text:
\section*{Abstract}
Since its formulation in the late 1940s, the Feynman-Kac formula has proven
to be an effective tool for both theoretical reformulations and practical
simulations of differential equations. The link it establishes between such
equations and stochastic processes can be exploited to develop Monte Carlo
sampling methods that are effective, especially in high dimensions. There exist many techniques of improving standard Monte Carlo sampling methods, a relatively new development being the so-called Multilevel Monte Carlo method. This paper investigates the applicability of multilevel ideas to the stochastic representation of partial
differential equations by the Feynman-Kac formula, using the Walk on Sphere algorithm to generate the required random paths.
We focus on the Laplace equation, the simplest elliptic PDE, while mentioning some extension possibilities.

\section{Introduction}
The Monte Carlo (MC) error normally converge like $1/\sqrt M$, where $M$ is
the number of samples. If all samples were equally expensive the error
versus work convergence rate would be $1/\sqrt {W}$, where $W$ is the total work.
Note that for a large class of problems the accuracy of the MC method is not only determined by the number of samples used, but also by the discretization error accepted in the computation of each sample.
A small discretization error, using e.g.~the Walk on Spheres (\texttt{WoS}\xspace) algorithm~\cite{Muller}, comes with larger computation costs per sample.
Hence the error
versus work convergence rate falls short of $1/\sqrt {W}$.
If applicable, Multilevel Monte Carlo (MLMC) methods may reach this
optimal $1/\sqrt {W}$ convergence rate.
This motivated us to evaluate the Feynman-Kac formula with MLMC methods instead of MC methods.

In this paper we present a procedure to evaluate a Feynman-Kac formula with MLMC using the Walk on Spheres method. As a model problem  we use the Laplace equation,
which is solved in high dimensions for instance for option pricing \cite[Chap. 8]{hilber2013}
 or in particle accelerator simulations~\cite{Adelmann2010}.
We prove that the error versus work converges with the optimal $1/\sqrt {W}$ convergence rate, and compute the suboptimal convergence when using MC methods. The  MLMC method outperforms the MC method only by a $\log(W)$ term. Our MLMC simulations, executed with an MPI parallel implementation, where up to twice as fast compared to the standard MC implementation.

First, in Section~\ref{sec:WoS} we introduce the Feynman-Kac formula, using the \texttt{WoS}\xspace algorithm, and compute the error versus work convergence rate when using MC. Then, we derive a method to use MLMC in this setting, and evaluate its error versus work convergence rate in Section~\ref{sec:multilevel_walk_on_spheres}.
In Section~\ref{sec:numResults} we present numerical results, which quantify the advantage of using MLMC.
Finally, we draw our conclusions in Section~\ref{cha:outlook_and_conclusion}.

\section{Standard Walk on Spheres} \label{sec:WoS}
\subsection{Brownian motion}
A Brownian motion~\cite{Chang2007} denoted by $X_t$, started at a point in a certain connected domain $D$, has several characteristic quantities. These include the first exit time and the first exit point, two important concepts in the application of the Feynman-Kac formula.
\begin{definition}[First exit time]\label{def:exit_time}
The time at which a realization of a Brownian motion $X_t$, started at some point $x\in D\subset\RR^d$, first hits the domain boundary $\partial D$ is called the first exit time $\tau \defeq \inf \{t>0: X_t \in \partial D\}$.
\end{definition}
\begin{definition}[First exit point]\label{def:exit_point}
    The point at which a realization of a Brownian motion $X_t$, started at some point $x\in D\subset\RR^d$, first hits the domain boundary $\partial D$ is called the first exit point $X_\tau$.
\end{definition}

Of great importance for the derivation of the Walk on Spheres algorithm is the \textit{distribution} of first exit points of a Brownian motion in a ball $B_{x}(R)$\newnot{symbol:Bxr} of radius $R$ centered at the point $x$.
If the starting point of the Brownian motion is the center of the ball, this distribution is uniform on its spherical surface, independent of the dimension
\cite[Theorem~3]{Wendel1980}.

In evaluating functionals of stochastic processes, one is often interested in their expectations and variances.
Throughout this paper, the $L^2$-norm over the sample space $\Omega$ is defined as the the expectation $\LL{X}:= \EE{|X|^2}^{\frac12}$.

\subsection{Feynman-Kac formula for the Laplace equation}

The Feynman-Kac formula, developed by Richard Feynman and Mark Kac \cite{Kac1949}, gives a probabilistic representation of the solution to certain PDEs at a single fixed evaluation point $x$.
The Feynman-Kac formula allows us to write a very general elliptic PDE
\begin{align*}
    Lu &= \frac12 \Delta u + \sum_{i=1}^d b_i(x) \partial_i u + c(x)u = -g(x), & &\text{ in }D\subset\RR^d, \\
    u(x) &= f(x), & &\text{ on } \partial D,
\end{align*}
along with the assumption that $b$, $c$ and $g$ are smooth and satisfy a Lipshitz growth condition \cite{Buchmann2003} as an expectation over a stochastic process $X_t$ beginning at the point $x$.
This process fulfills the stochastic differential equation $\dd{X}_t=b(x)\dd{t} + \!\dd{W}$ where $\dd{W}$ is a Brownian increment, and is stopped as soon as it hits the boundary.
Denoting the first exit time by $\tau$ and the condition that $X_0=x$ by the subscript $x$ in $\mathds{E}_x$, the formula can be written as
\begin{equation}\label{eq:feynman_kac}
    u(x) = \EEx{ f(X_\tau)\exp\left(\int_0^\tau\!c(X_s)\dd{s}\right) + \int_0^\tau\!g(X_t)\exp\left(\int_0^t\!c(X_s)\dd{s}\right)\dd{t} }.
\end{equation}
In this paper we are only interested in the Laplace equation $\Delta u = 0$ in $d$ dimensions on a domain $D\subset\RR^d$ with boundary values $u(x) = f(x)$ on $\partial D$. This simplifies the development of an MLMC method, which can be used as a basis for more general equations.
In this simple case the Feynman-Kac representation is given as an expectation over the exit point $X_\tau$ of a Brownian motion started at the point $x$. Denoting the condition $X_0=x$ by the subscript $x$ in $\mathds{E}_x$, the solution is written as~\cite{Buchmann2003}
\begin{equation}
	u(x)=\EEx{f(X_\tau)}=:\EEx{Y}.
	\label{eq:feynman_kac}
\end{equation}

\subsection{Walk on Spheres (\texttt{WoS}\xspace)}
\label{subsec:WoS}
The Walk on Spheres (\texttt{WoS}\xspace) algorithm \cite{Muller} can be viewed as an alternative to a conventional detailed simulation of the drift-free Brownian motion inside the domain $D$. Started at the center of a sphere the exit point distribution is known. This allows to simulate the Brownian motion using discrete jumps of a given size. Starting at $\tau=0$ with $X_0=x$ the algorithm measures the distance from the current position $X_t$ to $\partial D$ and jumps this distance in a uniformly random direction to the next position $X_{t+1}$. The algorithm terminates when it is $\varepsilon$ close to the boundary $\partial D$ at the point $X_N$, where $N$ denotes the number of \texttt{WoS}\xspace steps needed. The first exit point is approximated by the point $\overline{X}_N \in \partial D$ that is closest to $X_N$.
This \texttt{WoS}\xspace algorithm may be used to solve the Laplace equation~\cite{Muller}.
We recall that $\EE{Y}=u(x)$ when $Y$ is computed with a realization of a processes starting at $x$.
$Y_\varepsilon\defeq f(\overline{X}_N),$ is the estimator based on an \texttt{WoS}\xspace processes with the discretization parameter $\varepsilon$.
The discretization error presented in \cite{Mascagni2003} is given as
\begin{equation}\label{eq:err_discretization}
    e_\text{discr} = \LL{\EE{Y}-\EE{Y_\varepsilon}} \le \OO(\varepsilon).
\end{equation}

In \cite{Binder2009}, a worst-case upper bound of $\OO(\varepsilon^{4/d-2})$ for three or more dimensional problems ($d\ge3$) is derived for the expected \texttt{WoS}\xspace path length $\EE{N}$. For domains fulfilling certain regularity conditions tighter upper bounds of $\EE{N} = \OO(\log^p(\varepsilon^{-1}))$, $d\ge2$ are proven for $p=1$ or $p=2$ depending on the domain. These estimates are relevant for bounding the expected work per sample.

\subsection{Monte Carlo}

\subsubsection{Statistical error}
\label{ssub:statistical_error}
Given $M$ realizations $\{Y_\varepsilon^i\}_{i=1}^M$ of the random variable $Y_\varepsilon$ obtained from independent paths of the \texttt{WoS}\xspace process, the value for $\EE{Y_\varepsilon}$ can be approximated by the estimator $\EM{Y_\varepsilon} = \frac1M\sum_{i=1}^MY_\varepsilon^i$.
The error of this Monte Carlo estimator can be written in terms of the random variable $Y_\varepsilon$ as
\begin{align}\label{eq:err_statistical}
    \begin{split}
        e_\text{stat}   &= \LL{\EE{Y_\varepsilon}-\EM{Y_\varepsilon}} = \var{\frac1M\sum_{i=1}^M Y_\varepsilon^i}^\frac12 \\
                        &= \frac1{\sqrt{M}}\sqrt{\var{Y_\varepsilon}} \le \frac1{\sqrt{M}}\LL{Y_\varepsilon},
    \end{split}
\end{align}
where we use the fact that the samples $Y_\varepsilon^i$ are independent and identically distributed (i.i.d.)\ realizations of $Y_\varepsilon$.

\subsubsection{Total error}
\label{ssub:total_error}
The total error of an estimation $\EM{Y_\varepsilon}$ of $\EE{Y}$ by the Walk on Spheres algorithm can be written using the triangle inequality of the $L^2(\Omega)$ norm as
\begin{equation*}

    \LL{\EE{Y} - \EM{Y_\varepsilon}} \le \LL{\EE{Y}-\EE{Y_\varepsilon}} + \LL{\EE{Y_\varepsilon}-\EM{Y_\varepsilon}},
\end{equation*}
resulting in a total error of
\begin{equation}\label{eq:err_total}
    e_\text{tot} \defeq \LL{\EE{Y} - \EM{Y_\varepsilon}} \le \OO(\varepsilon) + \sqrt{\frac{\var{Y_\varepsilon}}{M}}.
\end{equation}
The two terms at the right follow by \eqref{eq:err_discretization} and \eqref{eq:err_statistical}, respectively.

\subsubsection{Error equilibration}
\label{ssub:error_equilibration}
For a fixed prescribed error, one way of choosing the sample size is by equilibrating the statistical and discretization errors in~\eqref{eq:err_total}, $\OO(\varepsilon) = \sqrt{\var{Y_\varepsilon}/M}$.
This yields the relationship
\begin{equation}\label{eq:M_equilibrium}
    M=\OO(\varepsilon^{-2}),
\end{equation}
giving a total error behavior of $\OO(\varepsilon)$.

\subsection{Error vs. expected work}
\label{sub:error_vs_work}
The expected total work $\EE{W}$ of a \texttt{WoS}\xspace simulation is the number of paths times the expected length of a path:
\begin{equation}
    \EE{W} = M\cdot \EE{N}.
\end{equation}

Each bound for the expected path length $\EE{N}$ shown in section~\ref{subsec:WoS} is multiplied with $M$ derived in \eqref{eq:M_equilibrium}, and the resulting expected work is solved for $\varepsilon$. Equation \eqref{eq:Wvseps_WOS_a} is valid for $d\ge3$, equation \eqref{eq:Wvseps_WOS_b} for well-behaved domains $d\ge2$.  To derive \eqref{eq:Wvseps_WOS_b} we must make use of the Lambert $W$-function, $\mathcal{W}_\text{lam}(\cdot)$, defined as the inverse  of the map $w\mapsto w\exp(w)$. It can be approximated with the truncated expansion $\mathcal{W}_\text{lam}(x) \approx \log(x)-\log\log(x)$ \cite{Bruijn1970}.
This yields the following relationships:
\begin{subequations}
    \begin{align}
        \EE{W} &= \OO(\varepsilon^{4/d-4}) &   e_\text{tot} = \OO(\varepsilon) &= \OO( \EE{W}^{\frac14\frac{d}{1-d}})\label{eq:Wvseps_WOS_a}\\
        \EE{W} &= \OO(\varepsilon^{-2}\log^2(\varepsilon^{-1})) & e_\text{tot} = \OO(\varepsilon) &= \OO(\EE{W}^{-\frac12}\log(\EE{W})) \label{eq:Wvseps_WOS_b}
    \end{align}
\end{subequations}
The total error is linear in $\varepsilon$ by the choice in \eqref{eq:M_equilibrium}. In both cases the \texttt{WoS}\xspace algorithm performs worse than the optimum possible in a Monte Carlo setting, namely $W^{-\frac12}$. This motivates the formulation of a multilevel version of the \texttt{WoS}\xspace algorithm, with the hopes of achieving the optimal convergence rate.

\section{Multilevel Walk on Spheres (\texttt{MLWoS}\xspace)}
\label{sec:multilevel_walk_on_spheres}
In this section, a multilevel version of the \texttt{WoS}\xspace algorithm is formulated and its error behavior analyzed.
The main idea is to execute the \texttt{WoS}\xspace algorithm on different ``discretization levels'', meaning for different values of the discretization parameter $\varepsilon$.
The subscript $\ell\in\{0,\ldots,L\}$ is used to denote a certain discretization level, where $\ell=0$ is the coarsest discretization level, corresponding to a \texttt{WoS}\xspace simulation with $\varepsilon_0$, and $\ell=L$ is the finest discretization level.

\subsection{Multilevel formulation}
\label{sub:multilevel_formulation}
On each discretization level $\ell=0,\ldots,L$, we define the discretization parameter $\varepsilon_\ell\defeq\eta^{-\ell}\varepsilon_0$, \newnot{symbol:epsl}\newnot{symbol:eta} where $\eta>1$, i.e.~the width of the stopping region is divided by $\eta$ between successive levels.

In the following definition, a single \texttt{WoS}\xspace process with $\varepsilon_\ell$ is used to define the multilevel process in order to incorporate the fact that stopping points on higher levels are continuations of previously stopped processes.
\begin{definition}[Multilevel Walk on Spheres process]\label{def:mlwos}
    Given a domain $D\subset\RR^d$, a point $x\in D$, a discretization parameter $\varepsilon_0$ and an $\eta>1$.
    Consider a \texttt{WoS}\xspace process $\{X_i\}_{i=0}^{N_\ell}$ started at $x$ with $\varepsilon_\ell = \eta^{-\ell}\varepsilon_0$ and the pair $\left( X_{N_{\ell-1}}, X_{N_\ell} \right)$ obtained by $N_{\ell-1} = \min \{N\in\NN : d_{\partial D}(X_N)<\varepsilon_{\ell-1}\}$, where $d_{\partial D}(X_N)=\min_{x'\in\partial D}|X_N-x'|$ is the distance to the boundary $\partial D$.
    The Multilevel Walk on Spheres process (\texttt{MLWoS}\xspace) on the level $\ell$ is the set of all such pairs.

\end{definition}

In the context of multilevel Monte Carlo, we have multiple estimators $\YY{\ell}$, one for each discretization level with discretization parameter $\varepsilon_\ell$.
These are obtained in the same way as for the non-multilevel case, e.g.~$\YY{\ell}=f(\overline{X}_{\!N_\ell})$. The expectation of the estimator $\YY{L}$ is written in a multilevel form as
\begin{equation}\label{eq:mlmc_exact}
    \EE{\YY{L}} = \EE{\YY{0}} + \sum_{\ell=1}^L \EE{\YY{\ell}-\YY{\ell-1}}.
\end{equation}
Replacing the expectation with the average over $M_\ell$ realizations on each discretization level $\ell$, we get the MLMC estimate
\begin{equation}\label{eq:mlmc_approx}
    E\left[\YY{L}\right] = \EMl{0}{\YY{0}} + \sum_{\ell=1}^L \EMl{\ell}{\YY{\ell}-\YY{\ell-1}}.
\end{equation}
The ``sample'' on level $\ell>0$ is now $(\YY{\ell}-\YY{\ell-1})$, for which it is assumed that the two values on discretization levels $\ell$ and $\ell-1$ come from the same \texttt{WoS}\xspace path.
The implication is that they should be simulated with the \texttt{MLWoS}\xspace process from \autoref{def:mlwos}.
The remaining estimator on discretization level $0$ is computed with an ordinary \texttt{WoS}\xspace simulation.

Note that the individual $\YY{\ell}$ are estimators on the \textit{discretization level} $\ell$, whereas $\YY{\ell}-\YY{\ell-1}$ are referred to as estimators on \textit{level} $\ell$.

\subsection{Multilevel error bounds}
\label{sub:multilevel_error_bounds}
The multilevel Monte Carlo error is defined as the difference between the expectation of the exact estimator $\EE{Y}$ and the MLMC approximation involving all levels, $E\left[\YY{L}\right]$, and is given by~\cite{Barth2011}
\begin{align*}
    \LL{\EE{Y}-E[\YY{L}]} &\le \LL{\EE{Y}-\EE{\YY{L}}} \\
    & \qquad + M_0^{-\frac12}\LL{\YY{0}} + \sum_{\ell=1}^L M_\ell^{-\frac12}\LL{ \YY{\ell}-\YY{\ell-1} }.
\end{align*}

\subsection{Asymptotic variance convergence rate}
\label{sub:variance_convergence_rate}
We want to bound the variance of the estimator on a level $\ell$ in the multilevel error bound.
In more exact terms, we want to determine for which functions $f$ and domains $D$ the relationship
\begin{equation}\label{eq:varconvrate}
    \LL{\YY{\ell}-\YY{\ell-1}} = \OO(\varepsilon_\ell^s) \quad\forall \ell>\ell_{min}
\end{equation}
holds for some $\ell_{min}>0,s>0$.
This behavior is desired since it allows a good multilevel performance as the variance of the estimators on fine levels is already small, thus requiring less fine-level realizations.

Conditioning the expectation on the current position $X_{N_{\ell-1}}$ incorporates the fact that the simulation continues with a given path from level $\ell-1$, not generating a completely independent one on the finer level $\ell$. Using $\overline{X}_{N_\ell} = \overline{X}_{N_{\ell-1}}+\Delta x$ yields
\begin{equation}\label{eq:varconv_1}
    \LL{\YY{\ell}-\YY{\ell-1}}^2 = \EE{\left.|f(\overline{X}_{N_{\ell-1}}+\Delta x)-f(\overline{X}_{N_{\ell-1}})|^2\right|X_{N_{\ell-1}}}.
\end{equation}
In order to write this expectation in terms of $|\Delta x|$, not in terms of a function of the process we make the following assumption:
\begin{assumption}[Hölder continuity]
    There exist $C,\alpha > 0$ such that
    \begin{equation*}
        |f(x)-f(y)| \le C|x-y|^\alpha
    \end{equation*}
    for all $x,y\in\partial D$.
\end{assumption}\newnot{symbol:alpha}
This assumption implies
\begin{align}\label{eq:varconv_2}
    \EE{\left.|f(x+\Delta x)-f(x)|^2\right|x} &\le C^2\EE{\left.|\Delta x|^{2\alpha}\right|x}.
\end{align}
The conditioning can be omitted by bounding the right-hand side of \eqref{eq:varconv_2} by the maximum over $x$, the stopping point of the process on level $\ell-1$.

Splitting the expectation into a converging part where $|\Delta x| \le R_\ell$ and a ``diverging'' part with $R_\ell<|\Delta x| \le |D|$, where $|D|$ is the diameter of the circumsphere of $D$, we get \newnot{symbol:absD}
\newcommand{p_{\text{div}}}{p_{\text{div}}}
\begin{align}\label{eq:Edx_ansatz}
    \begin{split}
    \EE{|\Delta x|^{2\alpha}} &\le \PP{|\Delta x_\ell| \le R_\ell}R_\ell^{2\alpha} + \PP{|\Delta x_\ell| > R_\ell}|D|^{2\alpha} \\
                              &= (1-p_{\text{div}})R_\ell^{2\alpha} + p_{\text{div}} |D|^{2\alpha}.
    \end{split}
\end{align}
Since the $|D|$ in the second term is a domain-dependent constant, we must find a bound for the divergence probability $p_{\text{div}}$ in terms of the discretization parameter $\varepsilon$.
As shown below, this is possible with the resulting behavior $p_{\text{div}}\propto\frac{\varepsilon_\ell}{R_\ell}$.

In order for the expectation to converge with a certain rate in $\varepsilon$, both terms $R_\ell^{2\alpha}$ and $p_{\text{div}}$ should be of the same order.
Equating the two, solving this equation for $R_\ell$ and inserting back into \eqref{eq:Edx_ansatz} yields
\begin{align}
    R_\ell &= C\varepsilon_\ell^{\frac1{2\alpha+1}} \nonumber \\
    \EE{|\Delta x|^{2\alpha}} &\le C\varepsilon_\ell^{\frac{2\alpha}{2\alpha+1}} \label{eq:varconv_result}, \\
    \LL{\YY{\ell}-\YY{\ell-1}}^2&\le C\varepsilon_\ell^{\frac{2\alpha}{2\alpha+1}}. \nonumber
\end{align}
For differentiable $f$, $\alpha=1$ this yields the convergence rate $\varepsilon_\ell^\frac13$ in \eqref{eq:varconvrate}.

\subsubsection{Convergence of divergence probability}
\label{ssub:convergence_of_divergence_probability}
Our goal here is to obtain a bound on the divergence probability $p_{\text{div}}$ on a certain level $\ell$ depending on $\varepsilon_\ell$.
Divergence here means that the process does not result in an estimate $\overline{X}_{N_{\ell+1}}$ that is located within a ball $B_{\overline{X}_{N_\ell}}\!(R_\ell)$ of radius $R_\ell$ around the current projected stopping point $\overline{X}_{N_{\ell}}$.

In the derivation we use the so-called harmonic measure.
There exist a few equivalent definitions of harmonic measure which are relevant to the current application.
The first defines the harmonic measure as a harmonic function satisfying certain boundary conditions,
the second is instructive in the context of Monte Carlo approximations of an integral and
the third uses a definition relying on the distribution of the first exit point given a Brownian motion. Consider the closed set $D\subset\RR^d$ with $d\ge2$ and let $\BB(\partial D)$ denote the $\sigma$-algebra of subsets of $\partial D$ and define:

\begin{definition}[Harmonic measure -- Dirichlet solution]\label{def:harmonicmeasure_dirichlet}
    The harmonic measure $\omega_D: D\times\BB(\partial D) \to [0,1]$, viewed as a function $\omega_D(x,E)$ of $x$ for every fixed $D$ and $E$, is the unique harmonic function that satisfies the boundary condition \cite[p. 39]{Garnett2005}
    \begin{equation*}
        f(x) =
            \begin{cases}
                1, &\text{ if }x\in E,\\
                0, &\text{ if }x\in \partial D \backslash E.
            \end{cases}
    \end{equation*}

\end{definition}

\begin{definition}[Harmonic measure -- integral representation]\label{def:harmonicmeasure_integral}
    The harmonic measure $\omega_D: D\times\BB(\partial D) \to [0,1]$ is the unique function that satisfies \cite[sec. 4.3]{Ransford1995}
    \begin{item}
        \item[(a)] for each $x\in D$, $E\mapsto\omega_D(x,E)$ is a probability measure on $\partial D$
        \item[(b)] if $f:\partial D\to\RR$ is continuous, the solution of the Laplace equation in $D$ is given by
            \begin{equation}\label{eq:u_harmonicmeasure}
                u(x) = \int_{\partial D}\! f(x') \,\omega_D(x,\!\dd{x'}).
            \end{equation}
    \end{item}

\end{definition}
\begin{definition}[Harmonic measure -- hitting probability]\label{def:harmonicmeasure_hitting}
Let $X_t^x$ denotes a Brownian motion started at $x$.
    The harmonic measure $\omega_D: D\times\BB(\partial D) \to [0,1]$ is given by
    \[ \omega_D(x,E) \defeq \PP{X_\tau^x \in E, \tau<\infty}, \]
    where $\tau = \inf\{t\ge0:X_t^x\in\partial D\}$ is the first exit time of $X_t^x$ from $D$.
    \cite[Theorem F.6.]{Garnett2005}
\end{definition}

Writing $\omega_D(x,x') = \frac{\partial {\omega}_D(x,x')}{\partial {x'}}$\newnot{symbol:omega_density} as a probability density function of the distribution of first exit points $x'$ of $\partial D$ by a Brownian motion started at $x$, \eqref{eq:u_harmonicmeasure} corresponds to the expectation value of $f(X_\tau)$.
This agrees with the Feynman-Kac formula \eqref{eq:feynman_kac} applied to the Laplace equation.
Simulating realizations of the stochastic process $X_t$ thus amounts to approximating the integral \eqref{eq:u_harmonicmeasure} by Monte Carlo integration, i.e. generating realizations of the exit points distributed according to the harmonic measure.

Often, the harmonic measure in a certain domain can't be given analytically.
Thus, it is important to be able to bound it from above and below.
This can be accomplished using the general inequalities popularized by Carleman, illustrated in \autoref{fig:carleman} and often referred to as \textit{Carleman's Principle of Monotonicity} or the \textit{Principle of Extension of Domain}.
\begin{definition}[Carleman's principle]\label{def:carlemans_principle}
    The harmonic measure of a portion $\alpha$ of the boundary of $G\subset\RR^d$ increases if the region is increased by modifying $\beta=\partial G \backslash \alpha$:
    \begin{equation}\label{eq:carleman_lower}
        \omega_G(x,\alpha) \le \omega_{G'}(x,\alpha).
    \end{equation}
    Since $\omega_G(x,\alpha)+\omega_G(x,\beta)=1$, it follows that the harmonic measure on the extended portion of the boundary decreases with respect to the new domain:
    \begin{equation}\label{eq:carleman_upper}
        \omega_G(x,\beta) \ge \omega_{G'}(x,\beta').
    \end{equation}

    \cite[p. 68]{nevanlinna1960analytic}, \cite[p. 131]{doob2001classical}
    \input{images/fig_carleman}.

\end{definition}

Writing the divergence probability $p_{\text{div}}$ in terms of a harmonic measure  $p_{\text{div}} = \PP{\overline{X}_{N_{\ell+1}}\notin B_{\overline{X}_{N_\ell}}\!(R_\ell)} = \omega_D(X_{N_\ell},\partial D \cap B_{\overline{X}_{N_\ell}}\!(R_\ell)^\mathsf{c})$, we can appeal to \textit{Carleman's Principle}.
This allows the transformation of a general bounded convex domain onto a semi-ball, the harmonic measure of which is known analytically.
Simple trigonometric observations then result in a bound depending on $\varepsilon_\ell$.

\input{images/fig_varianceconvergence}
The situation in two dimensions is sketched in \autoref{fig:varconv}: the intersection of the domain $D$ with a semi-ball $S$ of radius $R_\ell>\varepsilon_\ell$ is called $D'$.
In \autoref{fig:varconv_boundary}, the boundaries are split up such that Carleman's method can be applied to the harmonic measure of $\Gamma_d=\partial D \cap B_{\overline{X}_{N_\ell}}(R_\ell)^\mathsf{c}$, the portion of the boundary hit in the case that the process diverges to a point more than $R_\ell$ away from $\overline{X}_{N_\ell}$.
\input{images/fig_varianceconvergence_boundaries}

We start by bounding the harmonic measure of $\Gamma_d$ from above by ``shrinking'' $D$ to $D'$ and using the second bound \eqref{eq:carleman_upper}, where $\omega_D(x,\Gamma_d)$ corresponds to the larger domain and thus has smaller harmonic measure:
\begin{equation*}
    \omega_D(x,\Gamma_d) \le \omega_{D'}(x,\Gamma_d').
\end{equation*}
Now, we keep the portion $\Gamma_d'$ fixed and extend the remaining portion $\Gamma_c$ of the boundary of $D'$ to the semicircle boundary $\partial D\backslash\Gamma_d'$.
This lets us use the first inequality \eqref{eq:carleman_lower}, where $\Gamma_d'$ corresponds to $\alpha$ and $\Gamma_c$ corresponds to $\beta$ in the definition, yielding
\begin{equation*}
    \omega_{D'}(x,\Gamma_d') \le \omega_S(x,\Gamma_d') \le \omega_S(x,\partial S\backslash \Gamma_b),
\end{equation*}
where the second inequality comes from adding $\omega_S(x,\partial S\backslash(\Gamma_d'\cup\Gamma_b))$, the harmonic measure of the remaining portion of the arc.

We now assume that $R_\ell \gg \varepsilon_\ell$ and that $x$ lies on the normal of the base at its midpoint. Using that the shortest distance between $x$ and $\Gamma_b$ is less than $\varepsilon_\ell$ allows to use a series approximation of $\omega_S(x,\partial S\backslash \Gamma_b)$ around the midpoint of its base, by using the  Dirichlet solution representation of the harmonic measure Definition~\ref{def:harmonicmeasure_dirichlet}.
The boundary $\Gamma_b$ is zero by definition, as illustrated in \autoref{fig_semicircle_construction}, hence the same is valid for the first and second derivative in all directions of the base. As by definition the Laplacian is zero, the second derivative in normal direction of the base has to be zero as well. Hence the first derivative in this direction is a positive constant. This results in the final bound for the divergence probability in terms of $\varepsilon_\ell$
$$p_{\text{div}} \le C \frac{\varepsilon_\ell}{R_\ell},$$
for some constant $C$.
\input{images/fig_semicircle_construction}

\subsubsection{Optimal number of samples}
To simplify notation, we define the estimator $Y_\ell$ to be
\begin{equation*}
    Y_\ell  \defeq \begin{cases}
        \YY{0}, &\text{ if } \ell=0, \\
        \YY{\ell}-\YY{\ell-1}, & \text{ if } \ell>0,
    \end{cases}
\end{equation*}
and $w_\ell$ the expected work performed to compute a sample on level $\ell$.

Following \cite{Muller2012}, the optimization problem to solve is then
\begin{equation}\label{eq:Ml_optimizationproblem}
    \min  \EE{W_\text{tot}}  = \min \sum_{\ell=0}^L M_\ell w_\ell \quad \text{s.t.}\quad \varepsilon_L = \sqrt{\sum_{\ell=0}^L \frac{\var{Y_\ell}}{M_\ell} },
\end{equation}
hence minimizing the expected work, while maintaining a sampling error proportional to the discretization error $(\OO(\varepsilon_L))$.
\cite{Muller2012} provides the optimal number of samples per level
\begin{equation}
\label{eq:optimal_ML}
    M_\ell = \frac1{\varepsilon_L^2}\sqrt{\frac{\var{Y_\ell}}{w_\ell}}\sum_{\ell=0}^L\sqrt{\var{Y_\ell}w_\ell}.
\end{equation}

\subsection{Error vs. work}
The following basic relations will be used in the sequel.
\begin{subequations}
\begin{align}
    &\text{Refinement of }\varepsilon_\ell     & & \varepsilon_\ell = \eta^{-\ell}\varepsilon_0=\eta^{L-\ell}\varepsilon_L \\
    &\text{Scaling of work}             & & w_\ell = \begin{cases}
       \OO(\varepsilon_\ell^{-\gamma}) \\
       \OO(\log^p(\varepsilon_\ell^{-1})) = \OO(\ell^{{}^p})
    \end{cases} \\
  	&\text{Scaling of variance}             & &  \var{Y_\ell} = \OO(\varepsilon_\ell^{2s})
\end{align}
\end{subequations}

Determining $M_\ell$~\eqref{eq:optimal_ML} based on the estimates of variance and work lead to
\begin{align}
 M_\ell &= \begin{cases}
       \varepsilon_L^{-2} \sqrt{\varepsilon_\ell^{2s+\gamma}} \sum_{\ell=0}^L \sqrt{\varepsilon_\ell^{2s-\gamma}}  \\
       \varepsilon_L^{-2} \sqrt{\varepsilon_\ell^{2s}\cdot \ell^{-p}} \sum_{\ell=0}^L \sqrt{\varepsilon_\ell^{2s}\cdot \ell^p}
    \end{cases}      \nonumber \\
&= \begin{cases}
       \varepsilon_L^{-2} \sqrt{\varepsilon_\ell^{2s+\gamma}} \sqrt{\varepsilon_0^{2s-\gamma}} \underbrace{\sum_{\ell=0}^L \sqrt{\eta^{(\gamma-2s)\ell}}}_{\le C, \; \forall 2s>\gamma} \\
       \varepsilon_L^{-2} \sqrt{\varepsilon_\ell^{2s}\cdot \ell^{-p}} \sqrt{\varepsilon_0^2s}\underbrace{\sum_{\ell=0}^L \sqrt{\eta^{-2s\ell}\cdot \ell^p}}_{\le C}
    \end{cases}   \label{eq:Mlopt}
\end{align}
The summands of the remaining sum can be bounded by a convergent geometric sequence. Thus, by the comparison test the infinite series ($L\to\infty$) converges to a constant, in the first case provided $2s>\gamma$.

Inserting \eqref{eq:Mlopt} into the formula $\Wtot = \sum_{\ell=0}^L M_\ell w_\ell$, we can write the work as a function of $\varepsilon_L$, which scales linearly with $e_\text{tot}$
\begin{align*}
    \Wtot &\le  \begin{cases}
       C \varepsilon_L^{-2} \sqrt{\varepsilon_0^{2s-\gamma}} \sum_{\ell=0}^L \sqrt{\varepsilon_\ell^{2s-\gamma}}  \overset{\eqref{eq:Mlopt}}{\le} C^2\varepsilon_L^{-2} \varepsilon_0^{2s-\gamma}, \quad \forall 2s>\gamma\\
       C \varepsilon_L^{-2} \sqrt{\varepsilon_0^{2s}} \sum_{\ell=0}^L \sqrt{\varepsilon_\ell^{2s}\cdot \ell^{p}} \overset{\eqref{eq:Mlopt}}{\le} C^2 \varepsilon_L^{-2} \varepsilon_0^{2s}
    \end{cases} \\
    &= \OO(\varepsilon_L^{-2}).
\end{align*}
Thus, we obtain the optimal convergence rate
\begin{equation}\label{eq:work_vs_error}
    e_\text{tot} \propto (\Wtot)^{-\frac12},
\end{equation}
in the case $w_\ell = \OO(\varepsilon_\ell^{-\gamma})$ provided $2s>\gamma$.

\subsection{Measured values}
\label{sec:Measured_values}
The second approach to determining $M_\ell$ is based on estimating the work $w_\ell$ and the variance $\var{Y_\ell}$ on the different levels, rather than using their asymptotic convergence rates as in~\eqref{eq:Mlopt}. In the absence of alternatives we still use the asymptotic convergence rate of the discretization error $\OO(\varepsilon_L))$. As in~\cite{Muller} the required estimates are computed using the same samples already involved in the MLMC estimator. A certain minimum number of samples (so called warm-up samples) are needed on every level to provide an accurate estimate. Performance disadvantages arise if the number of required warm-up samples exceeds the optimal number of samples~\eqref{eq:optimal_ML}. This typically happens  for the finest level $L$. However, ~\eqref{eq:Mlopt} proves that asymptotically the optimal number of samples grow, which implies that for small $\varepsilon_L$ the required warm-up samples do not exceed the optimal number of samples. This technique complicates the implementation slightly, but does not increase the computational work for this application.

\section{Numerical results}\label{sec:numResults}

\subsection{Model problems}
In this section we formulate model problems, each posed on a different domain to test our algorithm.

\subsubsection{Square}
\label{sub:domain_rectangle}
Many of the simplest examples are formulated on  square domains. Here, we use the two-dimensional square domain  $D_\text{\Square} \defeq [0,2]^2$.
In order to more easily measure the convergence rate, a boundary condition resulting in a large discretization error is imposed with Hölder exponent $\alpha =1$.
\begin{equation}\label{eq:rectangle_BC}
    f(x,y) =
    \begin{cases}
        4(x-\frac12)^2, & \text{if } x \le \frac12, \; 0\le y \le 2,\\
        4(x-\frac32)^2, & \text{if } x \ge \frac32, \; 0\le y \le 2,\\
                     0, & \text{otherwise} , \; 0\le y \le 2.
    \end{cases}
\end{equation}
The midpoint $m=(1,1)$ of the square was chosen as the starting point.

\subsubsection{Hemisphere}
\label{sub:hemisphere}
A nice three-dimensional problem to consider is the Laplace equation on a hemisphere $D_{\!\!\!\semicircle}\!\!\defeq \{\ve{x}\subset\RR^3 \left|\;|\ve{x}|\le1, x_3\ge0\right.\}$,\newnot{symbol:Dhemisphere} as illustrated in \autoref{fig:domain_hemisphere}.
The boundary conditions of the Laplace equation are chosen such that the analytical solution is given by $u(\ve{x}) = \left[x_1^2+x_2^2+(x_3+1)^2\right]^{-\frac12}$, as stated in \cite{Mascagni2003}. This leads to the following formulation with Hölder exponent $\alpha =1$:
\begin{align}
    \begin{split}
        \Delta u &= 0 \text{ in } D_{\!\!\!\semicircle}\!\! \\
           u(\ve{x}) &= \left[2(x_3+1)\right]^{-\frac12} \text{ for } |\ve{x}| = 1 \\
           u(\ve{x}) &= [x_1^2 + x_2^2 + 1]^{-\frac12} \text{ for } x_3 = 0.
    \end{split}
\end{align}
The initial point was chosen to be $x_0 \defeq (0.2,0.3,0.1)$.

\input{images/fig_hemisphere}

\subsection{Implementation}
A generic C++ implementation for both the conventional and the multilevel version of the Walk on Spheres algorithm was created. The $M_\ell$ are either determined analytically or based on measurements during the simulation. The code uses MPI to generate samples in parallel, such that larger problems can be solved. The code was tested with up to 128 cores. Further information on the implementation is found in~\cite{Gantner2013}.

\subsection{Measurement methodology}
\label{sec:Measurement Methodology}
Monte Carlo methods are based on the approximation of an expectation by a sample mean.
The resulting estimator is itself a random variable -- thus, the total error given by
$$\LL{\EE{Y}-E[\YY{L}]}  = \EE{ \| \EE{Y}-E[\YY{L}] \|^2 }^{\frac12}.$$
This outer expectation can again be approximated by a sample mean, corresponding to a repeated call of the corresponding algorithm.
Thus, for a certain set of parameters one must call the implementation with different random number seeds and compute the sample mean over the resulting realizations.
An estimate for a confidence interval can also be computed using the results of these repeated calls.

\subsection{Measured variance convergence rate}
\label{sec:meas:variance_convergence_rate}
As stated in \eqref{eq:varconvrate}, the variance of the estimate on a level $\ell$ is assumed to converge like $\OO(\varepsilon_\ell^s)$.
In \eqref{eq:varconv_result}, it was shown that $s$ is bounded by $\frac{\alpha}{2\alpha+1}$, $\alpha$ being the exponent of Hölder continuity of the boundary condition.
The results presented here are empirical measurements showing the convergence of the variance of the estimator.
The measurements were averaged over ten calls for the square domain $D_\text{\Square}$.
In \autoref{fig:res:variance_convergence} the theoretical and the measured exponent $s$ are shown.
\begin{figure}[ht]
    \centering
        \includegraphics[width=0.47\textwidth]{images/results/rectangle/fig_varl}
    \caption[.]{ $D_\text{\Square}$   Convergence of the variance as the discretization parameter $\varepsilon_\ell$ is reduced for the  square domain $D_\text{\Square}$ with $\alpha =1$ using $\eta=2$.}
    \label{fig:res:variance_convergence}
\end{figure}
Shown only for the square domain, but valid for all described domains, the variance convergence rate is around $0.5$, which is better than the value $\frac13$ obtained from the analytical derivation assuming continuity of the boundary condition.
This is not a contradiction since the derivation yields an upper bound.
However, using the analytical upper bound will very likely result in suboptimal results, since obviously one can get away with fewer samples on the fine levels.

\subsection{Measured convergence rate}
\label{sec:meas:convergence_rate}

In this section we compare the error versus work of the proposed multilevel \texttt{WoS}\xspace algorithm with the plain \texttt{WoS}\xspace algorithm. For the multilevel \texttt{WoS}\xspace algorithm we distinguish between two strategies, one with analytically derived $M_\ell$, $w_\ell$ and $\var{Y_\ell}$ (MLWOS) and the other with $M_\ell$ based on measured  $w_\ell$ and $\var{Y_\ell}$ (MEAS) according to Section~\ref{sec:Measured_values}.

Measurements performed on the domain $D_\text{\Square}$ are shown in \autoref{fig:res:convergence_eta}. We performed measurements with $\eta=2$, $\eta=8$ and $\eta=16$. The measured values for the average error and the average work are shown, together with the $1\sigma$ confidence interval, for different algorithms. Sampling was performed until the confidence interval was very small. In all cases the multilevel \texttt{WoS}\xspace with analytically derived $M_\ell$ (MLWOS) performs poorly compared to the plain \texttt{WoS}\xspace algorithm. The MLWOS performance improves for $\eta=16$, but an improvement over the plain \texttt{WoS}\xspace algorithm is not measured. The multilevel \texttt{WoS}\xspace with $M_\ell$ using measured values (MEAS) always performs better than the plain \texttt{WoS}\xspace algorithm, especially for $\eta=16$ where the computation is up to 2 times faster.

\begin{figure}[ht]
    \centering
    \subfigure[][$\eta=2$]{
        \includegraphics[width=0.47\textwidth]{images/results/square/fig_convergence_2}
    }
    \subfigure[][$\eta=8$]{
        \includegraphics[width=0.47\textwidth]{images/results/square/fig_convergence_4}
    }
    \subfigure[][$\eta=16$]{
        \includegraphics[width=0.47\textwidth]{images/results/square/fig_convergence_16}
    }
    \caption[.]{Convergence of error vs.~work for the problem posed on $D_\text{\Square}$, for various values of the refinement parameter $\eta$.}
    \label{fig:res:convergence_eta}
\end{figure}

The analytically derived $M_\ell$ are based on the $\var{Y_\ell}$.  As observed in Section~\ref{sec:meas:variance_convergence_rate} $\var{Y_\ell}$ converges faster than predicted in our theory. Hence it is not surprising that the multilevel \texttt{WoS}\xspace algorithm with analytically derived $M_\ell$ performs suboptimally. The multilevel \texttt{WoS}\xspace algorithm with $M_\ell$ based on measurements does not suffer from this problem, therefore we observe a better performance.

The parameter $\eta$ does not influence the convergence rate of the multilevel scheme, but a clever choice may asymptotically reduce the work by a constant. Therefore it is expected that the performance of the multilevel \texttt{WoS}\xspace algorithm depends on $\eta$. For certain problems (see e.g.~\cite{Giles2008}), this parameter is optimized analytically. Here, various values are tried on the domain $D_\text{\Square}$ in order to empirically find good values. We observed that $\eta=16$ is a better choice than the other tested $\eta=8$ and $\eta=2$.

\autoref{fig:res:convergence_4} shows measurements for the domain $D_{\!\!\!\semicircle}\!\!$ for $\eta=16$. The results are similar to the ones seen for the domain $D_\text{\Square}$.
The multilevel \texttt{WoS}\xspace with analytically derived $M_\ell$ (MLWOS) performs poorly, where as the multilevel \texttt{WoS}\xspace with $M_\ell$ using measured values (MEAS) is up to 2 times better than the plain \texttt{WoS}\xspace algorithm.

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.47\textwidth]{images/results/hemisphere/fig_convergence_16}
    \caption[.]{ Convergence of the error vs.~work for the problem posed on $D_{\!\!\!\semicircle}\!\!$, using $\eta=16$.}
    \label{fig:res:convergence_4}
\end{figure}

\section{Conclusions and outlook}
\label{cha:outlook_and_conclusion}
In this work, a first application of the Multilevel Monte Carlo method to the Feynman-Kac formula was developed.
A novel version of the Walk on Spheres process, the ``Multilevel Walk on Spheres'' was formulated and central quantities were derived.
It was proven that the ``Multilevel Walk on Spheres'' algorithm convergence rate of error versus work is optimal.

In order to measure the actual performance gain of the proposed method, an implementation of both the conventional and the multilevel version of the Walk on Spheres algorithm was created.
By parallelizing the generation of samples, more thorough convergence results could be obtained.
Additionally, a version of the multilevel method that chooses the number of samples based on measurements was implemented.

In order to test a variety of different situations, multiple domains and boundary conditions were implemented.
The convergence rate of the error with respect to the work was measured, along with many other relevant quantities.
``Multilevel Walk on Spheres'' resulted in a reduction of the work by up to a factor 2.

Using tighter bounds, it would perhaps be possible to find an upper bound for the variance convergence rate that better fits the measured behavior.
This would hopefully allow to analytically determine the number of samples per level.

It would be beneficial to generalize the ``Multilevel Walk on Spheres'' to other elliptic equations, for example the Poisson equation with nonconstant right-hand side.
This equation has applications in many fields, such as particle accelerator modeling or biochemical electrostatics, in which a Multilevel Monte Carlo formulation may be of use.
This would require the formulation of a process that contains multiple discretization levels \textit{along the path}, i.e., with level dependent maximal sphere radius. This would increase the expected work especially for fine discretizations and presumably increase the benefit of MLMC over MC.

redpajama_set_name: RedPajamaArXiv


1. Click on the "Cannot access your account?" link. This will take you to the password recovery site.
2. Type your PayBook ID.
3. Select the option "I'm not a robot". It is possible that reCAPTCHA will activate to verify you are not a robot. Verify your identity selecting the images you are asked.
4. Once you verify your identity, the "Recover password" button will become active. Click the button. A message confirming an email has been sent to your account will appear.
5. Check the inbox of the mail you inserted. You will receive a Paybook email with the subject "Reestablish your password". If you cannot find the mail, check whether the mail was not sent to the Junk or Spam folder.
The mail will show a link to reestablish your password. Click on it.
6. The link will direct you to the PayBook password change website. Choose a new password and write it in the assigned fields.
7. Make sure both passwords coincide and the chosen password is ''Strong''. The app will indicate this.
8. One your passwords coincide, the button ''Create" will become active. Click on it.
All done! Now you will be able to use your new password in Glass.


text:
Nicolas Camille Flammarion (n. 26 februarie 1842, Montigny-le-Roi, Haute-Marne – d. 3 iunie 1925, Juvisy-sur-Orge, Essonne) a fost un astronom și scriitor francez.

Opere

Camille Flammarion este autorul a cincizeci de cărți, inclusiv:
 Les Habitants de l'autre monde ; révélations d'outre-tombe publiées par Camille Flammarion, communications dictées par coups frappés et par l'écriture médiumnique au salon Mont-Thabor, médium mademoiselle Huet, chez Ledoyen, libraire éditeur, Paris. 1862 en littérature
 La Pluralité des mondes habités, 1862
 Les Mondes imaginaires et les mondes réels, 1864
 Des Forces naturelles inconnues ; à propos des phénomènes produits par les frères Davenport et par les médiums en général, Étude critique par Hermès (pseudonyme de l'auteur), Didier et Cie, Paris, 1865
 Les Mondes célestes, 1865
 Les Merveilles célestes - Lectures du soir à l'usage de la jeunesse, 1865
 Études et lectures sur l'astronomie, 9 volume, 1866 - 1880
 Dieu dans la nature, 1866
 Contemplations scientifiques, 1870
 Voyages aériens, 1870
 L'Atmosphère, 1871
 Récits de l'infini, 1872
 Histoire du ciel, 1872
 Récits de l'infini, Lumen, histoire d'une comète, 1872
 Dans l'infini, 1872
 Vie de Copernic, 1873
 Les Terres du ciel, 1877
 Atlas céleste, 1877
 Cartes de la Lune et de la planète Mars, 1878
 Catalogue des étoiles doubles en mouvement, 1878
 Astronomie sidérale, catalogue des étoiles doubles et multiples, 1879
 Astronomie populaire, 1880
 Les Merveilles célestes, 1881
 Les Étoiles et les curiosités du ciel, 1881

Distincții
 Camille Flammarion a fost numit Comandor al Legiunii de Onoare la 12 august 1922.
 La data de 18 ianuarie 1881 a fost numit Cavaler al Legiunii de Onoare, iar la 2 aprilie 1912, Ofițer al Legiunii de Onoare.

Note

Legături externe
 Biografie detaliată a lui Camille Flammarion

Nașteri în 1842
Nașteri pe 26 februarie
Decese în 1925
Decese pe 3 iunie
Astronomi francezi
Scriitori francezi din secolul al XIX-lea
Scriitori francezi de literatură științifico-fantastică
Astronomi din secolul al XX-lea
Astronomi din secolul al XIX-lea
Cavaleri ai Legiunii de onoare
Ofițeri ai Legiunii de onoare
Comandori ai Legiunii de onoare
Scriitori francezi din secolul al XX-lea
Laureați ai Premiului Jules-Janssen
Eponime ale craterelor de pe Lună
Eponime ale asteroizilor

redpajama_set_name: RedPajamaWikipedia


text:
11 Actors Who Have Played The Doctor
BY Calli Arcale
There have been many actors who have portrayed the Doctor in various settings, but eleven have been the official Doctors. We'll look at all of them here.
1. William Hartnell
Veteran character actor William Hartnell was born in 1908 to humble beginnings; his mother was unwed, he never knew his father, and his first career move was into petty crime. A boxing instructor got him started on horse racing, but he found his real passion when he got a job as a stagehand at the age of 18. He quickly got into acting, working constantly with only a break to serve in World War II in an armored regiment. He ended up typecast in comic tough-guy roles (you can see one of them in The Mouse That Roared), and when Verity Lambert offered him the part of a mysterious time traveler in an educational show aimed at children, he jumped at the part. He created a character who was highly intelligent but not always as wise as he thought himself, brilliant but forgetful, cantankerous but with a deep compassion under the surface. He enjoyed the role tremendously, but by 1966, his health was deteriorating due to arteriosclerosis and he had to quit. The producers came up with the idea of having his character transform into a new actor, and Hartnell suggested Patrick Troughton, who was approached and accepted the part. Hartnell reprised his role once more for the tenth anniversary special, "The Three Doctors," but his health had deteriorated more than the production crew realized and his part had to be rewritten to accommodate his capabilities; it was his final work as an actor, and he passed away in 1974 at the age of 67.
2. Patrick Troughton
Born in 1920, Patrick Troughton went directly into an acting career and was undergoing formal training in New York City when World War II broke out. He returned to England and joined the Navy, where he had a decorated career before returning to the theater, gaining a reputation as a reliable and versatile character actor. In 1953, he became the first person to play Robin Hood on television and found a succession of television, film, and radio roles afterward before Innes Lloyd, the new producer of Doctor Who, approached him in 1966 about succeeding William Hartnell in the title role. He ended up playing the role as what series creator Sydney Newman called a "cosmic hobo," inspired partly by silent film star Charlie Chaplin — brilliant, a bit egotistical, and also a bit of a comedian. He'd sometimes play the recorder, a significant change from the First Doctor, who had no apparent musical talent, and it was during this era that the sonic screwdriver was first seen. After three years, he decided to move on, although he returned three more times to reprise the role, in "The Three Doctors," "The Five Doctors," and "The Two Doctors." He returned to his work as a character actor after his time on Doctor Who, working hard despite doctors' advice due to major heart problems. In 1987, he defied doctor's orders to stay in the country and recuperate and went on one more convention tour. He died on March 27, 1987, in Columbus, Georgia. (I actually saw him once, and got his autograph, earlier in the same U.S. tour. He seemed in good health, but, well, he was a very good actor.) Acting was in his blood; several of his children and grandchildren have gone into acting. The youngest of these is Harry Melling, whom Harry Potter fans know as Dudley Dursley.
3. Jon Pertwee
Born in 1919, and thus actually a year older than the man he would replace, Jon Pertwee was born into a family that already had a lot of actors in it. Like the first two Doctors, he joined the military in World War II; although his service wasn't as distinguished as Troughton's, he did acquire an interesting souvenir: he woke up one morning after a drunken shore leave to find a tattoo on his arm, which made a brief appearance in his debut episode of Doctor Who. After the war, he became known as a comic actor on stage, television, and film. When he heard that the part of the Doctor had become available, he inquired and discovered he was already on the shortlist, and ultimately was cast. He played the character as an action hero with an almost James Bond flair, wearing opera capes and driving souped up cars (including the spaceship-like Whomobile, which actually belonged to Pertwee himself, built on commission by a custom car builder). After five seasons, he departed the role. His career didn't falter afterwards, and in 1979 he found his second children's TV role in Worzel Gummidge. He returned to the part of the Doctor for "The Five Doctors." Like Troughton before him, he kept up the convention circuit, meeting with his fans frequently, and he died of a heart attack in Connecticut on May 20, 1996.
(One odd coincidence—Pertwee's godfather was the actor Henry Ainley, whose son Anthony later took on the part of the Master, originated by Roger Delgado during Pertwee's tenure.)
4. Tom Baker
Born in 1934 to a Catholic working class family in Liverpool, Baker first tried a career as a monk, then joined the military, serving as an orderly in a military hospital, before settling on acting as his career. In the '60s, he was a part of the National Theatre company under Lawrence Olivier, and in 1971 broke into film as Rasputin in Nicholas and Alexandra. In 1974, producer Barry Letts cast him as the Doctor. His interpretation was as arrogant and egotistical as all his predecessors, and just as determined to do good, but more eccentric, often described as "bohemian." The Fourth Doctor is particularly famous for his ridiculously long scarf, which resulted from a miscommunication between costume designer James Acheson and the knitter hired to produce it; Acheson never specified a length, and bought far too much yarn, so the knitter just kept going until it was all used up. Baker performed the part for a record-breaking seven seasons before retiring from it. After leaving, he had a brief marriage to costar Lalla Ward (the Second Romana), but it fell apart when both realized they'd really fallen in love with the other one's character, not the actor—an occupational hazard, unfortunately—and they parted amicably. He continued working on stage and screen, and is still active.
5. Peter Davison
Born Peter Moffett in 1951 (he chose the stage name "Davison" because there was already a Peter Moffatt on the English stage), Davison began work at Nottingham Playhouse and got into television in 1975 alongside the woman who would become his wife, Sandra Dickinson. (Sci-fi fans will remember her as Trillian in the BBC TV miniseries version of Hitchhiker's Guide to the Galaxy; it was her idea to get Davison into a big rubber suit to play the Dish of the Day in the same series.) But his first really big break was the role of Tristan Farnon, a young country veterinarian, in All Creatures Great and Small. In 1981, he got his next big break when he was signed on to succeed Tom Baker as the Doctor. Only 29 at the time, he was the youngest to play the role until Matt Smith in 2010. His Doctor had his little quirks, but was much less eccentric than his predecessor, save for a stick of celery he wore on his lapel and a preference for cricket attire. He left after three seasons. He returned for non-canon productions and the very short charity special "Time Crash," opposite David Tennant as the Tenth Doctor, the only "classic" Doctor to appear on the new series. After Doctor Who, he starred in A Very Peculiar Practice, Campion, and a revival of All Creatures Great and Small, as well as an assortment of other roles; he is still acting. His daughter from his first marriage, Georgia Moffett, later appeared on Doctor Who as well, the first child of a Doctor to appear on the show, in "The Doctor's Daughter" as, well, the Doctor's sort-of daughter. And that led, in a roundabout way, to Peter Davison becoming David Tennant's father-in-law, but more on that later...
6. Colin Baker
Born in 1943 (and of no relation to Tom Baker), Colin Baker initially studied law with the intention of becoming a solicitor, but at 23, found a different calling and became an actor. He had a smattering of television roles before appearing on Doctor Who in 1983 as Commander Maxil, commander of the chancellory guards on Gallifrey in "The Arc of Infinity." This got him onto producer John Nathan-Turner's radar, and he was cast as the Sixth Doctor after Peter Davison's departure. His tenure as the Doctor was a difficult one, marred by the battle of the production team with BBC leadership who hoped to see the series die. His costume was wildly garish, and he even attempted to kill his own companion in a fit of madness. But Baker put everything into the part and, although fans are mixed in their opinion of his Doctor, it cannot be denied that he threw himself into it, creating a Doctor who underwent a substantial amount of character development in two seasons. Unfortunately, the BBC1 Controller, Michael Grade, had never been a fan of the program and, after the troubled season 23, "Trial of a Time Lord," Colin Baker was fired despite having a full series left in his contract. He remains enthusiastic about the series despite that, however, and has lent his talents to numerous fan-made and non-canon productions. Following his time on Doctor Who, he moved primarily into theater, but still does occasional film and television work as well.
7. Sylvester McCoy
Born Percy James Patrick Kent-Smith in Dunoon, Scotland, in 1943, Sylvester McCoy would become the first non-English actor to play the part. (To date, there has been only one other: Scottish actor David Tennant.) He never knew his father, who died in World War II shortly before he was born, and he was raised in Dublin, Ireland. He tried a variety of careers before joining a comedy/vaudeville act called "The Ken Campbell Roadshow." One of the parts he played was a fictitious stuntman named Sylveste McCoy; confused reviewers thought it was his actual name, and he eventually adopted it as a stage name (adding an "r" to the first name to make it look better). In 1987, Doctor Who came out of a year-long hiatus following the firing of Colin Baker, and Sylvester McCoy was cast as the Seventh Doctor. His performance was evocative of the Second Doctor, and clearly informed by his comedy background, but in his second season became increasingly dark. The series was indefinitely suspended in 1989, ending the bulk of his tenure, although he returned for the American-produced Doctor Who movie in 1996, to film a regeneration scene to transition to Paul McGann. Following his work on Doctor Who, he worked extensively in theater and radio; he was nearly Governor Swann in Pirates of the Caribbean and even more nearly Bilbo Baggins in The Lord of the Rings. He recently traveled to New Zealand to film the part of Radagast the Brown for The Hobbit.
8. Paul McGann
Born in Liverpool in 1959, Paul McGann would ultimately have the shortest tenure as Doctor. He was born to a large family, and all four of the McGann boys went into acting. He played a series of roles on television before landing one of the two title roles in Withnail and I, playing Peter Marwood, the "I" of the title who is never named in the movie itself. He performed in a number of films after that, including American films, and was cast as Richard Sharpe, but a football injury just after filming started meant the number two, Sean Bean, got the part instead, and McGann ultimately walked away with a two million pound insurance settlement to compensate for the lost work and career advancement. In 1996, he was cast in an attempted revival of Doctor Who, filmed largely in Vancouver, British Columbia, and set in San Francisco, which was intended as a "back door pilot." It received very good ratings in the UK, but failed to interest US studios. That was the end of that effort, but Paul McGann went on to pursue a respectable film and television career. Recently, he has recorded audio plays featuring the Eighth Doctor — the BBC does not consider these plays canon — including a "do-over" of "Shada," a story written and partially recorded for the Fourth Doctor's tenure, but that was left uncompleted due to industrial action.
9. Christopher Eccleston
Born in 1964 to a working class family in Manchester, Eccleston pursued an acting career right out of school. He broke into film and television in the early '90s, keeping very busy and receiving multiple awards for his work in television before producer Russell T. Davies cast him as the Ninth Doctor in a newly revived Doctor Who. He played the part in more ordinary dress than his predecessors and with his natural Northern accent. Eccleston was the first Doctor younger than the series itself (by a few months) and the first other than Hartnell to have never seen it prior to being cast. He studied "The Talons of Weng-Chiang" (a Fourth Doctor serial), which was then newly released on DVD, as preparation, and came up with a Doctor who was every bit as egotistical as his predecessors, mischievous and impulsive, but also shadowed with massive grief—sometime in the untransmitted interim, his race had gone to war with the Daleks, and he was now the only Time Lord left. He was also in some way responsible for the fact that the Time Lords were now extinct. He was only contracted for one season, due to uncertainty whether the BBC would even be interested in commissioning a second season; miscommunications with the BBC marred his departure, as he was mistakenly reported to have quit due to issues with the crew when, in fact, it had been planned that way from the outset. He resumed his intense schedule after Doctor Who and, in 2011, earned the International Emmy Best Actor award for his role in Accursed.
10. David Tennant
David McDonald was born in 1971 in West Lothian, Scotland, later taking the name David Tennant as his given name was already in use by another performer. He was a born Whovian, and at the age of three announced his intention to go into acting because of it. A precocious actor, he managed to enter the Royal Scottish Academy of Music and Drama at 16. He performed a variety of roles, including many with the Royal Shakespeare Company, and began breaking into television in the 2000s. In 2005, he appeared in Harry Potter and the Goblet of Fire as Barty Crouch, Jr., and appeased his inner Whovian by appearing in the Big Finish audio productions that sort of worked around the edges of Doctor Who continuity and in the abortive attempt at an online animated series, The Scream of the Shalka. He achieved his lifelong dream in 2005 when he was cast as the Tenth Doctor. He elected not to use his natural Scots accent for the part, affecting an Estuary accent, and played it a more confident and more vengeful Doctor than Eccleston. During his first season, he became the first Doctor to appear with someone from the classic series, when Elizabeth Sladen reprised her role as Sarah Jane Smith in "School Reunion." That prompted a spinoff series called The Sarah Jane Adventures, and he appeared in an episode of Season Three of that show. He also provided voice work for two animated series, The Infinite Quest and Dreamland (the latter of which was referenced twice on The Sarah Jane Adventures). After three full seasons and a series of one-off specials in 2009, Tennant left the series, saying that he had to leave while he still could; any later and he wouldn't be able to bring himself to quit. After leaving the series, he became engaged to actress Georgia Moffett, Peter Davison's daughter. The two have since married and have a daughter together.
11. Matt Smith
Born in 1982 in Northhampton, Matt Smith initially dreamed of becoming a professional football player. ("Soccer" to us Yanks, of course.) A back injury put a stop to that, and his drama instructor at school pushed him into acting. He began to study drama and creative writing. He appeared in a variety of stage and television roles, but it was a complete surprise to most when he was cast as the Eleventh Doctor, the youngest ever to take the part. Producer Steven Moffatt had been going for someone in his mid-40s, but was particularly taken by Smith's oddball demeanor and ability to look very old indeed. His Doctor was played as an absent-minded professor, complete with tweed jacket and bowtie (which he obliviously insists is cool), with a sometimes mercurial disposition. He has also appeared on The Sarah Jane Adventures and has provided voice work for a series of video games. He has been signed for an additional 14 episodes, so will be the Doctor for at least three seasons. Like Eccleston, Smith was largely unfamiliar with the series before accepting the role, but he had an excuse—he was only seven when the series went on hiatus. (And now some of us can start feeling old!)

redpajama_set_name: RedPajamaCommonCrawl


text:
Diary - What a day, what a night!
What a day it turned out to be! As always, I began it on Portland, joining two volunteer watchkeepers at the Coastwatch station overlooking the bill. After a cup of tea, I headed through the constituency, visiting as many polling stations as I could to thank those manning them.
Blog - it's light or darkness today
Today is polling day. Today is the day the country either moves forward under Boris Johnson, or is taken to the darkest of places by Corbyn.
Campaign brief - time for the country to move on
Today is polling day and polling stations are open. Today, the British people can use their vote to break the gridlock and allow the country to move on.
A very wet and windy start to the day, but the weather improved as the day went on. Everyone to the polls tomorrow.
Campaign brief - so much to fight for
The Prime Minister has started the final day of the campaign before polling day delivering milk in Yorkshire. By the end of today, he will be door knocking in Essex and speaking directly to voters. Throughout today, he'll reinforce that:
Out & about on a wet and blustery day.
Campaign brief - let's get Brexit done
Two days before polling day, today, the Prime Minister will be in the West Midlands and the North West where he will outline why it is only the Conservatives who have a plan to get Brexit done.
A beautiful, clear and cold day. Out and about all over the patch, listening and chatting to voters.
Campaign brief - last push for our country's future
In the last three days before polling day, the Prime Minister will be visiting every region in England and Wales. His tour will include visits to North Wales, West Yorkshire, Cheshire, Leicestershire, East Anglia and the South West.
Campaign brief - a stark choice next Thursday
Sunday, 8 December, 2019
With four days to go the Prime Minister has written for the Sunday Telegraph and the Sunday Express reminding people of the choice they face in Thursday's election.


text:
Andreas Hofer
Why Smart People Hurt
Why Smart People Hurt: A Guide for the Bright, the Sensitive, and the Creative (2013) by Eric Maisel has very controversial ratings on Amazon. There are a lot of people raving about it and giving it five-star ratings, as well as a lot of people regretting having bought the book, many of whom didn't even bother to finish it. I can relate to both sides: the book certainly wasn't a fun read and I often struggled to go on as I felt there was a lot of meaningless babbling about meaning. However, there are a lot of passages I can totally relate to (not that I consider myself that smart).
Imagine the following sort of day in the life of a young girl with lots of native intelligence. Her household is in chaos, the kind of chaos that poverty, acrimony, addictions, and unfulfilled lives produce. Surrounded by threats, impulsivity, and zero tolerance for free thought, she somehow manages to get to school-and into another anti-thinking environment. At school, more chaos prevails and, despite the idea that school advocates for thinking, she is confronted with a shrink-wrapped, fact-based, topic-based, and test-driven curriculum that no adult with the freedom to leave would tolerate for an instant. After school, she goes off to parochial instruction and gets a narrow religious education that demands obedience, allegiance, and more thoughtlessness. Her evening involves her in more chaos, and to escape it she shuts her door, if she is lucky enough to have a room and a door, and finds some stress relief and some self-soothing by watching hours of ready-made, low-level television programming that further numbs her and dumbs her down. Finally she sleeps, only to awaken to another day just like this one .
Of course, most people have struggles in their lives, however, smart people tend to have their struggles with a particular set of challenges:
anti-intellectualism
authority and obedience
social struggles and finding our "tribe"
mental health problems (depression, anxiety, bipolar)
self-worth (switching between elevated and self-doubt)
mind-numbing routine work and boredom
making their lives (more) meaningful
While I think most smart people will be able to identify The author tries to provide recipes to cope with these struggles, like "thinking only useful thoughts". I am not sure how helpful that is to other people, but I find it pretty useless advice. First of all, I am hardly able to tell my brain to come up with useful thoughts. Secondly, sometimes the least useful thoughts are the most interesting ones (how useful was relativity to Einstein???) and thirdly if useful means being able to live a more "normal life" or "financially successful" then it smacks very much of "conformist thinking", something smart people are loath to.
Rather than getting some useless advice, I would have liked Eric Maisel to reflect on why smart people have this particular set of problems. It's certainly something that has more got to do with society than the individual. Why is it that high performing sports teens are treated like heroes and are highly popular whereas high performing cerebral teens are treated like freaks and often have next to no friends? Why is it that smart people need "more meaning" in their lives than other people? Why do they struggle with routine work, authority and conformity?
I have argued that these questions may be resolved by evolutionary thinking. Even though reality may not always be so clear-cut we can think of Maisel's "smart people" (a category that does by far not include all smart people) as evolutionary forager types living in a farmer world.
We tend to think of evolutionary changes as always leading to higher complexity. It's something that is simply not true. The transition from foraging to farming brought along less cerebral and more physical work. Hunter-gatherers have an encyclopaedic knowledge of flora and fauna and tracking involves such complex cognitive processes that Louis Liebenberg described it as the origin of science. Among our three ancestral subsistence strategies foraging, farming and herding, you can think of foragers as the "nerds" among them.
If you think of Maisel's "smart people" as evolutionary forager types in a farmer-herder world, it should become clear why they have the specific set of problems and why not all smart people have them. Foragers in a farmer society may have more problems with addictions like alcohol (or more modern forms like online and gaming addictions), mental health problems, such as ADHD, OCD, ODD and anxiety, and difficulty fitting in in general. These are all problems Native Americans face to a much higher extent than any other group in America. So, do many of the other remaining foragers all over the world.
On the flip side, you get lots of people who might even consider themselves not particularly intelligent because that is what they have been told in a school system that doesn't work well for forager types (authority, conformity, rote learning). This is one thing that Maisel actually points out in his book. Rather than "smart people" Maisel's book should have been about "alienated" people. People who don't find much meaning in acquiring material wealth and "keeping up with the neighbours". People who should think society is flawed, but only too often end up thinking it's them who are flawed, instead. There are many labels I have found for such people, like neurodiverse or indigo. Calling them the "hunter-gatherer neurotribe" makes the most sense to me.
For more on the hunter-gatherer neurotribe hypothesis check out my The hunter-gatherer neurotribe: gifted, geeks, aspies and other aliens in this world
And Understanding Genius: Sensibility, Social Awkwardness, Sleeplessness, Sex, Suicide and Sense of Humour
Originally published at http://the-big-ger-picture.blogspot.com on June 7, 2022.
More from Andreas Hofer
David Grace
David Grace Columns Organized By Topic
The Best Way To Stop Bad Conduct Is To Take Away The Rewards For It
Faizan Kazani
What is Neuromusicology?
💎MaVanessaCuencaDelich
Mind Of A Killer
RedFate
Laws 31–40 | Key Lessons
Astrid Scholtens
Uncovering the Secrets of Successful HUMINT Operations
Lily Yuan
Personality Psychology
Guide: 4 Main Types of Psychometric Data
Cosmic Garden
A Slightly Freaky Breakdown Of REM Sleep
dragnerz
Some Thoughts on Randomness, Determinism, and Dan Barker's "Free Will Explained"


During today's visit to my favorite French Wine cafe in Prague Na břehu Rhôny or Pas a pas I finally took few pictures. You would find this lovely place in the street Na Hrobci in Prague 2 near tram stop Vyton – currently you can take trams 17, 3 and 7. The owners buy wines from the South of France – especially from the area of Provence and they frequently organize wine tasting sessions.
The wine is delicious and you can get reasonably priced bottles as well as very expensive wine selection. If you feel like going there for breakfast you might try cappuccino or late with a large crispy croissant – chocolate, almond or just plain. The almond one is especially very nice. The waiter or waitress can also fix you a lovely baguette but that might take some time.


To work out the kinks with the author interview questions I am going to do myself first. So you can start fangirling before the first Shelly Ferguson book comes out this Spring.
What book do you recommend to everyone – even people who didn't ask: Agnes and the Hitman by Jennifer Cruise. If I ever find a pair of red glasses like hers I will so rock them.
Which of your quirks have you written into a character? Scarlett is a school teacher so I gave her my love of PG cusswords. Her favorite is Oh my giddy aunt or Heaven help me.
Have you written some one who was rude into a book so you could get eternal payback? Not yet but I have a list!
Which book boyfriend would you run away with? Curran from the Kate Daniels series by Ilona Andrews. Oh my giddy aunt. That is one fine man! The best part is when he decides he wants Kate he gets her. He literally rips apart anyone who tries to take her from him.
Have you ever taken a book-cation? I visited the Sherlock room in the Toronto library. Hidden on the top floor behind a gatekeeper and a glass wall is a room set up like an old English library. It has all the books by Sir Arthur Conan Doyle, other mysteries, and books of the time period. It was amazing.
What fandom do you geek out over? Doctor Who. 10 has always been my favorite – 13 is giving him a run for the money though. How big of a fangirl am I? I went to the Pensacola Comic-Con so I could meet 3 of the actors – John Barrowman (Capt. Jack), Arthur Darvill (Rory Williams), and Camille Coduri (Jackie Tyler).
Does anyone in your family read your books? If they do can they look you in the eye at Thanksgiving? So far just my mother-in-law. Mine are sweet though so I am not to worried.
How many more stories are rattling around your brain trying to get out? There are about a dozen couples fighting to get their stories out right not. I have the couples and the general idea what the problem they have to get past for their happily ever after. I just have to get the stories on paper.
What did you have the most fun with in your last novel? I needed a sidekick to put a little humor in the story. In came Fara. How could you not have fun with a little gremlin walking around in American Girl Doll Clothes with a pink mohawk? Especially when she likes to stab first and not bother with questions?
One crazy week will change Scarlett's life forever.
For the last ten years Scarlett thought her fated mate was dead … until he hit her with a buggy in the shampoo aisle of the grocery store three days after moving to a small town in the Georgia mountains.
That little bump led to a girlfriend roadtrip, a hot billionaire, a coworker who was a little to interested in her, and a gremlin sidekick with fashion flair.
When the Crazy Cat Book Club gets involved (takes over) they help her snag her man and plan the southern wedding of her dreams before she knew what happened.
You will love this book because who wouldn't love a woman finding the man of her dreams in a small town full of shifters, witches, and all of the things that go bump in the night.


text:
Poll: Half of California voters say state is…
Poll: Half of California voters say state is headed in the right direction
By Jessica Calefati |
PUBLISHED: February 25, 2015 at 12:00 am | UPDATED: September 8, 2018 at 12:00 am
SACRAMENTO — For the first time since 2000, when the dot-com bubble burst and Californians" view of the state soured, at least half of voters say they think it is headed in the right direction, a new Field Poll shows.
Between September and February, the share of people who endorse the state"s direction increased seven percentage points, to 50 percent, the poll found.
Even so, Democrats and Republicans remain deeply divided on how to grade the Golden State, with 73 percent of Democrats ready to give it an A and 75 percent of Republicans convinced the state is headed down the tubes.
"There"s been an absence of crisis emerging from Sacramento, and to an ordinary citizen, no news about state government is good news about state government," said Jack Pitney, a political expert at Claremont McKenna College.
Republicans are still stewing over Prop. 30, which temporarily increased the sales tax and raised taxes on top earners, and Gov. Jerry Brown"s push for costly projects like the bullet train, he said.
But the GOP"s displeasure is mostly griping, Pitney said. People from both parties can appreciate California"s job growth and on-time budgets.
One Democrat who was polled said she thinks the state is doing well because it is addressing her main concerns: education, infrastructure and water.
Oakland resident Jenny Veggersby said she"s especially worried about the drought but thinks Brown is tackling the bone-dry weather head-on.
"My sense is that Gov. Brown is looking to the future and trying to do something about water in this state," said Veggersby, a 66-year-old retired county mental health worker.
Even Assemblywoman Catherine Baker, the Bay Area"s only Republican state lawmaker, concedes that the state is in a much better position now than it was several years ago at the height of the Great Recession. Unlike Veggersby, however, she worries about infrastructure spending.
Brown"s insistence on building a $68 billion bullet train and spending $23 billion on twin tunnels beneath the Delta to siphon water south threatens that newfound stability, she said.
"I"m concerned that the governor is pushing projects the state can"t afford," Baker said.
Baker said the Contra Costans she meets want more state money spent on K-12 schools, and the poll indicates voters of different races, religions and parties statewide feel the same.
Asked how the state should spend extra tax revenue it picks up this fiscal year, 52 percent of registered voters favored K-12 public schools. Roughly a third said higher education should get some money.
The survey also found voters" support for the Legislature has improved since September when only 34 percent of people polled approved of the job the Legislature was doing. Now, 42 percent approve.
The survey was done between Jan. 26 and Feb. 16, with 1,241 registered voters. The margin of error is plus or minus 3.2 percentage points.
Contact Jessica Calefati at 916-441-2101.
Jessica Calefati


text:
Basewell®
INTRINSA™
Opinion & Memberships
Asia Pacific APAC
Europe, Middle-East and Africa
Challenges surrounding confirmation of EDVs
The UPOV Convention (International Union for the Protection of New Varieties of Plants) was changed in 1991 to extend the protection of Plant Breeders' Rights to 'essentially derived varieties', or EDVs. The aim was to ensure that the breeder receives proper compensation for the commercialisation of mutations (sports) of the protected initial variety. However, there is much confusion about how to identify an EDV. How much genetic similarity should exist between the derived and initial variety, considering the varying degree of relatedness between different crops?
'As Dümmen Orange we are fully committed to the EDV regulation. However, it would be useful if a method could be developed to prove unquestionably whether a variety is an EDV or not,' declares Biense Visser, CEO of Dümmen Orange. Dümmen Orange wants to work with other leading players in the ornamental horticulture sector to find a scientifically supported method for determining EDVs.
EDV and genetic relatedness
DNA analysis is an important tool for proving genetic relatedness. The International Community of Breeders of Asexually Reproduced Ornamental and Fruit Varieties (CIOPORA) states that the methods and required level of genetic conformity should be established for on a crop-by-crop basis using modern protocols.
'The Dutch association of plant breeders (Plantum) is basically in favour of crop-specific protocols, but thinks it is only realistic for the bigger ornamental species where the companies are willing to finance such work to obtain greater clarity. Plantum believes an EDV exists when the presumed derived variety looks more like the original variety than would normally be expected, based on variation within that species. Genetic similarity is important, but it is also about the phenotype, i.e. observable characteristics,' explains Judith de Roos, legal expert at Plantum.
UPOV intends to change the rules
The UPOV recently drafted new Explanatory Notes (EXN) on EDVs in an attempt to provide more clarity about the concept, which led to a discussion about the definition of 'essential characteristics'. Hans van den Heuvel, R&D Managing Director at Dümmen Orange: 'The criteria that an EDV is mainly derived from and conforms to the initial variety while retaining the expression of the essential characteristics, seem to have been abandoned and replaced by the presence or absence of at least one different essential characteristic. The presence of single different essential characteristic would then be sufficient for the plant not to be declared an EDV. However, it is still unclear what defines an essential characteristic in our industry, which is problematic for all concerned. Dümmen Orange believes that varieties which occur through spontaneous mutations without technical or human intervention should be treated according to current EDV guidelines, which does not seem to be in line with UPOV's latest position. However, UPOV's proposed changes would be easier to understand for varieties that came about through technical intervention.
Opinion Plantum & CIOPORA
CIOPORA does not support the new UPOV Explanatory Notes (EXN) on EDV. 'We are trying to convince UPOV to reconsider the EXN and not to restrict the EDV concept. We preserve that mutants (sports) and GMOs (genetically modified organisms) – as far as they are distinct from the initial variety – are EDVs, provided that they are genetically very similar to the initial variety, because mutants and GMOs by definition are largely derived from the initial variety. Many mutants exist in the ornamental and fruit world, so it important to have an effective EDV system with a broad scope.' Edgar Krieger, Secretary General of CIOPORA, explains.
The UPOV's explanatory notes have not succeeded in clarifying EDVs, states De Roos of Plantum. 'Like CIOPORA, we don't support the proposed changes. The EXN is written in such a vague manner that it can be interpreted in both ways. The matter will be further discussed at the upcoming UPOV meeting in October. According to Plantum (as well as ISF and ESA), the concept of EDV is based on the principle that it is fair to compensate the breeder of the initial variety for commercial exploitation of a derived variety that largely holds its essential characteristics. Up to now we haven't looked at the process – technical intervention of spontaneous mutation – in itself but at the end result (differences). But we will need to see how the UPOV member countries proceed with this discussion', concluded De Roos.
Room 505, No. 133 Shengang Avenue, Pudong New Area 201306,
info.cn@dummenorange.com
Subscribe for our press releases
Copyright © 2021 Disclaimer | Privacy
For more information about the cookies, or the update your preferences, please see: Privacy and Cookie statement
These cookies are necessary for the website to function.
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site.
These cookies allow us to prevent the abuse of forms on our website.
Save cookie preferences
Cookies: We use cookies to ensure you get the best browsing experience on our website. More information


text:
Veterans Day 2019: Where to find free food, deals and events in Knoxville
Maggie Jones | Knoxville News Sentinel
KARM honors service members with Veterans Day banquet
Knox Area Rescue Ministries opened its doors to nearly 60 veterans on Saturday to honor them and their service as part of its annual Veterans Day service.
Veterans Day arrives on Monday, and Knoxville businesses and events will commemorate the holiday in different ways.
Veterans can receive discounts or freebies at certain businesses, and events will take place to honor those who have served in the United States Armed Forces.
Here's a list of freebies, deals and events for veterans in the Knoxville area.
Area food freebies, deals for veterans
Applebee's will offer free meals to veterans and active military members on Monday at participating locations. Those interested must show proof of service. Offer is dine-in only. Knoxville Applebee's locations include 5316 Central Avenue Pike, 2912 Knoxville Center Drive, 261 North Peters Road and others. Info: applebees.com.
Bonefish Grill, 6610 Kingston Pike and 11395 Parkside Drive, will provide a complementary appetizer to veterans on Monday. The deal includes appetizers Mussels Josephine, Calamari, Imperial Dip and the famous Bang Bang Shrimp®. Active and retired military service members must present a valid military ID to get the deal. Info: bonefishgrill.com.
Bubba Gump Shrimp Co., 900 Parkway, Gatlinburg, will offer veterans 20% off their meals on Monday. The offer is valid for up to parties of four and on retail purchases at the restaurant. Active and retired military must show a valid military ID. Info: bubbagump.com.
Carrabba's Italian Grill, 324 N. Peter's Road, will offer a free order of Calamari with any purchase to all veterans, active military, police officers and firefighters with a valid military ID. The deal will be available from Nov. 8-11. Carrabba's locations will also provide 10% off to all active and retired service members Info: carrabbas.com.
Cracker Barrel will offer veterans and active duty military a free Pumpkin Pie Latte (available iced or hot) or a slice of our Double Chocolate Fudge Coca-Cola® Cake on Monday. Knoxville locations include 5001 Central Ave. Pike, 2920 South Mall Road, 1510 Cracker Barrel Lane, 9214 Park West Blvd. and others. Info: crackerbarrel.com.
Golden Corral will offer veterans and active duty military a free meal from 5-9 p.m. Monday. This offer is dine-in only. Area Golden Corral locations include 6612 Clinton Highway, 513 Winfield Dunn Parkway, Sevierville, and others. Info: goldencorral.com.
Logan's Roadhouse, 1560 Parkway, Sevierville, will offer a free meal to veterans and active military from 3-6 p.m. Monday. This offer is dine-in only. Info: logansroadhouse.com.
Menchie's Frozen Yogurt will offer veterans and active duty military a free frozen yogurt that's up to six ounces at participating locations on Monday. Those interested must present a valid military ID. Knoxville's Menchie's location is at 11162 Parkside Drive. Info: menchies.com.
Red Robin will offer veterans and active duty military a free Tavern Double® Burger and Bottomless Steak Fries at participating locations on Monday. Those interested must present a valid military ID. This offer is dine-in only. Knoxville's Red Robin is at 11433 Parkside Drive. Info: redrobin.com.
Shoney's Restaurants will offer veterans a free breakfast bar from 6-11 a.m. on Monday. This offer is dine-in only, and no other purchase is necessary. Knoxville area Shoney's locations include 4020 Rutledge Pike, 4032 N Broadway Street, 315 E. Emory Road and others. Info: shoneysknox.com.
Wild Wing Cafe,11335 Campbell Lakes Drive, Farragut, and 119 East Emory Road, Powell, will offer veterans and active military a free burger, chicken sandwich, wrap or salad on Monday. Info: 865-777-9464 (Farragut) or 865-339-4444 (Powell).
Other freebies for veterans
Dental Care of Powell, 574 E. Emory Road, will provide free dental services to veterans in Powell and surrounding areas on Nov. 16 as a part of Veteran's Free Dentistry Day. The date is dedicated to giving free dental care to veterans without dental insurance. Cleanings, fillings and extractions will be provided per patient from 8 a.m.-3 p.m. Patients will be accepted on a first-come, first-serve basis, and veterans are asked to bring a valid military ID. Info: 865-859-9553.
The East Tennessee History Center, 601 S. Gay St. is offering free admission to veterans and their families from 9 a.m.-4 p.m. Monday. The museum features exhibits including "Voices of the Land: The People of East Tennessee," "It'll Tickle Yore Innards!": A (Hillbilly) History of Mountain Dew" and others. Info: 865-215-8824 or EastTNHistory.org.
SafeShine Car Wash, 10561 Hardin Valley Road, will provide free car washes to veterans and active military from 8 a.m.-8 p.m. Monday. SafeShine is partnering with Grace for Vets for the event. Info: safeshinecarwash.com, 865-281-3931.
Simoniz Car Wash: 405 Merchants Drive and 7914 Kingston Pike, will provide free car washes to veterans and active military personnel from 8 a.m.-6 p.m. Monday. Those interested must offer proof of service through military ID, veterans organization membership card or vehicle plates that display POW, veterans or military metal/ribbon status. Info: 865-687-8432 (Merchants), 865-693-0020 (Kingston Pike).
Veterans Day events in Knoxville area
Anderson County Veterans Day Parade: The parade will go from 6-9 p.m. Monday in downtown Clinton on Market and Main streets, according to the event's Facebook page. Info: Leon Jaquet, 865-463-6803.
Gatlinburg's Veterans Day Ceremony: The event will take place at 11 a.m. Monday at the Ripley's Aquarium of the Smokies plaza. It will feature keynote speaker Lieutenant Colonel Steven Turner, Executive Officer of the 278th Armored Calvary Regiment, Gatlinburg-Pittman High School Band, Boy Scout Troop 111, Girl Scout Troop 20648, the American Legion Post 202 Honor Guard and others. Info: gatlinburg.com.
Knoxville Veteran's Day Parade: The parade will step off at 10:45 a.m. Monday at the Knoxville Civic Auditorium. Then, it will go west on Church Avenue, turn north on Gay Street and end at Magnolia Avenue, according to a City of Knoxville press release. Retired U.S. Army Capt. Mark Brogan will be the parade's Grand Marshal. It will also feature more than 114 parade units. At 11 a.m., the parade will stop near Mast General Store for a ceremony for attending veterans. Gay Street between Church and Magnolia avenues will close to on-street parking at 8 a.m. and to traffic at 10:40 a.m. Market Street and Union Avenue will close to parking and traffic at 9 a.m. Streets will reopen around 12:30 p.m.
Sevier County High School's "Smoky Salutes ... No Greater Love" Veterans Day Program: It will take place at 6:30 p.m. Thursday and 10 a.m. and 12:30 p.m. Friday at the school, 1200 Dolly Parton Parkway, Pigeon Forge. Veterans and the public are invited to the free program. Info: 865-453-5525.
Vet to Vet Tennessee event featuring Knox County Mayor Glenn Jacobs in the play "Worth It:" 6:30 p.m. Sunday, Nov. 10, at Faith Promise North Campus, 5830 Haynes-Sterchi Road. The play will teach suicide prevention awareness. Admission is free. Donations will be accepted for Vet to Vet Tennessee, which provides support to veterans who are recovering from mental health issues and substance abuse, according to a press release.
MAKE YOUR DOWN TIME YOUR BEST TIME
Subscribe and unlock full digital access to knoxnews.com | Find print delivery deals here.
Did you like this story? Follow Go Knoxville for more
Facebook | The 'Sneak Peek' Newsletter | GoKnoxville.com
Share your feedback to help improve our site!


text:
Banker's Association names new CEO
by Krizzel Canlas18 Dec 2018
Your office matters
7 ways to create a happier, more productive workplace
The Anti-Sales Method
How mortgage advisers can get clients without selling to them
The New Zealand Bankers' Association (NZBA) has appointed Roger Beaumont as its new chief executive.
Beaumont brings a wealth of experience in both the banking sector and in stakeholder management, his most recent role being the executive general manager of ASB Bank. He is currently Westpac Group's head of corporate affairs.
"Roger has an extensive career at an executive level in banking, corporate affairs and media, and is extremely well respected within both the public and private sectors of New Zealand, which is important in a role such as this," NZBA chair and Westpac CEO David McLean said.
McLean also commended the job done by current acting chief executive Antony Buick-Constable, who has recently worked through a number of serious issues and reviews.
"The banking industry as a whole is facing many ongoing challenges around competition, regulation and cybersecurity," McLean added. "The Bankers' Association has an extremely important job to do to ensure all stakeholders, and customers have confidence in the work we undertake."
Beaumont takes up his new role January 21, 2019.
DR scheme appoints new council member
Government unveils FMA board appointments


text:
Home › News and views › Blog › Driving a digital agenda
Driving a digital agenda
BLOG | UK | TRANSPORT COMMUNICATIONS | BAI COMMUNICATIONS
By Malcolm Keys Director, BAI Communications UK
London is the tech capital of Europe, and it offers a compelling mix of innovation, a thriving creative sector and a vibrant financial system. It is why when BAI Communications was looking to establish its European headquarters in 2016, London was our first choice.
Our decision to set up BAI's London office was vindicated when shortly afterwards, newly elected Mayor of London Sadiq Khan announced he would be appointing the capital's first Chief Digital Officer (CDO), to oversee digital development across London.
At London First's recent roundtable with CDO Theo Blackwell, I was reminded about why we are so enthusiastic about London, but also some of the challenges it faces to become a truly digital city.
Since being appointed, Theo has been a welcome champion for tech, setting out how it can improve the lives of Londoners and outcomes for policy makers. We have already seen positive developments such as the establishment of the London Office for Technology and Innovation (LOTI), the creation of the 'Not-Spot Team' tackling poor connectivity, and investment days to help innovative tech firms attract crucial funding.
The private sector has a huge role to play in helping the Greater London Authority (GLA) to achieve its aim of making London the smartest city in the world. The Smarter London Together Roadmap included some ambitious plans, including the development of 'smarter' lampposts throughout London, enhancing digital inclusion across the whole of London and improving data rights.
However, the recent roundtable was a reminder of the scale of task if London is to keep step with a fast moving tech marketplace. To this end, Theo explained his three priorities are:
To 'fix the plumbing', ensuring London has the necessary infrastructure in place
To enable better data collection and services across London
To improve the GLA's role in consolidating supply/demand and sharing information across the boroughs.
Drawing on our perspective as a global communications infrastructure provider, we see 'the plumbing' as a vital component to success. This includes both fixed line and wireless connectivity. However, progress is not always as fast as we would all like. Perhaps the best example Theo provides is that just five percent of homes in London are able to access full fibre services.
One of the key reasons for this is the historic urban environment of London. Like the other cities we have worked in, such as New York, it requires carefully thought out solutions which respect the built environment and communities, but also deliver the world class connectivity we need. It is encouraging to see Theo and the GLA seeking to tackle this through revised planning powers and supporting both boroughs and providers to improve the speed of implementation.
From our perspective as a global business, we are very pleased with the manner in which the GLA is driving the digital agenda, however such changes cannot be achieved by the public sector alone and it is vital that the private sector matches such enthusiasm and continues to work with Theo through organisations like London First to help make London the digital capital of the world.
This post originally appeared on the London First website.