $N\times n_e$. We call $\hat{\Psi}$ the
\textit{real space representation} of the Kohn-Sham orbitals and define diagonal matrix
$W=\mathrm{diag}[\omega_{1},\ldots,\omega_{N}]$.

\rr{Our method requires the Kohn-Sham orbitals to be be represented on a set of real space grid points. This is the case for a plane-wave basis set, as well as other representations such as finite differences, finite elements and wavelets. For instance, if
the Kohn-Sham orbitals are represented using the plane-wave basis
functions, their real space representation can be obtained on a uniform
grid efficiently with the fast Fourier transform (FFT) technique and in this case $\omega_{i}$ takes the same constant value for all $i$. It is in this setting that our method is of particular interest. However, this procedure is also applicable to other basis sets such as Gaussian type orbitals or numerical atomic orbitals when a real space representation of the basis functions is readily available. Therefore, our method is amenable to most electronic structure software packages.}

We define $\Psi=W^{\frac12} \hat{\Psi}$ such that the discrete orthonormality
condition in Eq.~\eqref{eqn:orthonormal_discrete} becomes
$\Psi^{*}\Psi=I$, where $I$ is an identity matrix of size $n_e$.
We now seek a compressed basis for the span of $\Psi$, denoted by the
set of vectors $\Phi=[\phi_{1}, \ldots,  \phi_{n_e}]$ where each
$\phi_{i}$ is a sparse vector with spatially localized support after truncating entries with small
magnitudes. In such case, $\phi_{i}$ is called a localized vector.

\subsection{Selected columns of the density matrix}

As opposed to widely-used procedures such as MLWFs \cite{WannierReview}, the key difference in
the SCDM procedure is that the localized orbitals $\phi_{i}$ are
obtained directly from columns of the density matrix $P = \Psi\Psi^*$.
The aforementioned nearsightedness principle states that, for
insulating systems, each column of the matrix $P$ is localized. As a result, selecting any linearly independent subset of $n_e$ of them
will yield a localized basis for the span of $\Psi.$ However, picking
$n_e$ random columns of $P$ may result in a poorly conditioned basis if,
for example, there is too much overlap between the selected
columns.  Therefore, we would like a means for choosing a well
conditioned set of columns, denoted $\mathcal{C} = \left\{c_1,c_2,\ldots,c_{n_e} \right\},$ to use as the localized
basis. Intuitively we expect such a basis to select columns to minimize
overlaps with each other when possible.

\rr{This is algorithmically accomplished with a QRCP factorization (see, \textit{e.g.,}{}~\cite{GVL}). More specifically, given a matrix $A$ a QRCP seeks to compute a permutation matrix $\Pi$ such that the leading sub-matrices $\left(A\Pi\right)_{1,\ldots,k,:}$ for any applicable $k$ are as well conditioned as possible. In particular, if we let $\mathcal{C}$ denote the columns selected by the first $n_e$ columns of $\Pi$ then $A_{:,\mathcal{C}}$ should be a well conditioned set of $n_e$ columns of $A.$}

\rr{In our setting, this means we would ideally compute an QRCP factorization of the matrix $P$ to identify $n_e$ well conditioned columns from which we may construct a localized basis. However, this
would be highly costly since $P$ is a large matrix of size
$N$. Fortunately, we may equivalently compute the
set $\mathcal{C}$ by computing a QRCP factorization of $\Psi^*$ \rr{or, in fact, any matrix $U$ with orthogonal columns such that $P=UU^*.$} More
specifically, we compute
\begin{equation}
\label{eqn:qrcp}
\Psi^*\Pi = Q\begin{bmatrix} R_1 & R_2 \end{bmatrix},
\end{equation}
and the first $n_e$ columns of $\Pi$ encode $\mathcal{C}.$}

The SCDM procedure to construct an orthonormal set of localized basis elements, denoted $\phi_i$ for $i=1,\ldots,n_e$, and collected as columns of the matrix $\Phi$ is presented in its simplest form in Algorithm \ref{alg:scdm}.
\begin{algorithm}
\caption{The SCDM algorithm}
\label{alg:scdm}
\begin{algorithmic}[1]
\Statex Given: the Kohn-Sham orbitals $\Psi$
\State Compute a column pivoted QR of $\Psi^*$, $\Psi^*\Pi = Q\begin{bmatrix} R_1 & R_2 \end{bmatrix}$
\State Compute $\Phi = \Psi Q$ or, alternatively, $\Phi^* = \begin{bmatrix} R_1 & R_2 \end{bmatrix}\Pi^*$
\Statex Output: a localized basis for the Kohn-Sham subspace $\Phi$
\end{algorithmic}
\end{algorithm}
In such form the algorithm requires knowledge of the orthogonal factor
from the QRCP. However, an alternative description simply requires the
column selection $\mathcal{C}.$ We may equivalently write the SCDM algorithm as
in Algorithm \ref{alg:scdm_no_q}. Note that in Algorithm
\ref{alg:scdm_no_q}, the cost of the QR factorization for the
matrix $\left(\Psi_{\mathcal{C},:}\right)^*$ is only $\mathcal{O}(n_{e}^{3})$.
\begin{algorithm}
\caption{An alternative version of the SCDM algorithm}
\label{alg:scdm_no_q}
\begin{algorithmic}[1]
\Statex Given: the Kohn-Sham orbitals $\Psi$
\State Compute $\mathcal{C}$ associated with a column pivoted QR of $\Psi^*$
\State Compute the QR factorization $\left(\Psi_{\mathcal{C},:}\right)^* = QR$
\State Compute $\Phi = \Psi Q$
\Statex Output: a localized basis for the Kohn-Sham subspace $\Phi$
\end{algorithmic}
\end{algorithm}

\begin{remark}
There are various equivalent ways to construct the SCDM
algorithm. While the simple presentation here differs slightly
from the original presentation \cite{SCDM} \rr{the two are mathematically equivalent up to a choice of sign for the columns of $\Phi$. The original presentation corresponds to the computation of a QR factorization of $\left(\Psi_{\mathcal{C},:}\right)^*$ via a Cholesky factorization of $\left(\Psi_{\mathcal{C},:}\right)\left(\Psi_{\mathcal{C},:}\right)^*.$ QR factorizations are not
unique, there is always ambiguity up to diagonal matrix with
entries on the unit circle. However, such an ambiguity does not have any
affect on the localization.}
\end{remark}

\rr{This second interpretation allows us to briefly explain why this algorithm constructs localized orbitals. Let $D$ be a diagonal matrix with $\pm 1$ on the diagonal such that $DR$ has positive diagonal entries. This means that
\[
Q = \left(\Psi_{\mathcal{C},:}\right)^*R^{-1}D
\]
where $R^{-1}$ is a Cholesky factor of $\left[\left(\Psi_{\mathcal{C},:}\right)\left(\Psi_{\mathcal{C},:}\right)^*\right]^{-1}.$ Importantly, $\left(\Psi_{\mathcal{C},:}\right)\left(\Psi_{\mathcal{C},:}\right)^* = P_{\mathcal{C},\mathcal{C}}$ and, therefore, exhibits off diagonal decay so long as $P_{\mathcal{C},\mathcal{C}}$ is well conditioned. This property is then inherited by $R^{-1}$ \cite{BenziBoitoRazouk2013}. Finally, since $P_{\mathcal{C},:} = \Psi\left(\Psi_{\mathcal{C},:}\right)^*$
\[
\Phi = P_{\mathcal{C},:}R^{-1}D
\]
we may conclude that it is well localized \textemdash $P_{\mathcal{C},:}$ is well localized and $R^{-1}D$ does not destroy that locality. Importantly, here we see that all the factorizations we are performing can be thought of as involving sub-matrices of $P$.}

The overall computational cost of the algorithm is $\mathcal{O}(N n_e^2),$ and
practically the cost is dominated by the single QRCP factorization regardless of the version used. Another key feature of the
algorithm, especially for our modifications later, is that because we
are effectively working with the spectral projector $P,$ the method
performs equivalently if a different orthonormal basis for the range of
$\Psi$ is used as input. In physics terminology, the SCDM procedure is
gauge-invariant. Lastly, the key
factor in forming a localized basis is the selection of a well
conditioned subset of columns. Small changes to the selected columns, provided they remain nearly as well conditioned, may
not significantly impact the overall quality of the basis.

\section{The approximate column selection algorithm}
\label{sec:algo}

When the SCDM procedure is used as a post-processing tool for a single
atomic configuration, the computational cost is usually affordable. In
fact in such a situation, the most time consuming part of the
computation is often the I/O related to the $\Psi$ matrices especially
for systems of large sizes.  However, when localized orbitals need to be
calculated repeatedly inside an electronic structure software package,
such as in the context of hybrid functional calculations with
geometry optimization or \textit{ab initio} molecular dynamics
simulations, the computational cost of SCDM can become relatively large.
Here we present an algorithm that significantly accelerates the SCDM
procedure.

The core aspect of the SCDM procedure is the column selection procedure. Given a set of appropriate columns the requisite orthogonal transform to construct the SCDM may be
computed from the corresponding rows of $\Psi$, as seen in Algorithm~\ref{alg:scdm_no_q}. Here we present a two stage procedure for accelerating this selection of columns and hence the computation
of $\Phi$. First, we construct a set of approximately localized orbitals
that span the range of $\Psi$ via a randomized method that requires only
$\Psi$ and the electron density $\rho,$ though if $\rho$ is not given it may be computed directly from $\Psi$ without increasing the asymptotic computational complexity.  We then use this approximately localized basis as the
input for a procedure that refines the selection of columns from which the localized basis is ultimately constructed. This is done by using the approximate locality to carefully partition the column selection
process into a number of small, local, QRCP factorizations.
Each small QRCP may be done in
parallel, and operates on matrices of much smaller dimension than
$\Psi.$

\subsection{Approximate localization}

The original SCDM procedure, through the QRCP, examines all $N$ columns
of $\Psi^*$ to decide which columns to use to construct $Q$. However,
physical intuition suggests that it is often not necessary to visit all
columns to find good pivots. For instance, for a molecular system in
vacuum, it is highly unlikely that a pivot comes from a column of the
density matrix corresponding to the vacuum space away from the molecule.
This inspires us to accelerate the column selection procedure by
restricting the candidate columns.

This is accomplished by generating $\mathcal{O}\left(n_e\log n_e\right)$
independent and identically distributed (i.i.d.) random sample columns,
using the normalized electron density as the probability distribution
function (pdf).  Indeed, if a column of the density matrix corresponds
to the vacuum, then the electron density is very small and hence the
probability of picking the column is very low. In statistics this
corresponds to leverage score sampling, see, \textit{e.g.,}{}~\cite{MahoneyDrineas}. This randomized version of the SCDM
algorithm is outlined in Algorithm~\ref{alg:rand}.

\begin{algorithm}
\caption{Computing an approximately localized collection of basis vectors}
\label{alg:rand}
\begin{algorithmic}[1]
\Statex Given: Kohn-Sham orbitals $\Psi,$ electron density $\rho,$ concentration $\gamma,$ and failure probability $\delta$
\State Sample $(n_e / \gamma) \log n_e / \delta$ elements from $\left\{1,\ldots,N\right\}$ based on the discrete distribution
\[
\mathbf{Pr}\left(\left\{j\right\}\right) = \rho(j)/n_e
\]
and denote this set $\wt{\CS}$
\State Compute the column pivoted QR factorization $$\left(\Psi_{\wt{\CS},:}\right)^*\Pi = QR$$
\State Form approximately localized basis $\wt{\Phi} = \Psi Q$
\end{algorithmic}
\end{algorithm}

To complete our discussion of Algorithm \ref{alg:rand} we must justify the sub-sampling procedure used to select $\wt{\CS}$. In order to do so we introduce a simple model for the column selection procedure based on the idea that columns ``similar'' to the ones selected by Algorithm~\ref{alg:scdm} will work well to compute an approximately localized basis. Our underlying assumption is that a set of columns will serve to approximately localize the basis if it contains at least one column in each region where one of the $\phi_i$ is large. Because the $\phi_i$ constructed via the SCDM procedure decay exponentially, this is analogous to saying that any column associated with a grid point close enough to the ``true'' grid point used will suffice. Though, by avoiding explicit use of spatial relations of grid points our algorithm and its parameters are not dependent on the physical geometry.

To codify this postulate, we let $\mathcal{I}_i \subset \left\{1,\ldots,N\right\}$ be the smallest non-empty set such that
\begin{equation}
\sum_{j\in\mathcal{I}_i} \lvert\phi_i(j)\rvert^2 \geq \gamma,
\label{eqn:gamma}
\end{equation}
where $\gamma \in (0,1)$. If multiple such sets exist we select the one
that maximizes ${\sum_{j\in\mathcal{I}_i} \lvert\phi_i(j)\rvert^2}$. Now, we may write our assumption more concretely: a column $c_i \in \left\{1,\ldots,N\right\}$ suffices to approximately construct $\phi_i$ if it is contained in $\mathcal{I}_i$. \rr{Taking $\gamma$ to be sufficiently small would enforce adequate sampling to ensure the columns selected by Algorithm~\ref{alg:scdm_no_q} are selected to be part of $\wt{\CS}.$ However, in practice this is not necessary for the construction of a localized basis and by choosing a larger $\gamma$ we allow for other columns near the peak of $\phi_i$ to act as good surrogates.}

Under this assumption, to approximately localize the basis, we must
simply ensure that $\wt{\CS}$ contains at least one distinct column in each of the sets
$\mathcal{I}_1,\mathcal{I}_2,\ldots,\mathcal{I}_{n_e},$ \textit{i.e.}{}~we need a one to one matching between sets and
columns. Theorem \ref{thm:sample} provides an upper bound on the
required cardinality of $\wt{\CS}$ to ensure it may be used to approximately
localize the basis with high probability. We do require an additional
mild assumption that ensures the sets $\mathcal{I}_i$ do not simultaneously
overlap significantly and have small support.
\begin{theorem}
\label{thm:sample}
Let $\eta$ be the largest constant such that there exist disjoint subsets $\mathcal{I}^s_i \subseteq \mathcal{I}_i, \; i=1,\ldots,n_e$ each satisfying
\[
\sum_{j\in\mathcal{I}^s_i} \lvert\phi_i(j)\rvert^2 \geq \eta\sum_{j\in\mathcal{I}_i}
\lvert\phi_i(j)\rvert^2.
\]
The set $\wt{\CS}$ constructed by sampling $$m \geq  \left(n_e / \eta\gamma \right) \log n_e / \delta$$ elements with replacement from $\left\{1,\ldots,N\right\}$ based on the discrete distribution
\[
\mathbf{Pr}\left(\left\{j\right\}\right) = \rho(j)/n_e
\]
contains an unique element in each $\mathcal{I}_i$ for $i=1,\ldots,n_e$ with probability $1-\delta.$
\end{theorem}
\begin{proof}
Let $\mathcal{F}$ be the event that $\wt{\CS}$ does not contain a distinct element in one of the sets $\mathcal{I}_i.$ We may write
\[
\mathbf{Pr}\left(\mathcal{F}\right) \leq \mathbf{Pr}\left(\left\{\wt{\CS} \cap \mathcal{I}^s_i = \emptyset \text{ for some } i\right\}\right)
\]
because requiring $\wt{\CS}$ to contain an element in each $\mathcal{I}^s_i$ implies that $\wt{\CS}$ contains a distinct element in each $\mathcal{I}_i$. Subsequently, by a union bound
\[
\mathbf{Pr}\left(\mathcal{F}\right)\leq \sum_{i=1}^{n_e} \mathbf{Pr}\left(\left\{\wt{\CS} \cap \mathcal{I}^s_i = \emptyset\right\}\right).
\]
The event $\left\{\wt{\CS} \cap \mathcal{I}^s_i = \emptyset\right\}$ is simply the probability that none of the $m$ samples fall in $\mathcal{I}^s_i.$ Because
\[
\rho(j) = \sum_{i=1}^{n_e} \lvert \phi_i(j)\rvert^2,
\]
we may find the lower bound for the probability of selecting an element $j$ in $\mathcal{I}_i$ as
\[
\mathbf{Pr}\left(\left\{j \in \mathcal{I}^s_i\right\}\right) \geq \eta\gamma/n_e.
\]
Consequently,
\[
\sum_{i=1}^{n_e} \mathbf{Pr}\left(\left\{\wt{\CS} \cap \mathcal{I}^s_i = \emptyset\right\}\right) \leq n_e (1-\eta\gamma/n_e)^m
\]
and to ensure the probability of missing any of the sets $\mathcal{I}^s_i$ is less than $\delta$ we may simply enforce
\[
m \geq \frac{n_e}{\eta\gamma} \log n_e / \delta.
\]
\end{proof}

\begin{remark}
The parameter $\eta$ is
necessary in the proof, but
algorithmically we simply assume it to be one, alternatively, one could
consider choosing $1 / (\gamma \eta)$ rather than just $1 / \gamma$ as the
oversampling factor.
\end{remark}

If, for example, we take $\gamma = 1/2$ and $\eta = 1/2$ this bound says
$4n_e\log n_e/\delta$ samples suffices for the approximate localization
procedure. As expected, when either the failure probability $\delta$ or
the cardinality of $\mathcal{I}_i$ go to one the required number of samples
grows. Furthermore, since $\gamma = \min_{i} \max_{j} \lvert \phi_i(j) \rvert^2$ corresponds to each of the sets $\mathcal{I}_i$ containing a single point, that is a lower bound on how small $\gamma$ can be theoretically. We remark that this theoretical bound may be pessimistic for two reasons. One is its use of the union bound. The other is the introduction of $\eta$. The disjoint requirement simplifies the assignment of selected columns to sets for the proof, but is a relaxation of what is really needed.

\subsection{Accelerating the SCDM procedure using an approximately
localized basis}

Once the approximate localized orbitals $\wt{\Phi}$ are obtained, we
would like to perform a refinement procedure to further localize the
basis. We do this by taking advantage of the locality of the input to
the SCDM procedure. \rr{In Algorithm~\ref{alg:rand} we approximate the behavior of the SCDM algorithm by restricting the number of columns of $\Psi^*$ that the QRCP factorization can select from. However, once we have a somewhat localized basis we can efficiently take more columns of $\Psi^*$ into consideration. This allows us to better approximate the original SCDM algorithm and, therefore, construct a more localized basis. We accomplish this with a procedure that resembles the tournament pivoting strategy for computing a QRCP. We first compute a bunch of small QRCP factorizations, each involving columns associated with the support of a subset of the rows of $\wt{\Phi}^*$. This is computationally feasible because the rows are already somewhat localized. Lastly, because this procedure generates more candidate columns than needed we perform one final QRCP to try and get as well conditioned a set of columns of $\Psi^*$ as possible.}

Algorithmically, we first need to select a superset of the ultimately
desired columns from which to select the final columns used in the
localization. To construct such a superset we consider each approximately localized
orbital and how we may refine it. Each orbital, once approximately
localized, only exerts influence on, \textit{i.e.}{}~significantly overlaps with,
nearby localized orbitals. Hence, we may refine the selected columns locally. This
means that for each orbital we may simply figure out which orbitals
it substantially overlaps with and compute a QRCP on just those orbitals (rows) of
$\wt{\Phi}^*$ while simultaneously omitting columns with small norm over
those rows. This process will yield a small number of selected columns
that we add to a list of potential candidate columns for the final
localization. However, because we repeat this process for each localized
orbital we might have more than $n_e$ total candidate columns by a small multiplicative factor. Therefore, we perform one final column pivoted QR factorization on these candidate columns to select the final set $\mathcal{C}$.

In principle, while the localized orbitals get small on large portions of the domain, they are not actually zero. Hence, for any given $\epsilon$ we say two orbitals substantially overlap if there is any spatial point where they both have relative magnitude greater than $\epsilon$. We find in practice that choosing $\epsilon$ close to the relative value at which one considers an orbital to have vanished suffices, though taking $\epsilon \rightarrow 0$ one recovers the original SCDM algorithm. Importantly, this parameter is completely independent of the geometry of the problem under consideration and thus does not require detailed knowledge of the system to set.

We detail our complete algorithm for computing the orthogonalized SCDM from approximately localized input in Algorithm~\ref{alg:refine}. The only parameter is how small a column is for it to be ignored. At a high level, the goal is simply to generate additional candidate
columns by allowing each orbital to interact with its immediate
neighbors. Provided that each orbital is only close to a small number of
others, and that the approximate localization is sufficient this procedure may be performed quickly.

\begin{algorithm}
\caption{Refining an approximately localized collection of basis vectors}
\label{alg:refine}
\begin{algorithmic}[1]
\Statex Given:  approximately localized Kohn-Sham orbitals $\wt{\Phi}$ and column tolerance $\epsilon$
\State $\JS_i = \left\{j\in \left\{1,\ldots,N\right\} \mid \lvert \wt{\phi_i}\rvert > \epsilon \max_{k} \lvert \wt{\phi_i} (\bvec{r}_k) \rvert \right\}$ for $i = 1,\ldots,n_e$
\For{$i = 1,\dots, n_e$}
  \State Set $\mathcal{R}_i = \left\{j \in \left\{1,\ldots,n_e\right\} \mid \JS_i \cap \JS_j \neq \emptyset \right\}$
  \State Set $\displaystyle \LS_i = \bigcup_{j\in \mathcal{R}_i} \JS_j$
  \State Compute a column pivoted QR factorization of $\left(\left.\rr{\wt{\Phi}}\right.^*\right)_{\mathcal{R}_i,\LS_i}$ and denote the pivot columns $\mathcal{C}_i$
\EndFor
\State Set $\wt{\CS} = \cup_i \mathcal{C}_i$
\State Compute the column pivoted QR factorization $$\left(\rr{\wt{\Phi}}_{\wt{\CS},:}\right)^*\Pi = QR$$
\State Form the localized basis $\Phi = \wt{\Phi} Q$
\end{algorithmic}
\end{algorithm}

To illustrate the behavior of this algorithm, we sketch the behavior of Algorithm~\ref{alg:refine} in two cases. In one case, after the approximate localization, we have two sets of orbitals whose support sets after truncation are disjoint. This is shown in Figure~\ref{fig:disjoint}. Here, simply computing two independent QRCP factorizations is actually equivalent to computing the QRCP of the entire matrix. As we see Algorithm~\ref{alg:refine} partitions the orbitals into two sets and then only considers the columns with significant norm over the orbital set. In the second case, \rr{illustrated in Figure~\ref{fig:chain}}, we have a chain of orbitals whose support set after truncation forms a connected region in the spatial domain. In this situation we do not actually replicate the computation of a QRCP of the whole matrix, but rather for each orbital we compute a local QRCP ignoring interactions with distant orbitals. More specifically, any column mostly supported on a given orbital will be minimally affected by orthogonalization against columns associated with distant orbitals. Therefore, we may simply ignore those orthogonalization steps and still closely match the column selection procedure.

\begin{figure}[ht!]
  \centering
  \includegraphics[width = 1\textwidth]{mat_disjoint.pdf}
  \includegraphics[width = .6\textwidth]{domain_disjoint.pdf}
  \caption{Matrix (top) and physical domain (bottom) associated with two collections of approximately localized orbitals whose support (lightly shaded region) is disjoint after truncation.}
  \label{fig:disjoint}
\end{figure}

\begin{figure}[ht!]
  \centering
  \includegraphics[width = 1\textwidth]{mat_chain.pdf}
  \includegraphics[width = .6\textwidth]{domain_chain.pdf}
  \caption{Matrix (top) and physical domain (bottom) associated with a chain of approximately localized orbitals whose support (lightly shaded region) is connected after truncation.}
  \label{fig:chain}
\end{figure}

\begin{remark}
In Algorithm~\ref{alg:refine} it may sometimes be the case that several of the $\mathcal{R}_i$ are identical (\textit{e.g.}{}~as in Figure~\ref{fig:disjoint}). In this case, as a matter
of practical performance optimization, one may simply skip instances of
the loop over $i$ that would be computing a QRCP of the exact same
matrix as a prior instance of the loop. Similarly, if there is a set
$\mathcal{S}\subset \left\{1,\ldots,n_e\right\}$ such that $\left(\cup_{i\in \mathcal{S}}
\mathcal{R}_i \right) \cap \left(\cup_{i\in\mathcal{S}^c} \mathcal{R}_i\right) = \emptyset$ and $\lvert \cup_{i\in \mathcal{S}} \mathcal{R}_i \rvert$ is small, we may combine the instances of the loop corresponding to $i\in\mathcal{S}$ into a single small QRCP by simply using $\cup_{i\in \mathcal{S}} \mathcal{R}_i$ as the set of rows. This avoids redundant work if a small collection of orbitals just interact amongst themselves and are disjoint from all others.
\end{remark}

\subsection{Computational complexity}

The computational cost of Algorithms \ref{alg:rand} and \ref{alg:refine}
is formally limited by the cost of computing a single general matrix-matrix
multiplication (GEMM) $\Psi Q$ at cost $\mathcal{O}(n_e^2 N).$ However, this single BLAS 3
operation must appear in any localization procedure starting with the
$\Psi$ matrix as an input.
This GEMM operation is also highly optimized both in sequential and parallel
computational environments. Furthermore, at large scale one can
approximate the product with $\mathcal{O}(n_e N)$ complexity using sparse linear algebra, since the support of the result
is sparse and one may infer the support based on the column indices from
which $Q$ is built. For these reasons, we let $T_{mult}$ represent the
cost of computing $\Psi Q$.

Using this notation, the computational cost of Algorithm~\ref{alg:rand}
is $$\mathcal{O}(n_e N \log n_e + n_e^3 \log n_e) + T_{mult}.$$ More
specifically, the random selection of columns costs $\mathcal{O}(N n_e \log
n_e)$ and the subsequent QR factorization costs $\mathcal{O}(n_e^3 \log n_e).$
If we assume that the support of the approximately localized basis used
as input for Algorithm~\ref{alg:refine} and the number of nearby orbital
centers are bounded by some constant independent of $N$ and $n_e,$ which
is reasonable for models where the molecule is growing and the
discretization quality remains constant, the computational cost is
$$\mathcal{O}(n_e N + n_e^3) +T_{mult}.$$ Under these assumptions each of the,
at most, $n_e$ small QR factorizations has constant cost.
However, the procedure for finding $\JS_i$ introduces a dependency on $N$.
While the computational cost of the
refinement procedure and randomized algorithms are broadly similar, the
randomized algorithm is significantly faster in practice because drawing
the random samples is cheaper than the support computations needed in
the refinement algorithm.

\rr{Lastly, our algorithms are memory efficient. In fact, for practical purposes their memory cost is bounded by the cost of storing $\Psi$ plus a few work arrays of size $N.$ Besides the storage for $\Psi,$ all of the matrices we operate on cost at most $\mathcal{O}(n_e^3 \log n_e)$ to store and may be discarded after use. Furthermore, the QR factorizations and matrix multiplication may be done in place excepting a few work arrays of length $N.$}

\section{Numerical examples}
\label{sec:numer}

To demonstrate the performance of our method we use three examples that
capture the different facets of our algorithm.  The first example is the
dissociation of a BH$_{3}$NH$_{3}$ molecule, and the second example is
the alkane chain. We select these two examples not because they are
computationally expensive, but that they clearly demonstrates that
our approximate localization algorithm is effective in two very
different regimes.  In particular, the effectiveness of the algorithm is
independent of whether localized
orbitals form one single group or multiple disconnected groups. Our third example is a large supercell with $256$ water
molecules. We demonstrate the performance gains over the existing SCDM
method and provide a comparison with Wannier90 \cite{wannier90}.

In all of the examples here we assume we have access to the electron density
$\rho$ from the electronic structure calculation, and therefore
exclude its computation from the timings of our randomized method. \rr{We use $3 n_e \log n_e$ samples in the randomized algorithm, corresponding to $\gamma = 1/3,$ for all of the experiments.}
Furthermore, to more clearly illustrate the advantages of our method, we
separately report timings for computing the orthogonal
transform $Q$ that localizes the orbitals and subsequent computation of the localized orbitals by a single matrix product. Here
we only consider the orthogonalized SCDM, as discussed in this paper.
For the refinement algorithm we set the relative truncation threshold at $\epsilon = 5\times 10^{-2}$ and we observe that this is sufficient for our new algorithm to closely match the results of the existing algorithm. Finally, in all of the experiments here we use $2.5 \times 10^{-2}$ as the relative truncation threshold of the localized orbitals when counting the fraction of entries that are non-zero. This measure of ``locality'' has the advantage of not depending on the geometry of the physical system. However, for completeness we also provide spread computations in the final example. \rr{Prior work validates the expected exponential decay of the orbitals by varying the truncation threshold \cite{SCDM}.}

All numerical results shown were run on a quad-socket
Intel Xeon E5-4640 processor clocked at 2.4 GHz with 1.5 TB of RAM and our algorithms were implemented in MATLAB R2015a. Our code is sequential and the only multi-threading is in the \mbox{LAPACK} and BLAS routines called by MATLAB. \rr{The storage cost of all the algorithms presented here is $\mathcal{O}(n_e N),$ which is also the cost to store $\Psi$ in memory. In our largest example a copy of $\Psi$ cost roughly 60 GB to store.} The Kohn-Sham orbitals were computed using Quantum ESPRESSO \cite{QE} and VMD \cite{VMD} was used for plotting orbitals and molecules in the alkane and water examples.

\subsection{BH$_{3}$NH$_{3}$}
First, we demonstrate the performance of the approximate column
selection method for the dissociation process of a BH$_{3}$NH$_{3}$
molecule. The main purpose of this example is to demonstrate that our
approximate localization algorithm is equally effective when localized
orbitals form disconnected groups or a single group.
Figure~\ref{fig:bh3nh3} shows the localized orbitals for three atomic configurations,
where the distance between B and N atoms is 1.18, 3.09, and 4.96 Bohr,
respectively and also shows the locality for orbitals
computed by Algorithms~\ref{alg:scdm}, \ref{alg:rand}, and
\ref{alg:refine} for each of the three atomic configurations.
Here we plot an isosurface of the orbitals at a value of $2.5 \times 10^{-2}.$

We find
that Algorithm~\ref{alg:refine} automatically identifies that the
localized orbitals should be treated as two disconnected groups for the
dissociated configuration, and as one single group for the bonded
configuration.
We see that in all cases, the randomized method works
quite well on its own. Furthermore, after applying
Algorithm~\ref{alg:refine} to the output of the randomized method, the
sparsity of the orbitals is nearly indistinguishable from that of the original SCDM
algorithm. \rr{In these three scenarios the condition number of $P_{:,\mathcal{C}},$ equivalently $\left(\Psi_{\mathcal{C},:}\right)^*,$ is never larger than three. Here we let $\mathcal{C} \subset \wt{\CS}$ denote the final set of columns we have selected via the last QRCP in Algorithm~\ref{alg:refine} as it corresponds to the columns of the density matrix from which we ultimately build the localized basis.}

\begin{figure}[ht!]
  \centering
  \includegraphics[width = .6\textwidth]{bh3nh3_90loc.pdf}
  \includegraphics[width = .3\textwidth]{bh3nh3_90iso.png}
  \includegraphics[width = .6\textwidth]{bh3nh3_45loc.pdf}
  \includegraphics[width = .3\textwidth]{bh3nh3_45iso.png}
  \includegraphics[width = .6\textwidth]{bh3nh3_1loc.pdf}
  \includegraphics[width = .3\textwidth]{bh3nh3_1iso.png}
  \caption{Sparsity (left) of localized orbitals computed by Algorithms~\ref{alg:scdm}, \ref{alg:rand}, and \ref{alg:refine} based on fraction of non-zero entries after
  truncation and orbital isosurfaces (right) at $2.5 \times 10^{-2}$ generated by Algorithm~\ref{alg:refine} when using the output of Algorithm~\ref{alg:rand} as input.
  Three different configurations moving from the bonded configuration
  (top) to the dissociated configuration (bottom).}
  \label{fig:bh3nh3}
\end{figure}

\subsection{Alkane chain}

Our second example is the alkane chain (atomic configuration shown in
Figure~\ref{fig:alkane}). Similar to the BH$_{3}$NH$_{3}$, this example
is not computationally expensive, but confirms that the approximate
localization algorithm is effective even when all localized orbitals
form one large connected group.
We demonstrate that the refinement process
still achieves the desired goal. In this example $N=820,125$ and $n_e =
100$.

\begin{figure}[ht!]
  \centering
  \includegraphics[width = .75\textwidth]{alkane.png}
  \caption{Atomic configuration of the alkane chain}
  \label{fig:alkane}
\end{figure}

Figure~\ref{fig:alkane_locality} shows histograms of the fraction of
non-zero entries after truncation for the randomized method, the
refinement procedure applied to the output of the randomized method, and
our original algorithm. We observe that the randomized method actually
serves to localize the orbitals rather well. However, the output is
clearly not as good as that produced by the original SCDM algorithm. However, once the
refinement algorithm is applied we see that, while not identical, the locality of the localized orbitals basically matches that of the localized orbitals generated by Algorithm~\ref{alg:scdm}. This is further
illustrated in Figure~\ref{fig:alkane_ex}, which shows isosurfaces for a localized orbital
generated by each of the three methods. \rr{Once again the columns of the density matrix we ultimately select are very well conditioned, in fact the condition number is less than two.}

\begin{figure}[ht!]
  \centering
  \includegraphics[width = 1\textwidth]{alkane_rand.pdf}
  \includegraphics[width = 1\textwidth]{alkane_refine.pdf}
  \includegraphics[width = 1\textwidth]{alkane_full.pdf}
  \caption{Histogram of localized orbitals for the alkane chain example computed by three different algorithms based on fraction of non-zero entries after truncation. (top) output of the randomized algorithm, (middle) output of the refinement algorithm applied to the output of the randomized algorithm, and (bottom) output of the original SCDM algorithm.}
  \label{fig:alkane_locality}
\end{figure}

\begin{figure}[ht!]
  \centering
  \includegraphics[width = .3\textwidth]{alkane_phi_rand.png}
  \includegraphics[width = .3\textwidth]{alkane_phi_refine.png}
  \includegraphics[width = .3\textwidth]{alkane_phi_full.png}
  \caption{The same (as determined by picking those maximally correlated) orbital as generated by the randomized algorithm (left), the refinement procedure (middle), and the original SCDM algorithm (right). Here an isosurface value of $5\times 10^{-3}$ was used and the colors delineate positive and negative regions.}
  \label{fig:alkane_ex}
\end{figure}

Table~\ref{tab:alkane} illustrates the computational cost of our new
algorithms as compared to the original version. The refinement
algorithm is over nine times faster than the original algorithm, though
it cannot be perform without first approximately localizing the
orbitals.
Hence, we also provide the time of getting the orthogonal transform out
of Algorithm~\ref{alg:refine}, which is the sum of the preceding
three lines in the table. Even taking the whole pipeline into account we
see a speed up of about a factor of close to seven.

\begin{table}
\caption{Runtime for localization algorithms as applied to an alkane chain.\label{tab:alkane}}
\centering
\begin{tabular}{|l|c|} \hline
 Operation & time (s) \\ \hhline{|=|=|}
 Matrix-matrix multiplication $\Psi Q$ & 0.2713 \\ \hline
 Randomized version, Algorithm \ref{alg:rand} & 0.0261 \\ \hline
 Refinement step, Algorithm \ref{alg:refine} & 0.7176 \\ \hline
 Total cost of our two stage algorithm & 1.0149 \\ \hline
 Original algorithm, Algorithm \ref{alg:scdm} & 6.9412 \\ \hline
\end{tabular}
\end{table}

\subsection{Water molecules}
We now consider a three dimensional system consisting of $256$ water molecules
(part of the atomic configuration shown in Figure~\ref{fig:water}).
In this example, $N=7,381,125$ and $n_e = 1024$.

\begin{figure}[ht!]
  \centering
  \includegraphics[width = .4\textwidth]{water64.png}
  \caption{Part of the atomic configuration of 256 water molecules}
  \label{fig:water}
\end{figure}

Figure~\ref{fig:water_locality256} shows histograms of the fraction of
non-zero entries after truncation for the randomized method, the
refinement procedure applied to the output of the randomized method, and
the original SCDM algorithm when applied to the 256 water
molecule system. As before, the randomized method actually serves to localize the
orbitals rather well.
However, there is still visible difference between the result of the
randomized algorithm and the original SCDM algorithm.
Application of the refinement
algorithm achieves a set of localized orbitals that broadly match the quality of the ones computed by the original SCDM algorithm. \rr{Similar to the previous two examples $P_{:,\mathcal{C}}$ is very well conditioned, its condition number is once again less than two.}

\begin{figure}[ht!]
  \centering
  \includegraphics[width = 1\textwidth]{water256_rand.pdf}
  \includegraphics[width = 1\textwidth]{water256_refine.pdf}
  \includegraphics[width = 1\textwidth]{water256_full.pdf}
  \caption{Histogram of localized orbitals for 256 water molecules computed by three different algorithms based on fraction of non-zero entries after truncation. (top) output of the randomized algorithm, (middle) output of the refinement algorithm applied to the output of the randomized algorithm, and (bottom) output of the original SCDM algorithm.}
  \label{fig:water_locality256}
\end{figure}

Table~\ref{tab:water256} illustrates the computational cost of our new
algorithms as compared to Algorithm~\ref{alg:scdm}. As
before, the randomized algorithm for computing is very fast and in
this case much faster than the matrix-matrix multiplication required to
construct the localized orbitals themselves. This makes the algorithm
particularly attractive in practice when $\rho$ is given and, as in many
electronic structure codes, the application of $Q$ to $\Psi$ may be
effectively parallelized. Furthermore, the complete procedure for getting orthogonal transform out of Algorithm~\ref{alg:refine}, the sum of the preceding three lines in the table, is more than 30 times faster than Algorithm~\ref{alg:scdm}.

\begin{table}
\caption{Runtime for localization algorithms as applied 256 water molecules.\label{tab:water256}}
\centering
\begin{tabular}{|l|c|c|} \hline
 Operation &  time (s) \\ \hhline{|=|=|}
 Matrix-matrix multiplication $\Psi Q$ & 47.831 \\ \hline
 Randomized version, Algorithm \ref{alg:rand} & 14.024 \\ \hline
 Refinement step, Algorithm \ref{alg:refine} & 78.361 \\ \hline
 Total cost of our two stage algorithm & 140.22 \\ \hline
 Original algorithm, Algorithm \ref{alg:scdm} & 4496.1 \\ \hline
\end{tabular}
\end{table}

Finally, we use this example to provide a comparison with the popular
Wannier90 \cite{wannier90} software package and further demonstrate the
quality of orbitals computed by our algorithms. While we have previously
been looking at sparsity after truncation to evaluate the quality of the
orbitals, we now also evaluate them based on the spread criteria Wannier90 tries to minimize. Loosely speaking this corresponds to the sum of
the variance of each orbital~\cite{MarzariVanderbilt1997}. All of the spreads here were computed by Wannier90 to ensure the same quantity was being measured in each case. Importantly, this is the quantity Wannier90 seeks to minimize and we therefore do not expect to do better under this metric. For example, given our localized orbitals as input, Wannier90 should always be able to at least slightly decrease the objective function value. For this comparison, we used the random initial guess option in Wannier90 and the default convergence tolerance of $10^{-10}.$ \rr{Notably, the random choice in Wannier90 does not correspond to a random gauge, rather an initial guess is constructed by randomly placing Gaussian functions in space.}

\begin{table}
\caption{Final spread of orbitals for 256 water molecules and time to
solution.}
\label{tab:wannier}
\centering
\begin{tabular}{|l|c|c|} \hline
 Algorithm & spread $\left(\text{\AA}^2\right)$ & time to solution (s) \\ \hhline{|=|=|=|}
 Algorithm~\ref{alg:scdm} & 589.91 & 4496.1 \\ \hline
 Randomized version, Algorithm \ref{alg:rand} & 636.60 & 14.024 \\ \hline
Two stage procedure using Algorithms~\ref{alg:rand} and~\ref{alg:refine} & 589.97 & 140.22 \\ \hline
 Wannier90 & 550.20 & 1715.2 \\ \hline
\end{tabular}
\end{table}

Table~\ref{tab:wannier} shows the spreads and time to solution for all of the algorithms. First, we observe that the spread using our two stage procedure almost exactly matches that of the output of Algorithm~\ref{alg:scdm} while being over an order of magnitude faster. Secondly, the output from our algorithms, even the randomized one, are close to the local minimum found by Wannier90. \rr{Figure~\ref{fig:wannier_spread} directly compares the spreads of our localized orbitals from the two stage algorithm and those at a local minimum found by Wannier90. As expected, there are discrepancies, but we are generally within approximately $0.05 \text{\AA}^2$ per orbital. This aligns with our expectations based on the objective function difference. For comparison, we computed a localized basis for an isolated $\text{H}_2\text{O}$ molecule. Using Algorithm~\ref{alg:scdm} yielded spreads of 0.46, 0.48, 0.56, and 0.57 $\text{\AA}^2$. Subsequently running Wannier90 to convergence yielded spreads of 0.47, 0.48, 0.55, and 0.55 $\text{\AA}^2$, which matched the results when starting with a Wannier90 generated random initial guess.}

\begin{figure}[ht!]
  \centering
  \includegraphics[width = 1\textwidth]{wannier_fast.pdf}
  \includegraphics[width = 1\textwidth]{wannier_conv.pdf}
  \caption{\rr{Histogram of localized orbitals for 256 water molecules computed by two different algorithms based on the Wannier90 spread functional. (top) output of the refinement algorithm applied to the output of the randomized algorithm, and (bottom) converged local minimum for Wannier90.}}
  \label{fig:wannier_spread}
\end{figure}

Admittedly, the time to solution for Wannier90 may depend strongly on the initial guess, \textit{e.g.,}{}~in this experiment the initial spread for Wannier90 was $5339 \text{ \AA}^2$ and convergence took 20 iterations. However, a poor initial guess could result in Wannier90 failing to converge, converging to a worse local minimum, \rr{or taking longer to converge}. Our algorithms are direct and have no such dependence on an initial guess. In this experiment, each iteration of Wannier90 took roughly 85 seconds. So even two iterations costs more than our two stage algorithm and we have omitted the cost of computing the overlap matrices based on $\Psi$, which Wannier90 requires as input and we computed using Quantum ESPRESSO on 256 processors. Collectively, these results make our algorithm an attractive alternative to Wannier90. If a local minimum of the objective is desired, the output from \rr{Algorithms~\ref{alg:rand} or~\ref{alg:refine}} may be used as an algorithmically constructed initial guess for Wannier90.

\begin{remark}
\rr{Seeding Wannier90 with our two stage algorithm convergence took 16 iterations and we appear to arrive at an equivalent local minima. For this system, it would appear the area around the local minimum is quite flat and the default absolute convergence criteria is rather stringent. In fact, after one iteration starting with our algorithm Wannier90 yields localized functions whose spread is withing $0.4\%$ of the converged value. To get to within $1\%$ starting from the random initial guess takes five iterations.}
\end{remark}

\section{Conclusion}
\label{sec:conclusion}

We have presented a two stage algorithm to accelerate the
computation of the SCDM for finding localized representation of the Kohn-Sham invariant
subspace.
We first utilize an
algorithm based on random sampling to approximately localize the basis
and then perform a subsequent refinement step. This method can achieve
computational gains of over an order of magnitude for systems of
relatively large sizes.
Furthermore, the orbitals computed are qualitatively and quantitatively similar to those generated by the
original SCDM algorithm. Lastly, for large systems we observe that our algorithm may provide an attractive alternative to Wannier90 and, at the very least, may provide a simple method for the construction of a good intial guess.

Rapid computation of a localized basis allows for its use within various
computations in electronic structure calculation where a localized basis
may need to be computed repeatedly. This includes computations such as
molecular dynamics and time dependent density functional theory with hybrid exchange-correlation functions.
Finally, the
ideas inherent to the SCDM procedure have potentially applicability in
problems outside of Kohn-Sham DFT because the structural and behavioral
properties we exploit are not necessarily unique to the problem. Our new
algorithms would admit faster computation in such contexts as well.

\section*{Acknowledgments}
A.D. is partially supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship under grant number DMS-1606277. L.L. is supported by the DOE Scientific
Discovery through the Advanced Computing (SciDAC) program, the DOE Center for Applied Mathematics for Energy Research Applications (CAMERA) program, and the Alfred P.  Sloan fellowship. L.Y. is supported by the National Science Foundation under award DMS-1328230 and DMS-1521830 and the U.S. Department of Energy's Advanced Scientific Computing Research program under award DE-FC02-13ER26134/DE-SC0009409.
The authors thank Stanford University and the Stanford Research Computing Center for providing computational resources and support that have contributed to these research results. The authors also thank the anonymous referees for their many helpful suggestions.

\bibliographystyle{siam}

redpajama_set_name: RedPajamaArXiv