text:
\section{Introduction}
Optimization methods have natural connections to sampling methods  \cite{Dalalyan2017FurtherAS} . Consider a potential function $V: \mathbb{R}^d \to \mathbb{R}$ that is twice differentiable and has only one simple global minimum ${\bm x}^\ast = \arg \min_{x \in \mathbb{R}^d} V({\bm x})$ for simplicity.
The first order Langevin Dynamics is defined by the following Stochastic Differential Equation (SDE):
\begin{equation}\label{langevin-dynamics}
    d{\bm x}_t = -\nabla V({\bm x}_t) + \sigma d\mathcal{B}_t,
\end{equation}
where $\sigma>0$ is the diffusion parameter proportional to the square of the temperature parameter, and $\{\mathcal{B}_t \}_{t\ge 0}$ is the standard Brownian motion in $\mathbb{R}^d$. Under certain assumptions on the drift coefficient $\nabla V$, it was shown that the distribution of ${\bm x}_t$ in Eq.\ref{langevin-dynamics} converges exponentially to its stationary distribution, the Gibbs measure $\nu_\sigma({\bm x}) \propto \exp( -2 V({\bm x}) /\sigma^2)$. With $\sigma \to 0$, $\nu_\sigma$ concentrates on the global minimum $x^\ast$, leading to various Langevin dynamics based algorithms like Gradient Langevin Dynamics (GLD) \cite{Xu2017GlobalCO,Dalalyan2014TheoreticalGF}, Stochastic Gradient Langevin Dynamics (SGLD) \cite{Welling2011BayesianLV} etc.

While Langevin dynamics based algorithms handle optimization tasks with slightly nonconvex potential function $V$ effectively  \cite{Raginsky2017NonconvexLV, Zhang2017AHT}, it encounters difficulties with global optimization for highly nonconvex $V$, as it requires $\sigma_n \to 0$ to guarantee convergence. However, a small $\sigma_n$ makes the probability of jumping across a certain potential barrier drastically diminish. Conventional Markov Chain Monte Carlo (MCMC) methods were proposed to alleviate such problems, such as
parallel tempering \cite{Swendsen1986ReplicaMC,Geyer1991MarkovCM}, flat histogram algorithms \cite{Berg1991MulticanonicalAF} and simulated annealing \cite{Kirkpatrick1983OptimizationBS}. Among them, Simulated Annealing (SA) provides a reasonable way to handle this problem by maintaining a population of particles to sequentially approximate $\nu_{\tilde \sigma_n}({\bm x})$  with $\tilde \sigma_n \to 0$. However, it requires numerious evaluations of $\nabla V$, which can be computationally costly, making it less attractive for optimization tasks. Therefore in this paper, we aim at accelerating Langevin dynamics based algorithms by modifying the underlying continuous Langevin dynamics.

Recently, an Ensemble Kalman Sampler (EKS) \cite{GarbunoIigo2019InteractingLD,Schillings2016AnalysisOT} method was suggested to accelerate the convergence rate toward the equilibrium $\nu_\sigma({\bm x})$. This method approximates a preconditioning-modified Langevin dynamic that shares the same equilibrium with the original one, but enjoys a faster convergence rate for ill-posed problems. Besides, this method has another benefit when the potential function $V({\bm x}) = \Vert
\mathcal{G}({\bm x}) - y\Vert^2$ is a least square functional \cite{Engl1996RegularizationOI}, i.e. the gradient of $V$, or the Jacobean matrix of $\mathcal{G}$ does not need to be calculated explicitly, as those terms involving them can be approximated by the $0^{th}$ order information in current steps, see Eq. ~\eqref{ls-langevin-diffusion-der-free}.

However, the EKS method has little improvement when solving highly non-convex optimizations. Besides,  as our purposes is to find the global minimum of $V$, restricting the stationary distribution of the underlying dynamic to be precisely $\nu_\sigma({\bm x})$ is not necessary. Suppose we have another  $\sigma$-dependent dynamic with limit distribution approximate $\delta_{{\bm x}^\ast}({\bm x})$
, the Delta distribution at the minimum point $\bm x\ast$, then this dynamic can also be chosen to approximate the global minimum. This inspires us to further modify the Langevin dynamics for faster convergence. Specifically, we modify the Fokker-Planck Equation related to Langevin Dynamics by adding a linear source term, which can be proven, by the spectral approach \cite{Pankov2001IntroductionTS}, to improve the convergence rate and the mass concentration near the global minimum of the invariant measure. The new process belongs to a type of nonlinear operators called Feynman-Kac Semigroup which was developed in Large Deviation Theory for calculating generating functions \cite{Varadhan2010LARGED} and also used in important practical applications such as the Diffusion Monte Carlo (DMC) method \cite{Foulkes2001QuantumMC}.
To design a practical algorithm, we use the Interacting Particle methods \cite{Moral2000BranchingAI,Moral2013MeanFS}, introducing the reweighting and resampling technique to simulate the effect of this source term and get the so-called Reweighted Interacting Langevin Diffusion (RILD) algorithm. This is, as far as we know, the first time using Interacting Particle methods to solve optimization tasks. Many numerical experiments are tested to show the effect in accelerating convergence and flatten the potential barriers.

The main contributions are summarized as follows:
\begin{itemize}
    \item We provide a simple but effective way to modify the Langevin Dynamics in Section \ref{sec:continuous}  for faster convergence and flatter potential barriers.
    \item We provide a feasible discretization way to design algorithms in Section \ref{sec:discrete}, and compare the new algorithm Alg. \ref{alg:RILD}  with several existing methods, showing its advantages.
    \item We provide theoretical results in Section \ref{theoreticalproperty} based on spectral analysis to guarantee the advantages.
\end{itemize}

\section{Preliminary}
\subsection{Overdamped Langevin Dynamics}
In this section, we introduce the classical Langevin Dynamics which is related to GLD and SGLD algorithms.

Suppose $V: \mathbb{R}^d \to \mathbb{R}$ that is twice differentiable and has a simple global minimum ${\bm x}^\ast = \arg \min_{x \in \mathbb{R}^d} V({\bm x})$ for simplicity.

The overdamped Langevin Dynamics is defined as in \ref{langevin-dynamics}.
Such a dynamic has plenty of relationships with the Fokker-Planck equation. Denote \begin{equation}\mathcal{L}\cdot := -\langle \nabla V , \nabla \cdot \rangle + \frac{\sigma^2}{2} \Delta \cdot\end{equation} the infinitesimal generator \cite{ksendal1987StochasticDE} of the Markov process $({\bm x}_t)_{t\ge 0}$, and its $L^2$ adjoint operator:
\begin{equation} \mathcal{L}^\dagger \cdot := \text{div}( \nabla V \cdot + \frac{\sigma^2}{2} \nabla \cdot ) \end{equation}
Let us assume the law of  ${\bm x}_t$ at time $t$ has a density $p_t( x)$ for Lebesgure measure. Then $p_t({\bm x})$ satisfies the Fokker-Planck equation:
\begin{equation}\frac{\partial}{\partial t}p_t({\bm x}) =\mathcal{L}^\dagger p_t({\bm x}) = \text{div} \big( p_t({\bm x}) \nabla V({\bm x})  + \frac{\sigma^2}{2} \nabla p_t({\bm x}) \big) \label{Langevin-Diffusion}\end{equation}
We may denote $p_t({\bm x}) = e^{\mathcal{L}^\dagger t} p_0({\bm x})$ where we denote $p_0({\bm x}) $ the initial distribution of overdamped Langevin dynamics, and it is well-known that the Markov operator $e^{\mathcal{L}^\dagger t}$ admits a unique invariant probability measure $\nu({\bm x}) = Z_\nu^{-1}e^{-2\sigma^{-2}V({\bm x})}, Z_\nu = \int_{\mathbb{R}^d}e^{-2\sigma^{-2}V({\bm x})} dx$. The rate converging to $\nu({\bm x})$ has been wildly studied,

\begin{prop}[Proposition 2.3 in \citet{Lelivre2016PartialDE}]
    Under specific condition (Assumption \ref{Asmp:VW}),  \footnote{ We define $L^2(\mu):= \{f: \Vert f \Vert_{L^2(\mu)} < \infty\}$,  where $\langle f,g \rangle_{L^2(\mu)} := \int_{\mathbb{R}^d} f(x)g(x) \mu(x) dx$,  $\Vert f \Vert_{L^2(\mu)} = \langle f,f \rangle_{L^2(\mu)}^{\frac{1}{2}}$}for all $p_0$ such that $p_0 / \nu \in L^2(\nu)$, and for all $t \ge 0$,
    $$\Vert e^{\mathcal{L}^\dagger t}p_0/ \nu - 1\Vert_{L^2(\nu)} \le \Vert p_0 / \nu - 1\Vert_{L^2(\nu)} e^{-\delta t},$$
    where $\delta$ is the first non-zero eigenvalue $-\lambda_1$ of the operator $-\mathcal{L}$ on $L^2(\nu)$.
\end{prop}
Note that $\lambda_1$ is real because $\mathcal{L}$ is self-adjoint\footnote{We say a linear operator $A$ is self-adjoint on a Hilbert space $\{ \mathcal{H}, \langle \cdot \rangle_\mathcal{H} \}$, if for any $f, g \in \mathcal{H}, \langle f, Ag \rangle_\mathcal{H} = \langle Af, g \rangle_\mathcal{H}$} on $L^2(\nu)$.

When for sampling purposes, individual samples are sampled from independent paths by Monte Carlo Markov Chain (MCMC, \cite{Berg2004MarkovCM}) corresponding to the discretized Langevin dynamics. This method is somehow slow for large dimensional, non-convex problems. Such a weakness make it less attractive for optimization purposes. Fortunately, we can explore the connection between individual samples, as they did in \citet{GarbunoIigo2019InteractingLD}. We will introduce their work in the next subsection.

\subsection{Interacting Langevin Diffusion}
In \citet{GarbunoIigo2019InteractingLD}, the authors introduced a modified Langevin dynamics and analyzed its property, proving its superiority in designing sampling algorithms. We now introduce the main result of their work.

The convergence rate of classical Langevin dynamic based algorithm can be really slow when $V$ varies highly anisotropic, . A common approach for canceling out this effect is to introduce a $d\times d$ preconditioning positive semi-definite matrix $C$ in the corresponding gradient scheme,
\begin{equation} d {\bm x}_t = -C\nabla V({\bm x}_t)dt + \sigma\sqrt{C}d\mathcal{B}_t.\end{equation}
Here $\sqrt{C}$ can be any $d\times r$ real matrix $U$ such that $U U^T = C$ (Note that in this case, $W_t$ can be reduced to $r$-dimensional standard Brownian motion as the essential rank of $C$ is no larger than r).

The corresponding Fokker-Planck equation now becomes
\begin{equation}\frac{\partial}{\partial t}p_t({\bm x}) = \text{div} \big(p_t({\bm x})  C\nabla V({\bm x}) +  \frac{\sigma^2}{2} C \nabla p_t({\bm x}) \big) \label{Preconditioned-Langevin}\end{equation}
and the infinitesimal generator
\begin{align}\mathcal{L}_C\cdot &:= -\langle C\nabla V, \nabla \cdot \rangle + \frac{\sigma^2}{2} \text{div}(C\nabla \cdot),\\
\mathcal{L}_C^\dagger\cdot &:= \text{div} \big( C\nabla V \cdot  +  \frac{\sigma^2}{2} C \nabla\cdot \big).\end{align}
One can easily verify that $e^{-2\sigma^{-2}V({\bm x})}$ is the invariant measure to the above system for all semi-definite $C$, and the only one positive invariant measure if $C$ is strictly positive definite.

To find a suitable $C$ is of general interest. One of the best choices is to let $C = \text{Hess} V$, as a counterpart of Newton's scheme in optimization, which is unfriendly for computation. One intrinsic benefit of choosing $\text{Hess}V$ is its affine-invariant property when designing numerical schemes.

Such a property can also be preserved if we take $C = \mathcal{C}(p)$, the covariance matrix under the probability measure $p({\bm x})$, \begin{equation} \mathcal{C}(p):= \int_{\mathbb{R}^d} \big({\bm x}- m(p)\big) \otimes \big({\bm x}- m(p)\big) p({\bm x}) d{\bm x},\end{equation} \begin{equation}m(p):= \int_{\mathbb{R}^d} {\bm x}p({\bm x}) dx.\end{equation}

The dynamic now becomes a nonlinear flow
\begin{align} d{\bm x}_t &= -\mathcal{C}(p_t) \nabla V({\bm x}_t)dt + \sigma \sqrt{\mathcal{C}(p_t)} d\mathcal{B}_t,\\
\label{Inteacting-Langevin-Diffusion}\frac{\partial}{\partial t}p_t &= \text{div} \big(p_t  \mathcal{C}(p_t)\nabla V +  \frac{\sigma^2}{2} \mathcal{C}(p_t) \nabla p_t \big)\end{align}
To simulate from such a mean-field model, the finite interacting particle system $\mathcal{X}_t = \{ {\bm x}^i_t \}_{i=1}^N$ is introduced:
\begin{equation} d{\bm x}^i_t = - \mathcal{C}(\mathcal{X}_t)\nabla V({\bm x}_t^i)dt + \sigma \sqrt{\mathcal{C}(\mathcal{X}_t)}d \mathcal{B}_t^i, \label{particle-langevin-diffusion}\end{equation}
where $\mathcal{B}_t^i$ are i.i.d. standard Brownian motions, and
\begin{align*} \label{covarianceupdate} &\mathcal{C}(\mathcal{X}_t) := {\bm X}_t {\bm X}_t^T, \sqrt{\mathcal{C}(\mathcal{X}_t)} := {\bm X}_t,\\
&{\bm X}_t := \frac{1}{\sqrt{N}}\large( {\bm x}_t^1 - \bar {\bm x}_t, \cdots, {\bm x}_t^N - \bar {\bm x}_t\large), \bar {\bm x}_t = \frac{1}{N} \sum_{i=1}^N {\bm x}_t^i\end{align*}
Particles in this system are then no longer independent to each other, in contrast to independent simulations in SGLD algorithm.

Now suppose $V$ is a least squares functional\footnote{For any positive-definite matrix $A$, we define $\langle a, a'\rangle_A = \langle a, A^{-1} a'\rangle =  a^T A^{-1} a'$, and $\Vert a \Vert_A = \Vert A^{-\frac{1}{2}}a\Vert$.  } with Tikhonov-Phillips regularization \cite{Engl1996RegularizationOI}:
\begin{equation} V(x) = \frac{1}{2} \Vert y - \mathcal{G}(x)\Vert_\Gamma^2 + \frac{1}{2}\Vert x \Vert_{\Gamma_0}^2,\label{Least-square-functional}\end{equation}
where $y \in \mathbb{R}^k, \mathcal{G}: \mathbb{R}^d \to \mathbb{R}^k$ is the forward map, $\Gamma$ and $\Gamma_0$ are two positive definite matrixs. In this situation, the system (\ref{particle-langevin-diffusion}) can be re-written as
\begin{align}
d{\bm x}^i_t =& - \frac{1}{N}\sum_{j=1}^N \left \langle D\mathcal{G}(\bm{x}^i_t)(\bm{x}^j_t - \bm{\bar x}_t), \mathcal{G}(\bm{x}^i_t) - y \right\rangle_\Gamma \bm{x}_t^j dt \nonumber\\
 &- \mathcal{C}(\mathcal{X}_t) \Gamma_0^{-1} \bm{x}_t^i dt + \sigma \sqrt{\mathcal{C}(\mathcal{X}_t)}d W_t^i. \label{ls-langevin-diffusion}\end{align}
Using the $1^{\text{st}}$ order Taylor approximation
$$D\mathcal{G}(\bm{x}^i_t)(\bm{x}^j_t - \bm{\bar x}_t) \approx \mathcal{G}(\bm{x}^j_t) - \mathcal{\bar G}_t, \quad \mathcal{\bar G}_t := \frac{1}{N}\sum_{k=1}^N \mathcal{G}(\bm{x}^k_t),$$
one may approximate Eq. \ref{ls-langevin-diffusion} in a derivative-free manner as
\begin{align}
d{\bm x}^i_t =& - \frac{1}{N}\sum_{j=1}^N \left \langle \mathcal{G}(\bm{x}^j_t) - \mathcal{\bar G}_t, \mathcal{G}(\bm{x}^i_t) - y \right\rangle_\Gamma \bm{x}_t^j dt \nonumber\\
 &- \mathcal{C}(\mathcal{X}_t) \Gamma_0^{-1} \bm{x}_t^i dt + \sigma \sqrt{\mathcal{C}(\mathcal{X}_t)}d W_t^i. \label{ls-langevin-diffusion-der-free}\end{align}

\section{Reweighted Interacting Langevin Diffusion}
\subsection{Continuous process analysis}\label{sec:continuous}
Keeping the invariant measure to be exactly $e^{-2\sigma^{-2}V({\bm x})}$ is not necessary for optimization. It would be preferable if a new process can converge faster to its invariant measure with its invariant measure still concentrating on the global optimum as $\sigma \to 0$.

Now we introduce an additional source term to modify Eq. \eqref{Preconditioned-Langevin}  into
\begin{align}\frac{\partial}{\partial t}\tilde p_t &= \text{div} \big(\tilde p_t C\nabla V +  \frac{\sigma^2}{2} C \nabla \tilde p_t \big) + W \tilde p_t \label{Inteacting-Langevin-Diffusion-reweight1}\\
p_t({\bm x}) &= \frac{\tilde p_t({\bm x})}{\int_{\tilde {\bm x}\in \mathbb{R}^d} \tilde p_t(\tilde {\bm x}) d \tilde {\bm x}}\label{Inteacting-Langevin-Diffusion-reweight2}
\end{align}

Here we mainly consider the situation when $C = I$ corresponding to Eq. \ref{Langevin-Diffusion}, or $C = \mathcal{C}(p_t)$ corresponding to Eq. \ref{Inteacting-Langevin-Diffusion}, but the analysis remains valid for all positive-definite $C$. We assume in this paper that the function $W$ is smooth and upper bounded.

Let us look inside what the role $W$ plays in the evolution of this process. If we take $W$ as a function to $f$ such that $W({\bm x})$ becomes larger when $f({\bm x})$ becomes smaller, intuitively the ratio of the mass in the better-fitness region (closer to the global minimum) becomes larger, thus we expect the invariant measure concentrates more on the global optimum.

Note that such a process (\ref{Inteacting-Langevin-Diffusion-reweight1}), (\ref{Inteacting-Langevin-Diffusion-reweight2})  is no longer a gradient flow structure, which does not preserve total mass,  thus a normalization is added to keep the total mass equal to 1. Such a normalizing process has been well-studied as the so-called Feynman-Kac Semigroup when considering the linear case: $C$ is a constant matrix. As the spectral analysis is wildly studied for linear operators \cite{Kato1966PerturbationTF}, and for numerical discretization the $C$ is fixed at each time step, we thus conduct analysis for any fixed matrix $C$.

We introduce the solution operator corresponding to Eq. \ref{Inteacting-Langevin-Diffusion-reweight1} and Eq. \ref{Inteacting-Langevin-Diffusion-reweight2}: recall the infinite generator $\mathcal{L}^\dagger_C$, the solution in Eq. \ref{Inteacting-Langevin-Diffusion-reweight1} can be represented by $\tilde p_t = e^{t(\mathcal{L}^\dagger_C + W)}p_0$. We denote the corresponding reweighted operator, which is also well-known as Feynman-Kac Semigroup, as $\Phi^t_{\mathcal{L}_C+W}:$

$$ \Phi^t_{\mathcal{L}_C+W}(p_0) := \frac{ e^{t(\mathcal{L}^\dagger_C + W)} p_0}{\int_{\mathbb{R}^d}e^{t(\mathcal{L}^\dagger_C+W)} p_0(\tilde {\bm x}) d\tilde {\bm x} },$$

then $p_t =  \Phi^t_{\mathcal{L}_C+W}(p_0)$ is exactly the solution to Eq. \ref{Inteacting-Langevin-Diffusion-reweight2}.

The operator $\Phi^t_{\mathcal{L}_C+W}$ is nonlinear in general, but it has many similarity with the linear operator $e^{t(\mathcal{L}^\dagger_C + W)}$. In specific, the unique positive fixed-point $p_\infty$ of $\Phi^t_{\mathcal{L}_C+W}$ is proportional to the principle eigenfunction of $(\mathcal{L}^\dagger_C + W)$, and the convergence rate of $\Phi^t_{\mathcal{L}_C+W}(p_0)$ to $p_\infty$ is controlled by the spectral gap of $(\mathcal{L}_C + W)$. For the Feynman-Kac semigroup, one can refer to \citet{Ferr2017ErrorEO} for time-invariant case, \citet{Lyu2021ACI} for time-periodic case, and \citet{Moral2004FeynmanKacFG} for
systematical details.

We are interested in what benefits the source term $W$ can contribute. We show that, when taking $W = \varepsilon m(V)$ for some small $\varepsilon >0$ where $m:\mathbb{R}\to\mathbb{R}$ is a monotonic decreasing function of $V$, this brings mainly two benefits:
\begin{itemize}
    \item the spectral gap to the operator $\mathcal{L}_C + W$ is larger than $\mathcal{L}_C$ considered in same functional space;
    \item the invariant measure of $ \Phi^t_{\mathcal{L}_C+W}$ concentrates more on the global minimum compared to the invariant measure of $e^{t(\mathcal{L}^\dagger_C)}$.
\end{itemize}
These two benefits show that, in the same noise level, process \eqref{Inteacting-Langevin-Diffusion-reweight2} can converge faster and have more concentrated invariant measure than process \eqref{Langevin-Diffusion} or \eqref{Inteacting-Diffusion-reweight}. We will further explain the benefits, giving proof in Section  \ref{Interacing-Langevin-Diffusion-Theory}.

\subsection{Discrete algorithm design}\label{sec:discrete}
Now let us use interacting particle methods \cite{Moral2000BranchingAI,Moral2013MeanFS} and mean field theory \cite{Kac1960ProbabilityAR} to design algorithms. We introduce the so-called Reweighted Interacting Langevin Diffusion algorithm, by simply approximating $p_t$ with a population of particles, and discretize the evolution of time by operator-splitting technique and forward-Euler scheme.

First, we convert Eq. \ref{Inteacting-Langevin-Diffusion-reweight1} and \ref{Inteacting-Langevin-Diffusion-reweight2} into a discrete-time version: let $\tau >0$ be a fixed timestep:
$$p_{(n+1)\tau} = \frac{ e^{\tau(\mathcal{L}^\dagger_C + W)} p_{n\tau}}{\int_{\mathbb{R}^d}e^{\tau(\mathcal{L}^\dagger_C+W)} p_{n\tau}(\tilde {\bm x}) d\tilde {\bm x} }$$

then we use the operator splitting technique\footnote{Note that higher order splitting technique can be applied, such as Strange splitting \cite{Strang1968OnTC} $e^{\frac{\tau}{2} W} \circ e^{\tau \mathcal{L}^\dagger_C} \circ e^{\frac{\tau}{2} W} $ with splitting error $O(\tau^3)$. However, the approximation error can still be induced in later steps, thus we only choose a simplest one here.}: approximate $e^{\tau(\mathcal{L}^\dagger_C + W)}$ by $e^{\tau W}\circ e^{\tau \mathcal{L}^\dagger_C}$ with splitting error $O(\tau^2)$ .

Let's now approximate $e^{\tau \mathcal{L}^\dagger_C} p_{n\tau}$. Suppose $p_{n\tau}$ is approximated by a weighted empirical measure that can be expressed as  $\hat p_{n\tau} ({\bm x}) =  \sum_{i=1}^{N}w^i_n\delta_{{\bm x}_n^i}({\bm x})$ generated from a sort of sample-weight pair $\{ {\bm x}_n^i, w_n^i \}_{i=1}^N$. We use the simplest Euler-Maruyama \cite{Faniran2015NumericalSO} approximation,
\begin{align}
&e^{\tau \mathcal{L}^\dagger_C}\hat p_{n\tau}({\bm x}) \approx \frac{1}{N} \sum_{i=1}^{N}w_n^i\delta_{ {\bm x}_{n+1}^i}({\bm x}), \\
&{\bm x}_{n+1}^i = {\bm x}_n^i - C \nabla V({\bm x}_n^i)\tau + \sqrt{\tau\sigma^2C}\xi^i_n \\
&e^{\tau W} e^{\tau \mathcal{L}^\dagger_C}\hat p_{n\tau}({\bm x}) \approx  \frac{1}{N} \sum_{i=1}^{N} w_n^i e^{\tau W( {\bm x}_{n+1}^i)} \delta_{ {\bm x}_{n+1}^i}({\bm x}), \label{discretescheme-reweighted-Langevin}\end{align}
Here $C$ can be $I$ or the covariance matrix of current step: if we take the matrix $C$ to be the covariance matrix, it is computed as (denote $\Lambda = \text{diag}(w_n^i), \quad\bar {\bm x}_n = \frac{1}{N} \sum_{i=1}^N w_n^i {\bm x}_n^i$)
\begin{align} C = \mathcal{C}(\hat p_{n\tau}) =  {\bm X}_n \Lambda {\bm X}_n^T, \sqrt{C} = {\bm X}_n\Lambda^{\frac{1}{2}},  \\
{\bm X}_n := \frac{1}{\sqrt{N}}\large( {\bm x}_n^1 - \bar {\bm x}_n, \cdots, {\bm x}_n^N - \bar {\bm x}_n\large). \label{Reweight-Covariance-Update}\end{align}

Then, after normalizing, we get the natural approximation
\begin{align}  p_{(n+1)\tau}({\bm x}) \approx    \sum_{i=1}^{N}w^i_{n+1}\delta_{{\bm x}_{n+1}^i}({\bm x}), \\
 w_{n+1}^i = \frac{ w_n^i e^{\tau W( {\bm x}_{n+1}^i)}}{\sum_{j=1}^N w_n^je^{\tau W( {\bm x}_{n+1}^j)}} \label{weight-update}\end{align}
Note that elements in $\{w^i_{n}\}_{i=1}^N$ may be polarized, we use a resampling technique for better approximation: If $\frac{\max_i w^i_n}{\min_i w^i_n}$ reaches to a threshold, we resample the replicas $\{ \bm{x}_n^i \}_{i=1}^N $ according to the multinomial distribution associated with $\{w_n^i\}_{i=1}^N $, which defines a new set of replicas $\{ \bm{\tilde x}_n^i \}_{i=1}^N$ and the empirical distribution $$\tilde p_{n\tau}(\bm{x}) = \frac{1}{N}\sum_{i=1}^N \delta_{\bm{\tilde x}_n^i}(\bm{x}).$$
Then we replace $\hat p_{n\tau}$ by $\tilde p_{n\tau}$ in Eq. \ref{discretescheme-reweighted-Langevin} and conduct further computation.

When $V$ is a least square functional with Tikhonov-Phillips regularization like in Eq. \ref{Least-square-functional}, we can follow the same analysis, approximating $\mathcal{C}(\hat p_{n\tau}) \nabla V({\bm x}_n^i)$ in a similar derivative-free manner as in Eq. \ref{ls-langevin-diffusion-der-free}: ( $\mathcal{\bar G}_n := \frac{1}{N}\sum_{k=1}^N w_n^i\mathcal{G}(\bm{x}^k_n) $ )
\begin{align}\label{rw-langevin-diffusion-der-free}
    &\mathcal{C}(\hat p_{n\tau}) \nabla V({\bm x}_n^i)\\
    \approx &\frac{1}{N}\sum_{j=1}^N w_n^j\left \langle \mathcal{G}(\bm{x}^j_n) - \mathcal{\bar G}_n, \mathcal{G}(\bm{x}^i_n) - y \right\rangle_\Gamma \bm{x}_n^j
     +\mathcal{C}(\hat p_{n\tau}) \Gamma_0 \bm{x}_n^i. \nonumber
\end{align}
 We now conclude in algorithm \ref{alg:RILD}. Note that we fix the population $N$, stepsize $\tau$, and noise level $\sigma$ for simple, but these can be adjusted dynamically for faster convergence.

Note that our methods  can also be interpreted as a mutation-selection genetic particle algorithm with MCMC mutations: the update rule for $\bm{x}_n^i \to \bm{x}_{n+1}^i$ can be regarded as mutation, and the resampling step can be regarded as selection. Such a type of algorithm acts like a ridge connecting Genetic Algorithm and PDEs, which can conduct better convergence analysis.

\begin{algorithm*}
\caption{Reweighted Interacting Langevin Diffusion (RILD) }\label{alg:RILD}
\begin{algorithmic}
\STATE {\bfseries Input:} Functional to minimize $V({\bm x})\in \mathbb{R}, x \in X \subseteq \mathbb{R}^d$, gradient $\nabla V({\bm x}) \in \mathbb{R}^d$, Fitness  $W(\bm{x})\in \mathbb{R}$, population size $N$, Maximum iteration $Iter$, i.i.d. d-dimensional Standard Gaussian samples $\{ \xi^i_n \}_{1\le i\le N, 0 \le n < Iter}$, Stepsize $\tau$, noise $\sigma$.
\STATE {\bfseries Onput:} Samples nearby global minimum.
\STATE Sample ${\bm x}_0^i, i = 1,\cdots,N$ under initial distribution $p_0$; $w_0^i = \frac{1}{N},i = 1,\cdots,N$, $n = 0$.
\WHILE {$n < Iter$}
\STATE Calculate $C$ by \eqref{Reweight-Covariance-Update} or just $C=I$, depends on if one use covariance modification.
\STATE If use covariance modification, directly calculate $C \nabla V({\bm x}_n^i)$ or approximate it by \eqref{rw-langevin-diffusion-der-free}, depends on if $V$ is a least square functional \eqref{Least-square-functional}.
\STATE ${\bm x}_{n+1}^i = {\bm x}_n^i - C \nabla V({\bm x}_n^i)\tau + \sqrt{\tau\sigma^2C}\xi^i_n$, $\quad w_{n+1}^i = \frac{ w_n^i e^{\tau W( {\bm x}_{n+1}^i)}}{\sum_{j=1}^N w_n^je^{\tau W( {\bm x}_{n+1}^j)}}$,  $\quad 1\le i \le N$

\IF {$\frac{\max_i w^i_{n+1}}{\min_i w^i_{n+1}} > Threshold$}
    \STATE Sample $\{ \bm{\tilde x}_{n+1}^i \}_{i=1}^N $ from $\{ \bm{x}_{n+1}^i \}_{i=1}^N $ according to the multinomial probability $\{ w_{n+1}^i\}_{i=1}^N $.
    \STATE $\bm{x}_{n+1}^i = \bm{\tilde x}_{n+1}^i, w_{n+1}^i = \frac{1}{N}, 1\le i \le N$
\ENDIF
\STATE $n = n+1$
\ENDWHILE
\STATE {\bf Output} $\{ \bm{x}_{n}^i \}_{i=1}^N$
\end{algorithmic}
\end{algorithm*}

\subsection{Gradient-Free variants}

In practice, many problems are hard to get the exact gradients for optimizing: gradient information is infeasible, or computationally expensive. We thus suggest a gradient-free variant, which corresponds to a process only with diffusion and source term. We will prove that such a  process can exponentially converge to its invariant measure, and it also has an invariant measure that generally concentrates on the global minimum as $\sigma \to 0$.

We introduce the formula of the modified process as
\begin{align}\label{Inteacting-Diffusion-reweight}\frac{\partial}{\partial t}\tilde p_t &= \text{div} \big(  \frac{\sigma^2}{2} C \nabla \tilde p_t \big) + W \tilde p_t \\
p_t({\bm x}) &= \frac{\tilde p_t({\bm x})}{\int_{\tilde {\bm x}\in \mathbb{R}^d} \tilde p_t(\tilde {\bm x}) d \tilde {\bm x}}
\end{align}

Note that we only delete the term related to $\nabla V$,  the term $W$ that can still be chosen to relate to $V$. We will state in Thm. \ref{Thm:Gradfree} that, the system will finally converge to a distribution concentrating on the maximum of $W$.

The discrete algorithm is designed similarly as in Algorithm \ref{alg:RILD}, but just ignores the gradient term $C\nabla V$. This can be seen by taking $V(\bm x) \equiv \text{const}$.

\section{Theoretical properties} \label{theoreticalproperty}

In this section, we state the main theoretical results associated with our RILD algorithm \ref{alg:RILD}. The proofs are offered in Appendix \ref{sec:proofs}.

\subsection{Spectral Gap enhancement of the reweighting modification}\label{Interacing-Langevin-Diffusion-Theory}
Now we analyze how $W$ helps in improving the convergence rate, as well as the sharpness of the invariant measure.

First let us restrict the comparison in the $L^2(\nu)$, where $\nu(x) = e^{-\frac{2V(x)}{\sigma^2}}$. The reason to choose this space is mainly because the following property,
\begin{lem}\label{Lem:symmetric}
    For any  positive definite matrix $C$, $L_C + W$ and $L_C$ are self-adjoint over $L^2(\nu)$.
\end{lem}
To continue our analysis, we need to make following assumption for $V$ and $W$, which is necessary for $L_C+W$ and $L_C$ have a discrete and up-bounded spectrum (see \citet{Pankov2001IntroductionTS}).

\begin{assumption}\label{Asmp:VW} The functions $V$ and $W$ are assumed to satisfy:
    $$\lim_{|\bm x| \to \infty} V = +\infty, W < A \text{ for a constant A} \in \mathbb{R},$$
    and
    $$\lim_{\vert \bm x\vert \to \infty} \frac{ \vert \nabla V\vert^2}{2\sigma^2} - \frac{\Delta V}{2} - W =   \lim_{\vert \bm x\vert \to \infty} \frac{ \vert \nabla V\vert^2}{2\sigma^2} - \frac{\Delta V}{2} = +\infty$$
\end{assumption}

Next, we need to prove that the Feynman-Kac semigroup $\Phi^t_{\mathcal{L}_C+W}$ does map any initial density $p_0 \in L^2(\nu)$ to a limit density. Such a result is concluded as follows:

\begin{thm}\label{Thm:convergence1}
    Under Asmp. \ref{Asmp:VW}, For any $C$ that is positive-definite, there exists a principle eigenvalue $\lambda_0$ of $\mathcal{L}_C + W$ over $L^2(\nu)$ with the corresponding normalized eigenfunction $\phi({\bm x})$. Furthermore, for any positive density $p_0$, the normalized probability $p_t := \Phi^t_{\mathcal{L}_C+W}(p_0)$ has a limit equal to $\phi({\bm x}) \nu({\bm x}) $, that is, $\lim_{t\to \infty} ||p_t/\nu -  \phi||_{L^2(\nu)} = 0$.
\end{thm}

\begin{thm}\label{Thm:convergence2}
     Under the same assumption as in Lem. \ref{Thm:convergence1}, the convergence rate of the system \eqref{Inteacting-Langevin-Diffusion-reweight1}, \eqref{Inteacting-Langevin-Diffusion-reweight2}  can be evaluated by the spectral gap of $\mathcal{L}_C + W$ over $L^2(\nu)$:  let $\lambda_0$ and $\lambda_1$ be the first two eigenvalues of $\mathcal{L}_C + W$, then  $ ||p_t/\nu -  \phi||_{L^2(\nu)} \le C || p_0/\nu - \phi||_{L^2(\nu)} e^{-(\lambda_0 - \lambda_1)t}$.
\end{thm}

Next, let us analyze how the convergence speed of $\Phi^t_{\mathcal{L}_C+W}p_0$ is improved compared to the original process $e^{t\mathcal{L}_C}(p_0)$, which is heavily related to the spectral gap. To exactly analyze the spectral gap to a differential operator is hard in general. Many existing analysis of spectral gap only consider the simplest case: $\nabla V$ is a constant matrix. In contrast to these existing techniques, we analyze the contribution of $W$ to the spectral gap by perturbation theory:  we will prove that the new process has a better convergence rate compared to the old one, if $W = \varepsilon m(V)$ for a small $\varepsilon >0$, where $m: \mathbb{R} \to \mathbb{R}$ is a monotonic decreasing function to $V$.

\begin{thm}\label{Thm:SpectralGapEnhancement}
    Suppose in addition $V$ satisfies the condition the same as in \citet{Nier2004QuantitativeAO}. If we take $W = \varepsilon m(V)$, consider under the space $L^2(\nu)$, where $\nu(x) = e^{-\frac{2 V(x)}{\sigma^2}}$, then when $\sigma$ small enough,  the spectral gap of $\mathcal{L}_C+\varepsilon m(V)$ is locally increasing  v.s. $\varepsilon$ for small enough $\varepsilon>0$. Besides, the principle eigenfunction of $\mathcal{L}_C+\varepsilon m(V)$ concentrates more on global minimum than the principle eigenfunction of $\mathcal{L}_C$ for small enough $\varepsilon>0$.
\end{thm}

\subsection{Convergence of the Gradient-Free variants}

In the gradient-free situation, the original operator  $\mathcal{D}_C:= \text{div} \big(  \frac{\sigma^2}{2} C \nabla \cdot \big) $ has a trivial invariant measure: uniform distribution, thus no optimization property can be expected when $W$ is not included. We thus turn our results to another way, showing that $\Phi_{\mathcal{D}_C + W}^t p_0$ exponentially converges to a distribution that concentrates on the neighborhood of the global minimum, and as $\sigma \to 0$, the invariant distribution gets more and more concentrate on the global minimum. We now state the result as follows:
\begin{thm}\label{Thm:Gradfree}
    Consider under the space $L^2$, assuming $W$ is bounded on $\mathbb{R}^d$, then $\Phi_{\mathcal{D}_C + W}^t p_0$ converges exponentially to the principle normalized eigenfunction $\mu_\sigma({\bm x})$ of the operator $\mathcal{D}_C + W$. In addition, for any $f \in L^2$ that's smooth compactly supported, $\lim_{\sigma \to 0} \langle\mu_\sigma(\bm x), f\rangle = f(\bm x^\ast) $, where ${\bm x}^\ast$ is the global maximum of $W$, or to say, $\mu_\sigma(\bm x)$ generally
   converges to $\delta_{\bm x^\ast}(\bm x)$.
\end{thm}

\section{Numerical experiments}
In this section, we first present Fourier Spectral analysis to verify our theoretical analysis of the proposed method. Then, we conduct two inverse problem tests, showing the positive effect as a sampling method when introducing the reweighting/resampling procedure. Finally, we test an optimization task, showing that the resampling technique can help escaping from the local minimum.
\subsection{Fourier spectral method analysis for enhancing spectral gap and concentrating invariant measure }
Let us consider a 1 $d$ periodic problem. We first verify our results in Thm. \ref{Thm:SpectralGapEnhancement}. Let $x \in [0,1)$\footnote{Although our analysis considers the space $\mathbb{R}$, the results can be easily transferred to the periodic case with Asmp. \ref{Asmp:VW} be removed.}, we use Fourier Spectral method \cite{Shen2011SpectralMA} to discretize the differential operator $\mathcal{L}$ and $\mathcal{L} - \varepsilon V$,where
\begin{equation}
    \mathcal{L}f(x) := -\frac{d}{dx}V(x)\frac{d}{dx}f(x) + \frac{d^2}{dx^2} f(x)
\end{equation}
for any periodic $f \in C^\infty([0, 1))$,  and
\begin{equation}\label{Vfunction}
    V(x) = \cos(9\pi x) - \cos(11\pi x)
\end{equation}
with boundary smoothly modified.
\begin{figure}[h!]\label{SpectralGap}
\subfigure[]{\label{Fig:Eigfunction}
    \includegraphics[width = 0.47\linewidth]{lapeigspectrum.pdf}
    }
\subfigure[]{\label{Fig:EigGap}
    \includegraphics[width = 0.47\linewidth]{lapeigspectrumgap.pdf}
    }
\vspace{-1em}
\caption{\ref{Fig:Eigfunction} are the graphs to the $1^{st}$ eigfun. $\nu$ to $\mathcal{L}$, $1^{st}$ eigfun. $\mu$ to  $\mathcal{L} - 0.1V$, function $V$, and the quotient of $\mu$ and $\nu$ got by Fourier Spectral method. \ref{Fig:EigGap} is the graph to the first two eigenvalues $\lambda_0(\varepsilon),\lambda_1(\varepsilon)$ of $\mathcal{L} - \varepsilon V$ v.s. $\varepsilon$. }
\vspace{-1em}
\end{figure}

In Fig. \ref{Fig:Eigfunction}, We plot the graphs of the principal eigenfunction $\nu$ to the operator $\mathcal{L}$, the principal eigenfunction $\mu$ to the operator $\mathcal{L}-0.1V$, function $V$ and $\frac{\mu}{\nu}$ respectively. As we expected, $\mu$ concentrates more on the global minimum of $V$ than $\nu$. In Fig. \ref{Fig:EigGap}, we can see $\lambda_0(\varepsilon) - \lambda_1(\varepsilon)$ increasing as $\varepsilon$ increasing, showing the enhancement of spectral gap. This experiment gives a
visual explanation to Thm. \ref{Thm:SpectralGapEnhancement}: specifically, the eigenfunction  to $\mathcal{L}$ is the Gibbs measure $\nu(\bm x) = \exp(-2V(\bm x)/\sigma^2)$, getting larger when $V $ gets smaller; the eigenfunction $\mu$ to $\mathcal{L}- 0.1V$ has no doubt more mass than $\nu$ that concentrating nearby the global minimum of $V$, and the staircase-like graph of the quotient of $\mu/\nu$ gives a direct explanation to \eqref{eigenfunctionpurturbation}.

Next, let us verify our result in Thm. \ref{Thm:Gradfree}. We use the same space $x\in [0,1)$ and the source term $W(\bm x)  = -V(\bm x)$ where $V(x)$ is the same as in  \eqref{Vfunction}, and test the  analytical property of the following operator
\begin{equation}(\mathcal{D_\sigma} + W)f(x):= \frac{\sigma^2}{2} \frac{d^2}{dx^2}f(x) + W(x) f(x) \end{equation}
for any periodic $f \in C^\infty([0, 1))$. We plot in Fig. \ref{Fig:gradfreeInvariantmeasure} the principal eigenfunction, or invariant distribution density $\mu_\sigma(x)$ of $\mathcal{D}_\sigma$ with different $\sigma$. We also calculate the mass nearby the global minimum: we take the interval $I = [0.44,0.68]$, and calculate $\int_I \mu_\sigma(x) dx$. The results is plotted in Fig. \ref{Fig:gradfreeConcentrateratio}, showing the concentrating tendency when $\sigma \to 0$. This coincides with the result in Thm. \ref{Thm:Gradfree}: the invariant measure concentrates more and more on the global maximum of $W$ as $\sigma \to 0$.

\begin{figure}[h!]
\vspace{-1em}
\subfigure[]{\label{Fig:gradfreeInvariantmeasure}
    \includegraphics[width = 0.47\linewidth]{diffreweightinvariantmeasure.pdf}
    }
\subfigure[]{\label{Fig:gradfreeConcentrateratio}
    \includegraphics[width = 0.47\linewidth]{diffreweightconcentrateratio.pdf}
    }
\vspace{-1em}
\caption{\ref{Fig:gradfreeInvariantmeasure} is the invariant density $\mu_\sigma(x)$  got by Fourier Spectral method under different $\sigma$. \ref{Fig:gradfreeConcentrateratio} plots the ratio $\int_{0.44}^{0.68} \mu_{\sigma}(x) dx$.}
\vspace{-1em}
\end{figure}

\subsection{Numerical tests for inverse problem derivative-free solving and sampling}
In this section, we design some numerical tests in inverse problem solving fields. We compare our RILD algorithm with EKS \cite{GarbunoIigo2019InteractingLD} and EKI \cite{Kovachki2018EnsembleKI}. These problems can be seen as an optimization problem of a least square functional with Tickhonov-Phillips regularization \eqref{Least-square-functional}, thus a derivative-free approximation to the updating rule (\eqref{ls-langevin-diffusion-der-free} for EKS and EKI, or \eqref{rw-langevin-diffusion-der-free} for RILD) can be used to design derivative-free schemes.

We first try to solve a low-dimensional inverse problem. The numerical experiment considered here is the example originally presented by \citet{Ernst2015AnalysisOT}, and also used in \citet{Herty2018KineticMF}. We compare with the result from \citet{GarbunoIigo2019InteractingLD}, and the experimental settings are exactly the same. The forward map is given by the solution of a one-dimensional elliptic boundary value problem as defined in \citet{GarbunoIigo2019InteractingLD},
\begin{equation}
    -\frac{d}{du}\left( \exp(x_1) \frac{d}{du} f(u)\right) = 1, u \in [0,1]
\end{equation}
with $f(0) = 0, f(1) = x_2$. The explicit solution is given by
\begin{equation}
    f(u) = x_1 u + \exp(-x_2)\left( -\frac{u^2}{2} + \frac{u}{2}\right).
\end{equation}
Thus we define the forward map
\begin{equation}\mathcal{G}(\bm x) = \left(f(u_1),f(u_2)\right)^T.\end{equation}
Here $\bm x = (x_1,x_2)^T$ is a constant vector we want to solve, and we have noisy measurements $\bm y = (27.5,79.7)^T$ of $f(\cdot)$ at locations $u_1 = 0.25, u_2 = 0.75$. This can be solved by minimizing the least-square functional defined as in Eq. \ref{Least-square-functional}. We assume observation noise $\Gamma = 0.1^2 I_2$, prior matrix $\Gamma_0 = 10^2 I_2$, and initial ensemble drawn from $N(0,1) \times
U(90,110).$  The ensemble size is $N = 10^3$. We fix $\sigma = \sqrt{2}$. The stepsize $\tau$ is updated adaptively as in \citet{GarbunoIigo2019InteractingLD}. We take $W(\bm x) = -\Vert \mathcal{G}(\bm x) - \bm y  \Vert^2_\Gamma$.

We compare our RILD algorithm \ref{alg:RILD} with EKS and EKI algorithms. The key difference between our RILD and EKS in this situation is the use of reweighting/resampling. The results are plotted in Fig. \ref{Fig:EKCompare1}, \ref{Fig:EKCompare2}, and \ref{Fig:EKCompare3}.  From the figure, we can see that our RILD algorithm converges much faster than EKI or EKS algorithms using the same super-parameters, and as our problem is settled as a posterior sampling problem, the ensemble of RILD stops shrinking to the minimum point after some iterations, unless one decrease the diffusion parameter $\sigma$. One could expect, using a decay schedule to $\sigma$, RILD algorithm will perform much better than EKS or EKI algorithms for optimizations.
\begin{figure}[h!]
\vspace{-1em}
\subfigure[]{\label{Fig:EKCompare1}
    \includegraphics[width = 0.3\linewidth]{EensembleKalmantests1.png}
    }
\subfigure[]{\label{Fig:EKCompare2}
    \includegraphics[width = 0.3\linewidth]{EensembleKalmantests2.png}
    }
\subfigure[]{\label{Fig:EKCompare3}
    \includegraphics[width = 0.3\linewidth]{EensembleKalmantests3.png}
    }
\vspace{-1em}
\caption{The convergence comparison between RILD, EKS and EKI algorithms. \ref{Fig:EKCompare1} is the mean loss versus iterations, \ref{Fig:EKCompare2} is the ensembles at $15^{th}$ iteration, \ref{Fig:EKCompare3} is the ensembles at $30^{th}$ iteration.}
\vspace{-1em}
\end{figure}

We also tested in a high-dimensional case. Specifically, we define the map $\mathcal{G}: \mathbb{R}^d \to \mathbb{R}^{2d-2}$
\begin{equation*}
    \mathcal{G}(\bm x) = \left(10(x_2 - x_1^2),\cdots,10(x_d - x_{d-1}^2), x_1 , \cdots, x_d \right)^T,
\end{equation*}
$\bm y = (0,\cdots,0, 1, \cdots,1)^T$ with $0$ repeats $d-1$ times and $1$ repeats $d-1$ times. One can verify that
$$\Vert \mathcal{G}(\bm x) - \bm y \Vert^2 = \sum_{i=1}^{d-1} \left(100(x_{i+1} - x_i^2)^2 + (x_i-1)^2\right) $$
is exactly a $Rosenbrock$ function.  We choose $d = 100$, observation noise matrix $\Gamma = 0.1^2 I_{198}$, prior distribution matrix $\Gamma_0 = 10^2 I_{100}$, and initial ensemble drawn from $N(2,0.3^2 I_{100})$. The global solution of $\mathcal{G}(\bm x) = \bm y$ is $\bm x = (1,\cdots,1)^T.$ The ensemble size is fixed to $N = 10^3$. We take $\sigma = \sqrt{2}$ and the stepsize $\tau$ is updated adaptively as before. For RILD algorithm, we choose $W(\bm x) = -5*10^{-3} \Vert \mathcal{G}(\bm x) - \bm y\Vert^2$. Similar to the test before, one can find RILD converges much faster than EKI or EKS in these high-dimensional sampling tasks, and thus perform better in optimization situations.

\begin{figure}[h!]
\vspace{-1em}
\subfigure[]{\label{Fig:RB1}
    \includegraphics[width = 0.3\linewidth]{rocenbrocktest1.png}
    }
\subfigure[]{\label{Fig:RB2}
    \includegraphics[width = 0.3\linewidth]{rocenbrocktest2.png}
    }
\subfigure[]{\label{Fig:RB3}
    \includegraphics[width = 0.3\linewidth]{rocenbrocktest3.png}
    }
\vspace{-1em}
\caption{The comparison between RILD, EKS, and EKI algorithms. \ref{Fig:RB1} is the mean loss v.s. iterrations, \ref{Fig:RB2} is the ensembles at $150^{th}$ iteration, \ref{Fig:RB3} is the ensembles at $400^{th}$ iteration.}
\vspace{-1em}
\end{figure}

\subsection{Numerical tests for highly nonconvex high-dimensional optimization}

We now test our RILD algorithm in a highly nonconvex high-dimensional situation. we test with 100 dimensional $Ackley$ function, which is defined as follows:
\begin{equation}
    V(\bm x) = -a e^ { -b \sqrt{ \frac{1}{d} \sum_{i=1}^{d} x_i^2}} - e^{ \frac{1}{d} \sum_{i=1}^{d} \cos(c x_i)} + a+ e,
\end{equation}
where $\bm x = (x_1,\cdots,x_d)^T, d = 100, a = 20, b = 0.2, c = 2\pi$. As the difficulties mainly raise from the numerious local minimum, covariance modification that is designed for ill-posed problems is not suitable here, we just take $C = I$.

We compare our algorithm RILD in Alg. \ref{alg:RILD} with the classical Gradient Langevin Dynamics (GLD) algorithm. GLD algorithm discretizes a single path of the Langevin Dynamics:
$$\bm x_{n+1} = \bm x_n - \nabla V(\bm x_{n})\tau + \sqrt{\tau \sigma^2} \xi_n. $$
The difference between RILD and GLD is: RILD maintains an ensemble of size $N$  while GLD only maintains 1 individual, and at each step, RILD calculates a weight associated with each individual, then resamples the ensemble according to the weight, see Alg. \ref{alg:RILD}. Now we test if RILD has better ability getting out of local minimums, compared to GLD.

For RILD, we take ensemble size $N = 50$, and randomly pick the initial ensemble\footnote{Such an initial setting creates many difficulties to find the global minimum, as the first term in the $Ackley$ function becomes quickly dominated when $\bm x$ gets away from the origin. } from $N(0,30^2I_{100})$. For GLD, the initial point is randomly chosen from the initial ensemble of RILD. We test a wide range of the stepsize $\tau \in [2,32]$, $\sigma \in [1,16]$, and for each fixed $\tau $ and $\sigma$, we repeat 10 trials to calculate the pass rate: we say one trial is passed, if the RILD or GLD algorithm can find a point $\bm x$ that $V(\bm x)<17$ in $5*10^4$ evaluations\footnote{If one finds a point smaller than 17, the remaining task will be trivial as the first term in the $Ackley$ function gets dominant.}. All trials in all super-parameter settings begin with the same initial ensemble. In \ref{Fig:passrate1} One can see that RILD has a wide range of super-parameter settings to find the true decay directions, while GLD algorithm cannot make it in any tested settings. We also tested GA and PSO under the same initial condition with different super-parameters, and reported the best searching result in Fig. \ref{Fig:passrate2} together with RILD and GLD.

\begin{figure}[h!]
\vspace{-1em}
\subfigure[]{\label{Fig:passrate1}
    \includegraphics[width = 0.6\linewidth, trim = 120 0 180 60 clip]{heatmap-passratio.png}
    }
\subfigure[]{\label{Fig:passrate2}
    \includegraphics[width = 0.34\linewidth]{ConvergeCompare.png}
    }
\vspace{-1em}
\caption{\ref{Fig:passrate1}: pass rates heat map for RILD(left) and GLD(right). \ref{Fig:passrate2}: decay graph for PSO, GA, RILD and GLD where $\tau = 10,\sigma = 5$ for RILD and GLD.  }
\vspace{-1em}
\end{figure}

\section{Conclusion}
In this work, we have demonstrated a methodology for accelerating Langevin Dynamics based algorithms by the addition of the source term $W$ and the use of reweighting/resampling technique -- the RILD algorithm. Our algorithm and analyses shed some light on combining gradient algorithms and genetic algorithms using Partial Differential Equations (PDEs) with provable guarantees.

In the future, we will combine the reweighting technique with higher-order optimization schemes such as momentum accelerated gradient. We will also conduct a finer analysis for convergence with finite particles, which is somehow more important as asymptotic results are only suitable for a large enough ensemble. We expect these studies will bring some insights to design new numerical algorithms.

\clearpage


A Japanese professional wrestler-turned-lawmaker touched base in North Korea with a previous NFL lineman and more than twelve military specialists on Thursday for the first huge games occasion offering great known outsiders since Dennis Rodman's dubious b-ball amusement recently.
Japanese parliamentarian Kanji "Antonio" Inoki said he trusted the occasion would open an entryway of games discretion with the North.
Sway "The Beast" Sapp, who quickly played American expert football before exchanging to blended hand to hand fighting and picking up VIP status in Japan, will lead the outsiders in the show, to be held in a 15,000-seat coliseum in Pyongyang on Saturday and Sunday. It will likewise offer nearby military specialists and musical acts.
Around 20 wrestlers and military specialists from far and wide are required to go to. Coordinators say the International Pro-Wrestling Festival in Pyongyang will be telecast on the web, publicized on Japanese TV and demonstrated on North Korean state-run TV.
Inoki is an insightful, magnetic artist – he generally wears a brilliant red scarf in broad daylight – and one of the main parts of Japan's parliament who backs and takes part in trades of any sort with North Korea. He has gone to North Korea about 30 times, however was suspended by parliament for a month in the wake of making an unapproved excursion to the North a year ago.
In the ring, the square-jawed, 6ft 3in Inoki is associated with battling Muhammad Ali in Tokyo in 1976. In 1995, he battled American Ric Flair in the "Crash in Korea," a two-day occasion held in Pyongyang's tremendous May Day Stadium that drew a reported 380,000 onlookers. It was the greatest pay-for every perspective occasion in master wrestling history. Ali was among the visitor actively present people.
Inoki's association with North Korea dates to his guide, an ace wrestler named Rikidozan who was perhaps the best-known games figure in post bellum Japan. Rikidozan was Korean, and his name and endeavors in the ring are still recollected in North Korea.
In spite of the fact that Inoki won't be battling this time, Sapp told the Associated Press before landing in Pyongyang that he was anticipating putting on a decent show for the North Korean group of onlookers.
Sapp – who is 6ft 5in and weighs 350lbs – played quickly as a lineman with the Chicago Bears and Minnesota Vikings before turning into one of Japan's best-known blended hand to hand fighting identities.
The display will be the greatest games show with a marquee American since Rodman and a group of other ex-NBA players took to a Pyongyang b-ball court in January. Rodman called that occasion "noteworthy", yet was panned by parts of the US Congress, the NBA and human rights bunches who said he had turned into an advertising device for North Korea's administration.
When he returned to the US, Rodman apologized openly for his behavior and entered recovery.
Sapp said he will guide as far from contention and legislative issues as he can.


I love cheese. I love melted cheese. I really love Mozzarealla Sticks. I also love Keto, how it makes me feel, and what it is doing for my body, but Mozzarella Sticks have lots of carbs. No kidding, like about 8 carbs for one stick of melty goodness. The fat is on point for Keto with about 6 grams of fat (over 50% of the about 100 calories come from fat).
I tried a couple of recipes that I found online, but I was not impressed so I came up with one that I think is pretty damn good.


text:
« Antonelli, Giacomo Antonians Antoninus Pius »
Antonians
ANTONIANS, an-tō´ni-ans, or ANTONINES, an"to-nainz´: 1. Religious orders among the Roman Catholic Chaldeans, Maronites, and Armenians, which follow a rule called the rule of St. Anthony. In reality St. Anthony (251-356), although he is justly styled the father of cenobitic life, left no rule to his followers save those scattered directions found in his writings. The so-called rule of St. Anthony is, therefore, the work of some later writer who took its substance, however, from the teachings of the saint. At the present time the Antonians are grouped in four congregations; the Chaldean Antonians of St. Hormisdas, founded in Mesopotamia in 1809 for missionary work, with about one hundred members; the Maronite Antonians of Aleppo, with 120 members; the Maronite Baladite Antonians, the most numerous of all, with 700 members; and the Maronite Congregation of St. Isaiah, with 240 members.
John T. Creagh.
A fifth congregation called after St. Anthony, now almost extinct, was founded among the Roman Catholic Armenians by Abraham Attar-Muradian, a merchant, who in 1705, with his brother James, a priest, retired to Mount Lebanon to lead an ascetic life. Here, in 1721, they established the monastery of Kerem, followed by another at Beit-Khasbo near Beirut. In 1761 a third community was founded in Rome, near the Vatican. About 1740 the exiled bishop of Haleb (Aleppo), Abraham Ardzivian, who had found refuge at Kerem, took advantage of a long vacancy in the Cilician patriarchate to set himself up as catholicos of Cilicia, and secured papal confirmation in 1742. His first successor was the above-mentioned James, who was followed by Michael and Basil, also Antonians. In 1866 the patriarch of the Catholic Armenians, Anthony Hasun, residing in Constantinople, adopted the title "Patriarch of Cilicia," and put an end to the nominal Antonian patriarchate. The Antonians usually numbered fifty or sixty, and served the Roman Catholic mission in Turkey. In 1834 they transferred their novitiate and school to Rome, only the abbot and a few brothers remaining in the Lebanon. In 1865 Sukias Gazanjian was chosen abbot and was consecrated by the last Lebanon patriarch. He lived in Constantinople as head of the anti-Hasun party. On Hasun's charges, he was summoned to Rome in 1869; but before his case could be heard, the Vatican council met. He and his monks were among the first to reject papal infallibility, and were obliged to escape by night, with the help of the French ambassador. In 1876 Malachi Ormanian, the best-known and best-educated of the Antonians, went to Rome and finally closed their house there. (He afterward joined the Armenian Church, and has published Le Vatican et les Arméniens and other works.) The present members of the congregation, having made their submission to the pope, are concentrated in one community in Constantinople.
2. An antinomian sect which originated in the canton of Bern, Switzerland, early in the nineteenth century, founded by Anton Unternährer (b. at Schüpfheim, in the canton of Lucerne, Sept. 5, 1759; d. in the jail of Lucerne June 29, 1824). Unternährer was educated and confirmed in the Roman Catholic Church; after a varied career as cowherd, cabinet-maker, private teacher, and quack doctor, he settled in 1800 at Amsoldingen, near Thun, and began to hold religious meetings, to preach, and to issue books. He announced himself as the Son of God, come to fulfil the incomplete work of Jesus, to judge mankind (especially rulers and judges, who were all to be abolished), and to cancel all debts. On Apr. 16, 1802, he appeared before the Minster of Bern with a crowd of adherents, to whom he had predicted the occurrence of some great event. The tumult was suppressed, and Unternährer was condemned to two years' imprisonment. On his release he was received by his adherents with enthusiasm, and riots again occurred. For five years Unternährer was confined in Lucerne as a lunatic. He returned to the world more collected and more serious, but by no means cured, and in 1820 he was permanently confined in the jail.
Unternährer's publications comprise about fifteen pamphlets, including, with others, Gerichtsbüchlein; Buch der Erfüllung; and Geheimniss der Liebe. He taught that the primitive relation between God and man was expressed in the two commandments, to love and multiply, and to abstain from the tree of knowledge. Tempted by Satan, man violated the second commandment and attained great wisdom, which is the curse of mankind. It began with the distinction between good and evil, and ends in institutions innumerable—State, Church, courts, schools, and the like. From the curse there is only one means of salvation; namely, through the fulfilment of the first commandment, to love and multiply; and for this purpose all restraints arising from such ideas as marriage, family, etc., must be thrown off. The principal seat of the sect was Amsoldingen, whence it spread to Gsteig, near Interlaken. Suppressed here in 1821, it reappeared at Wohlen, near Bern, in 1830, under the leadership of Benedict Schori, and again at Gsteig, in 1838-40, under the leadership of Christian Michel. Severe measures were necessary to suppress its excesses.
Bibliography: J. Ziegler, Aktenmässige Nachricten über die sogenannten Antonisekte im Kanton Bern, in Trechsel, Beiträge zur Geschichte der schweizerischen reformirten Kirche, iii. 70 sqq., Bern, 1842; G. Joss, Das Sektenwesen im Kanton Bern, ib. 1881.

redpajama_set_name: RedPajamaCommonCrawl


As mentioned here, one of my 2014 resolutions is to stay healthy and be fit. I am not in any way, shape or form trying to lose weight. My goal is to adopt a healthier lifestyle and become more active. We all know diet and exercise go hand-in-hand. You just simply cannot achieve a healthy lifestyle without personal fitness and nutrition. Three months ago, it was not unusual for us to make 3,4 trips to a local grocery store a week and believe me, it can be frustrating from time to time. Planning is key when it comes to building a healthy diet. I started meal planning recently and wanted to share a few tips.
- Eliminate the question "What's for dinner tonight?"
1. Plan the menu: I usually spend 30 minutes on a Sunday morning and plan our weekly meals. There are a handful of trusty favorites I keep on rotating. A few of my go-to cooking sites are: The skinny taste, The pioneer woman cooks and Food network.
2. Take inventory: I look through my pantry and fridge first to make sure I only buy what I need. I use a free app called "Shop List" as my shopping list.
3. Shop: It's important to go grocery shopping on a weekly basis. If possible, try your best to only buy what you need.
4. Cook: We try to stick to lean meats (i.e: chicken and fish). They provide plenty of protein, without much saturated fat. I will talk about ways to cook healthy meals in a later post. I use an app called My Fitness Pal for calorie tracking.
5. Store: Keep prepared meals in containers to use throughout the week.
I'll have to use that grocery app you featured! I always have to email myself my list and I hate not being able to check things off! Your meals look scrumptious, Sydney! What a lucky H!
Great post Sydney. I'm a big fan of meal planning myself and I love seeing your meal ideas. The teriyaki salmon looks delicious. I use the "Our Groceries" app because you can sync your list with your husband so if he uses up the last of something, he goes into his phone and enters it and it shows up on your list too. Very convenient for keeping track of what we need in our house!
Awesome tips! I totally have to meal plan, too, before I go to the grocery store, otherwise how will you know if you've gotten everything! I hadn't heard of the Shop List app before - thanks so much for sharing!
Wow, you even know the calorie count? That's impressive!
I tried out the app you suggested today. Pretty neat! I like that I can use the categories and enter an amount.
Meal planning for me is usually come up with 3-4 dinner ideas. I usually do not make lists, I just know what to get based off of what I want to make for dinner. Most of the other groceries are so habitual (oatmeal, yogurt, berries...), so I do not forget stuff too often. But I like the idea of using an app when I have a longer list of things that I am not used to getting.
Love this post! I find it fascinating how people grocery shop and prepare daily meals. Love your organizational techniques!
Hi Sydney! I'm a new blogger but a long time reader of your blog! This is my first comment, and I cannot agree with you more about weekly meal planning. Like you, I used to go to the grocery 3-4 times a week as well and I always decided last minute on what to make. Ever since I had to adapt a gluten free and dairy free diet, I find that it's helping me a lot by planning out my weekly menu and buying what I need. I cook a lot of Asian food so I will go to the Asian market and the grocery store for my other staples like milk, etc.
@Miss Hot Couture Hope you enjoy blogging so far. It takes some time for meal planning but saves so much time and money in the long run.
These are some pretty great tips Sydney.


The IIBA Ottawa-Outaouais Chapter is pleased to host a special, structured evening that lets job seekers, recruiters and employers in our community meet one another. You can network with like-minded peers and meet with local recruiting agencies and employers in a "Speed Recruiting" setting. "Speed Recruiting" works a lot like "Speed Dating" but without the 'Dating'. Participants engage in a series of short conversations in order to determine whether there is potential for employment opportunities.
Each of the employers/recruiters will be stationed at their own table. Job seekers will initially be split amongst the tables and will have a set amount of time (5 – 6 minutes) to speak with the employers / recruiters. Job seekers will rotate, meeting with other employers / recruiters. All job seekers and employers / recruiters will have a chance to meet each other over the course of the evening. Once the "Speed Recruiting" has completed, all participants will be able to enjoy the rest of the evening meeting peers and networking.
6:45 to 7:00 pm – Door Prize Draw!
IIBA Members $11 | Non-Members $16 | Monthly Meeting Pass-holders are pre-paid.
This event is sponsored by the IIBA Ottawa-Outaouais Chapter.


An absolute scandal. That's what it is.
"Dancing pig". "A dancing dis-arse-ter". How are you supposed to carry on blithely with such comments ringing in your ears?
The fact beaming John Sergeant has managed to appear onscreen each Saturday sporting a smile above the spangles – and, in the process, brought a national feelgood factor unseen since Jonathan Ross took three months' leave - is testament alone to his strength of personality.
The four Strictly judges should hang their heads in shame. They've driven the people's choice out of the competition.
Their tactless championing of fancy footwork over feelgood frippery has cast a huge cloud over what is – no matter what anybody says – an ENTERTAINMENT show.
John says he feels forced to step down because there's now a real danger he might win the show. So what? His performances are often more entertaining than some of his self-regarding rivals' combined.
Besides, no matter what the judges said John COULD dance and we could see it. Perhaps he wasn't so technically brilliant as Luvvie Lisa Snowdon and co but the vicious criticism and lowly marks he received were wholly unjust and prompted his massive popularity.
For the past few series, Strictly's voting viewers and po-faced judges (who decide, between them, who goes) have been locked in an arm wrestle. There have even been rule changes to suit Craig, Arlene, Len and Bruno, with rumours of more next year.
Matters came to a head in the case of John. The meaner they were, the more we loved him. He was even blamed for good dancers falling by the wayside… but surely his continued presence only gave them a week's grace, at best?
It's purely the presence of ordinary Joe dancers like John that keep millions tuning in – and voting - week after week. In fact, could the BBC please offer us a refund on the cash we spent supporting him?
If Strictly were just about the "darnce"… like its predecessor Come Dancing, most of us would waltz off to the X Factor. Now John's gone… well, we may do just that.


1 Review from SA-Venues.com travellers about this Cape Flats destination. Learn about Langa from other travellers or allow your experiences to guide future visitors to this town by submitting a review.
Langa is a great community full of inspirational entrepreneurs! Everything they do helps their community and they fully support one another. Everyone we met in the township was very friendly and enjoyed telling their personal stories. I did a tour with Ubizo Tours and finished the tour at Mzansi Restaurant. Had a great experience with both! The restaurant made accommodations for our group members that were vegetarian and it was very affordable! During our meal, there was a live band playing and at the end our group learned how to play the instruments and danced with them. Overall it was a great experience and will be going back soon!
Have you been to Langa in Western Cape? Let us know how you enjoyed your visit to the Cape Flats region and what you thought of the attractions, restaurants and accommodation options in Langa. All reviews are subject to our review guidelines.


text:
MICHAEL B./CLIMATE SOLUTIONS
Third time's a charm for clean fuels in Washington
The Washington State Senate voted to pass a Clean Fuel Standard late Thursday night, 27-20, marking a victory for people's health, the environment, and economic recovery. This is the first time the Senate has voted on the policy, after a majority of the House of Representatives passed an earlier version of the governor-request legislation on February 27th. The House also passed it in the 2019 and 2020 legislative sessions. Two-thirds of Washingtonians support a Clean Fuel Standard.
The Senate vote was the culmination of a long day of floor debate on several meaningful climate policies, and followed hours of Senators rejecting dozens of amendments attempting to significantly weaken the policy.
"Climate change is profoundly affecting our lives and our quality of life," said Senator Reuven Carlyle (D-36). "And it is real, and it is happening on every level. At the same time, it can feel like we can't affect change, but we can. There is something extraordinary in our opportunity to lead, our opportunity to be, as my people, the Jewish people, say, 'a light among the nations,' and to show how we can perform good deeds in the world."
"To say we cannot enact our own climate policies here at home until another country does their part is unconscionable," said Senator Liz Lovelett (D-40) during floor debate Thursday evening. "We have to do our part here in Washington."
"We are doing this for the kids, for kids like Jessie, who loves soccer more than anything else in the world. She tries to go to soccer practice every day after school, but she just can't do it because she doesn't have the breath to play soccer because she has asthma. Her parents can only afford an apartment close to the freeway," explained Senator June Robinson (D-38), who volunteers in the Everett School District when not working in the Legislature. "We're doing this for the kids who cannot go to school, who cannot breathe when the air is unhealthy. Their childhoods are compromised. These are the kids we are passing this bill for tonight."
Senator Carlyle also added, "The work of taking meaningful action is not easy. It feels overwhelming at times... I want to thank my good friend Representative Joe Fitzgibbon (D-34) for his decade-long work on this issue and Governor Inslee for his support."
Because the House passed a stronger version of the policy, that chamber next needs to either concur on the bill in its current amended form or enter conference to negotiate with the Senate to finalize the bill. Several legislators aim to ensure a strong standard that can quickly work alongside the Clean Fuel Programs in California, Oregon and British Columbia and meet the state's carbon reduction goals set into law last year. Oregon doubled its standard last year, and New Mexico and New York are also considering similar policies.
Earlier in the day, Representative Fitzgibbon and Senator Rebecca Saldaña joined a virtual rally with clean fuels supporters highlighting the delivery of close to 5,000 names of Washingtonians calling on the Legislature to pass a strong standard this session. There were signatories from over 250 cities and towns, all 49 legislative districts, and 38 out of 39 counties.
A Clean Fuel Standard reduces air and climate pollution by requiring transportation fuel producers to reduce the carbon intensity in their products or invest in the production of cleaner fuels, such as sustainable biofuels and electricity. The transportation sector is the largest source of carbon pollution in the state.
ABOUT THE CLEAN FUELS COALITION
Hundreds of organizations, local elected officials and jurisdictions, health associations and hospitals, small businesses, unions and advocacy groups support clean fuels and have been working for several years to see the solution come to Washington. @cleanfuelswork cleanfuelswork.com
ABOUT THE ENVIRONMENTAL PRIORITIES COALITION
The Environmental Priorities Coalition is a network of over twenty leading environmental groups in Washington state that influence policy at the state level. For over a decade, the Coalition has selected joint priority issues to work on during the legislative session to help focus environmental community resources and best achieve our shared goals. @epctweets environmentalpriorities.org
Connect with Climate Solutions
The article was written by Climate Solutions staff. Our individual bios can be found here.
What's next for climate in Washington?
Protect Northwest farmworkers from extreme heat
Oregon commits to 100% clean and renewable electricity
Climate action at an inflection point
Give for a brighter future
Please email development@climatesolutions.org or contact Savitha Reddy Pathi, Deputy Director, at 206-854-6630.
Washington State Government


Is ScalaMock thread-safe? Can mocks be used in a multi-threaded environment?
but any such calls must be completed before the test completes - it's an error to call a mock afterwards.
Can I mock final / private methods or final classes?
This is not supported, as mocks generated with macros are implemented as subclasses of the type to mock. So private and final methods cannot be overridden. You may want to try using an adapter or façade in your code to make it testable. It is better to test against a trait/interface instead of a concrete implementation. There are libraries that support this kind of mocking, such as PowerMock. Be aware that this kind of mocking involves Bytecode manipulation, which has the risk that your test double diverges from the actual implementation.
Can I mock val / lazy val?
No, the Scala compiler will not allow overriding a val with a def, so with ScalaMock this is not possible. An option we are looking at for a possible future verion of ScalaMock is scala.meta, but this is not yet available for all build targets we want. If you can, it is better to design a trait with a def and mock that instead. The concrete implementation can still override that def with a val to give invariant behaviour.
No, macros cannot generate a subclass of a singleton object, sorry. A way to refactor your code may be to create a trait with the function(s) you want to mock and extend that with the object. Or use Dependency Injection (with classes or functions) to abstract your code and make it testable.
Can I mock Java APIs with raw types?
Can I mock hashcode, equals, clone or toString ?
No - all methods from java.lang.Object are filtered out during Mock creation. toString is particularly problematic as it can cause an infinite loop blowing up the stack on test failure.
override def clone(cloneName: String): Resource = ???
Can I mock static Java calls?
We recommend you use Dependency Injection, higher-order functions or OO patterns (interfaces, facades, etc) instead of depending directly of implementations. We are not considering adding support for this feature as static calls are effectively procedural composition and not good style.


text:
Iran begins producing rare earth
Iran has launched a pilot project for extraction of rare earth after obtaining the technology for deep-level mining, the head of Iran's largest holding in metals sector known as IMIDRO says.
The achievement will create significant added value for Iran, Khodadad Gharibpour said at the inauguration of the landmark project on Tuesday.
Every year, Iran imports 180 tonnes of rare earth elements which are used in a wide range of consumer products such as catalysts in cars and oil refineries, televisions, superconductors and fiber optics.
Gharibpour was thankful to Ministry of Industry, Mine and Trade's support for technology and research, saying Iran was able to obtain the know-how for extraction of these elements which require deep-level mining and are costly to process.
"The price of each kilo of these elements is between $5 and $60, the production of which can bring in a lot of hard currency," he said.
Deputy head of the Atomic Energy Organization of Iran Pejman Rahimian said the launch of the pilot plan for extraction of rare earth elements was a "big event".
The Islamic Republic, he said, has accommodated "the best facilities, exploration equipment and investment" which will be provided to those active in the field.
"Rare earth elements are of particular importance and they are also important in the nuclear industry," he said.
Iran, Rahimian said, had found a rare earth deposit in Saghand which is already known for its uranium mine in the central province of Yazd.
"If the processing plant is set up in this mine, good resources will be provided to the country," he said.
Strategic use
Rare earth elements are a group of mostly strategic metals which appear in low concentrations in the ground. They have lately emerged as one of the fronts in an escalating trade war between China and the United States.
"During the recent US trade war with China, one of the places where China threatened America with is the embargo on the export of rare earth elements," Rahimian said.
The metals are used in a broad range of consumer products, from iPhones, advanced ceramics, computers, DVD players, wind turbines, televisions, lighting and glass polishing to electric car motors.
Some rare earth minerals are essential in military equipment such as jet engines, missile guidance systems, missile defense systems, satellites, as well as in lasers.
Companies such as Raytheon, Lockheed Martin and BAE Systems all make sophisticated missiles that use rare earths metals in their guidance systems, and sensors.
Iran has already been trying its hand in the production of titanium, joining an elite club of countries which produce the strategic metal used in a range of high-tech manufacturing from military aircraft to artificial limbs.
Iran to become strategic metal producer
Mining in depths of 3,000 meters
Iran is tapping its mines and metals deposits like never before as it feels the pressure of the American sanctions that have hampered its access to normal oil revenues.
That comes as the United States has specifically targeted Iran's trade and production of metals with a series of bans since May 2018.
Gharibpour said Iran currently has in place deep-level mining on its agenda. IMIDRO subsidiary Iran Minerals Production and Supply Company (IMPASCO) is now exploring minerals in depths of up to 3,000 meters in central Iran, he said.
"The methodology for deep-level exploration is being developed in the country," said the official who also cited treatment of large mineral tailings as another key focus of Iran's mining operations.
The country possesses 7% of the world's total mineral reserves worth about $700 billion but officials say this figure could rise to $1.4 trillion with new discoveries.
After the lifting of sanctions in 2016, officials unveiled plans for $29 billion of mining investment for a number of projects ranging from steel to aluminum, copper, gold, rare earth elements and coal.
Several multi-billion mining projects with the participation of Italian, French and Chinese companies were in the pipeline. Italy's Danieli had signed a joint venture and agreed orders worth about 5.7 billion euros during President Hassan Rouhani's tour of Europe in January 2016.
However, unilateral US sanctions upended many of them as European companies turned their back on their governments which had pledged to safeguard trade connections with Iran.
Officials say while the sanctions have slowed the ambitious plans, they have not taken the wind out of Iran's sails in mining.
The country's precious metals sector has emerged as the key frontline in the battle, with the Trump administration firing the latest shot early this month.
Washington imposed new sanctions on the sector, targeting the construction, manufacturing, textiles, mining, aluminum, copper, iron and steel industries, Treasury Secretary Steven Mnuchin said.
A senior official dismissed them out of hand, however, saying Iran was on course to export 10 million tonnes of steel in the year to March 21, 2020.
The country is a leading producer of steel in the world with 35 million tonnes of production capacity. It plans to raise this figure to 55 million tonnes a year by 2025, of which 20 to 25 million tonnes would be earmarked for export.
Iran's steel exports increased up to 24% German's BHF-Bank begins long-term coop. in Iran Iran, Russia ink $1b oil rig deal Iran, Poland sign health cooperation document in Warsaw
February 1, 2020 IRAN THIS WAY BussinessIMIDRO , IRAN , Mining , rare earth
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4QFY17 net profit off 53% y‐y: SCCC put in a weak performance in 4QFY17 as net profit tumbled 53% y‐y largely because of a dramatic drop in exceptional gains. In this period, SCCC recorded a FX gain of only Bt1.8mn against the Bt342mn divestment gain it earned in the like year‐ago period. Stripping out ex‐items, its core profit still declined 22% y‐y as sustained high levels of operating expenses took a bite out of margins though sales jumped 18% y‐y on the back of rising sales volumes following the acquisition of Vietnamese cement manufacturer Holcim. Operating expenses especially interest expenses shot up due chiefly to the increased debt load to fund the above acquisition deal. Despite a weaker profit performance, SCCC declared a final dividend of another Bt4/share from its 2HFY17 operations, taking the full‐year dividend to as high as Bt10/share after it paid an interim dividend of Bt6/share in 1HFY17. In our view, the dividend was pretty high compared to its full‐year EPS of Bt6.59/share. The dividend would be paid out of retained profits. Nonetheless, given its current profitability levels and financial position, it seems to us that huge dividend payout may not be good for SCCC in terms of sufficient financial capacity to fund its future investment plans.


What crosses your mind when you hear people talking about paperless offices? Sounds like science fiction, right. The fact is that we can't go completely paperless, but some huge steps have been made in that direction.
The majority of us have witnessed the decline of paper usage in offices, our desks are becoming less cluttered and we rely on digital documents more than before. The most used digital document type is a PDF, standing for the portable document file. The whole idea of this document type is that it can be shared on every operating system and device. It doesn't matter if you are using Windows, Mac or your Smartphone, the document will look the same.
The drawback of using PDFs is that they can be hardly edited. When you're working with multiple PDF documents on a daily basis, you often need to make some changes on them. It would be very inconvenient if you need to write a new document from the beginning every time because you can't edit the old one. There are some software and online tools that can help you with those tasks by converting PDF to more suitable file type, but now there is a specially dedicated solution for your mobile phone, as well.
The developers that know a thing or two about PDFs, like Cometdocs, are dedicated to improving the way you handle your digital documents. They are making converters tools and apps that are helping millions of users to improve the way they handle PDF documents.
One of their most popular apps is most certainly PDF to Word Converter. As the name says, it can convert any PDF, scanned or regular, to an editable Word document. When you think about it, that is a neat solution and the possibility to do that from your phone sounds even better.
This app is free to download and free to use. You have no limit on free conversions, although you would need to convert one document at the time. Also, it doesn't matter how big the file for conversion is. That means you can convert even whole book or manual that has over 100 pages. This is an amazing feat, the time and effort that you save by using this app are enormous.
PDF to Word is using one of the best converting servers enabling to convert complex PDF with ease. The quality of conversion is in most cases exactly similar to the original file and better the quality, the less time you spend correcting errors. While there is still no app or tool that can do the perfect conversions every time, PDF to Word converter is on a good path of becoming one such app.
Optical Character Recognition Technology, abbreviated OCR, has become standard in converter apps and cam scanning apps. It enables that scanned PDFs can be converted as well, apart from original PDFs. You can basically take a picture of any document with text, as long as you save it as a PDF format, the OCR technology will recognize the character from the text and will convert them to a Word document.
The Cloud Services allow us to store our files in the cloud instead of files being stored in our device's memory. Immediate access to files from Google Drive, Onedrive, Dropbox and Box directly from the app is a great example how mobile apps are becoming easier to use rather than some robust software or online tools. It makes document manipulation a lot easier when you can just sign in to your Google Drive and select the file you want to convert, instead of downloading the file first and then uploading it.
To find out more about this app and how to use it, please follow our detailed step-by-step tutorial below.
First, you need to download the app, so head to the Play Store and type "PDF to Word" and you should see this result.
Once you install the app, you will see an uncluttered interface that offers a few information. You will only have two options to add a new file for the conversion and to see the old files that had been converted before. Since this was the first time we used the app we had a screen like this.
The first step in using this app is to select the file for conversion. There are two options, to upload the file from the device or to upload the file from DropBox, Google Drive, Onedrive and other cloud services.
After selecting the file for conversion, the upload will begin. It takes a couple of seconds for the file to upload, depending on the size of the file and the Internet connection. Then you can close the app and simply continue using your phone like you did before because the conversion isn't performed on your phone, but on the conversion server, so your device's memory and battery aren't used in the process.
The conversion time varies, depending on the size and complexity of the PDF. In the end, you will get a Word document that looks like the original PDF, with identical pictures, text, and tables. The next step is to simply open the file and edit it further using MS Word app on your phone.
And that is how you convert PDF to Word on your smartphone. If you find this app useful you can download it from Play Store.
The app is also available in the App Store.
When dealing with PDFs we face difficulties in terms of editing files. People have a need for good and reliable tools to help them overcome these problems and PDF to Word Converter is the right one. It can convert any file for free with excellent conversion quality and most importantly you can do all that from your phone on the go.


STRADAL is a company fully dedicated to concrete products.
STRADAL is part of CRH, international group focused on construction materials and products, world leader for concrete precast products.
STRADAL has been manufacturing concrete sleepers for more than 50 years in its 2 plants located in Migennes and Bergerac : twin-blocks since the 50's, prestressed monoblock since 1973, bearers for switches and crossings since 1991.
STRADAL operates on following markets : renovation and modernization of classical tracks, construction of new lines, mainly high speed lines, construction of new tramway and metro lines.


text:
NOWCAST NewsCenter 5 at Noon
4th human case of West Nile Virus diagnosed in Massachusetts
Risk level elevated to "high" in 11 communities
Mary Saladna
WEBVTT MARY? REPORTER: YEAH, EMILY AND BEN, NEWTON IS ONE OF THOSE 11 COMMUNITIES ALTHOUGH OFFICIALS HERE CANNOT CONFIRM THAT THE LATEST CASE IS ACTUALLY FROM NEWTON, BECAUSE OF PRIVACY RESTRICTIONS. WE DO KNOW THAT THE FOURTH CASE IS A WOMAN IN HER 50'S FROM MIDDLESEX COUNTY. WE ALSO KNOW THAT SEVERAL OF THE FOUR REPORTED CASES HAVE COME FROM THE SAME GENERAL AREA, WHICH IS WHY THAT THREAT LEVEL HAS BEEN RAISED IN NEWTON AND SURROUNDING COMMUNITIES. THE MOST FOLKS WHO GET WEST NILE WILL NEVER EVEN KNOW IT, OR THEY'LL EXPERIENCE MILD SYSTEMS, BUT IT CAN POSE A SERIOUS HEALTH RISK TO CERTAIN POPULATIONS. >> WE WORRY ABOUT THE SAME PEOPLE THAT WE WORRY ABOUT WITH OTHER TYPES OF ILLNESSES, OUR VERY YOUNG AND THEN OUR OLDER ADULTS OR PEOPLE WHO ARE IMMUNOCOMPROMISED. THOSE ARE THE FOLKS WHO WANT TO THINK ABOUT THESE PRECAUTIONS. REPORTER: AGAIN, THESE ARE THE 11 COMMUNITIES AT HIGH RISK FOR WEST NILE VIRUS. THAT MEANS EXTRA PRECAUTIONS ARE IN ORDER SO YOU DON'T GET BIT BY MOSQUITOS. STAY INSIDE AT DAWN AND DUSK. WEAR LONG SLEEVED CLOTHING. GET RID OF STANDING WATER ON YOUR PROPERTY AND CARRY AROUND A LOT OF THIS AND WEAR IT. WE'RE LIVE IN N
Eleven communities are at high risk for West Nile Virus and four human cases have been diagnosed so far this year, according to the Massachusetts Department of Public Health.The new patient is a woman in her 50s from Middlesex County, officials said. Three other cases were reported Friday. Mass. DPH officials said at least two of the four patients were exposed to the virus in greater Boston area, which led them to raise the risk level for 11 communities from moderate to high.Included communities are Arlington, Boston, Belmont, Brookline, Cambridge, Chelsea, Everett, Medford, Newton, Somerville and Watertown. Residents in those communities are advised to exercise precautions, such as wearing insect repellent, draining standing water and limiting exposed skin. Last year, health officials identified six human cases of the virus, which is usually transmitted through mosquito bites."Several individuals from the same area have developed West Nile virus," said Public Health Commissioner Dr. Monica Bharel. "That means that there is an increased risk in this specific area and that additional people could become infected. We are particularly concerned about people over 50 and those who are immunocompromised as they are the ones most likely to develop WNV disease."
NEWTON, Mass. —
Eleven communities are at high risk for West Nile Virus and four human cases have been diagnosed so far this year, according to the Massachusetts Department of Public Health.
The new patient is a woman in her 50s from Middlesex County, officials said. Three other cases were reported Friday.
Mass. DPH officials said at least two of the four patients were exposed to the virus in greater Boston area, which led them to raise the risk level for 11 communities from moderate to high.
Included communities are Arlington, Boston, Belmont, Brookline, Cambridge, Chelsea, Everett, Medford, Newton, Somerville and Watertown. Residents in those communities are advised to exercise precautions, such as wearing insect repellent, draining standing water and limiting exposed skin.
Last year, health officials identified six human cases of the virus, which is usually transmitted through mosquito bites.
"Several individuals from the same area have developed West Nile virus," said Public Health Commissioner Dr. Monica Bharel. "That means that there is an increased risk in this specific area and that additional people could become infected. We are particularly concerned about people over 50 and those who are immunocompromised as they are the ones most likely to develop WNV disease."


My nephew sent me a text informing me that this week Sermon Audio featured a recent sermon of mine as their "Staff Pick." This is probably the most transparent sermon I've ever preached, which includes lessons I've learned/am learning about bitterness and its remedy, forgiveness.
Thank you Pastor Paul for your wondeful sermon "Bitter Root, Rotten Fruit" – I am a 50 year old father of 4 from Canberra Australia and I was moved this morning to download your sermon from SermonAudio.com – I listened to it on the way to work this morning. Wow – I am so glad that I did. Such honest, personal, Biblical and compeling preaching. My prayer is that the Holy Spirit continues to bless you, your preaching gift, your family and your Church. Yours in Christ, Michael.


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Animal Kingdom Season 5 Episode 9: What to Expect?
Nishitha Dutta
In the eighth episode of 'Animal Kingdom' season 5, the Codys fight over an issue related to the mishandling of their latest coke shipment. As a result, harsh words and punches are thrown around. J is just about to leave when Pope interrupts and forces him not to. The highlights of episode 8 are mentioned in the recap section. Now, here is what episode 9 might bring to the table!
Animal Kingdom Season 5 Episode 9 Release Date
'Animal Kingdom' season 5 episode 9 is set to release on September 5, 2021, at 9 pm ET on TNT. The fifth season has 13 episodes that run for 45–55 minutes each. New episodes land every week on Sundays.
Where to Watch Animal Kingdom Season 5 Episode 9 Online?
You can watch 'Animal Kingdom' season 5 episode 9 on TNT at the aforementioned date and time. You can also catch the episode on the official website of TNT and the TNT app. The episode can be streamed live through Hulu+Live TV. If you do not have access to cable television, you can watch the episode on DirecTV, YouTube TV, Spectrum on Demand, Xfinity, and Sling TV. Additionally, you can purchase individual episodes or an entire season on Amazon Prime Video, iTunes, Google Play, Vudu, and Microsoft Store.
Animal Kingdom Season 5 Episode 9 Spoilers
In the ninth episode, titled 'Let It Ride,' the Codys will find themselves under a lot of pressure as local authorities actively pursue them. There are secrets that could fracture the dynamic among the family members. Meanwhile, Deran will hear news about Adrian while Craig will realize that Renn hasn't been entirely transparent with him.
Craig has also been extremely domineering and authoritative in his relationship with Renn. This could imply that they might part ways. In the upcoming episode, we might see Craig, J, Deran, and Pope might leave their differences aside and join hands now that the police are constantly on their tail. Here's a promo for episode 9!
Animal Kingdom Season 5 Episode 8 Recap
The eighth episode of 'Animal Kingdom' season 5 is titled 'Gladiators.' In the episode, we see an old memory featuring young Smurf in a jewelry store when three men barge in and rob the place. They steal the security footage before leaving and climb into a getaway truck, revealing Janine as one of the masked robbers. Smurf quickly gets inside the same vehicle. Back in the present, we see Pope pulling his gun on Liam before leaving him stranded in the desert. J looks everywhere but is unable to find Deran. He then meets Craig and tells him that Livengood's card has ended up with Pete.
Deran gets the coke transferred. However, Craig later informs him that the drugs reached Pete, who flushed it down. In the past, the gang discards the robbery vehicle in the desert, and Craig's motivation to keep stealing pisses off Jake's wife. Henry confronts Pope about having killed his father, but the former denies it. He is then taken to Cassandra, who takes some money to go past the Canadian border. In the present, Deran, Craig, and J finally gather at the Cody headquarters.
Deran is disappointed in J for moving the coke to Pete's, and the altercation slowly escalates into violence. Pope almost loses his life, and Deran later asks him if he intends to get better. He nods, and they both hug. J, however, is not ready to reconcile with the others and is just about to leave when Pope stops him. When J does not listen, Pope threatens to track him down and drag him back home.
Read More: Is Animal Kingdom a True Story?


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The Rumor Game (Hardcover)
By Dhonielle Clayton, Sona Charaipotra
Forthcoming - Available for Pre-Order, Check Format + Release Date
"A juicy, elegant, absolutely flawless thriller with a twist. It's the diverse Gossip Girl we've been waiting for!" —Tiffany D. Jackson, New York Times best-selling author of Grown and White Smoke
"Explosive." —Entertainment Weekly
All it takes is one spark to start a blaze.
At Foxham Prep, a posh private school for the children of DC's elite, a single rumor has the power to ruin a life.
Nobody knows that better than Bryn. She used to have it all—the perfect boyfriend, a bright future in politics, and even popularity thanks to her best friend, cheer captain Cora. Then one mistake sparked a scandal that burned it all to the ground.
Now it's the start of a new school year and the spotlight has shifted: It's geeky Georgie, newly hot after a summer makeover, whose name is on everyone's lips. When a rumor ignites, Georgie rockets up the school's social hierarchy, pitting her and Cora against each other. It grants her Foxham stardom . . . but it also makes her a target.
As the rumors grow and morph, blazing like wildfire through the school's social media, all three girls' lives begin to unravel. But one person close to the drama has the power to stop the gossip in its tracks. The question is—do they even want to?
From Dhonielle Clayton and Sona Charaipotra, authors of the Tiny Pretty Things duology (now a Netflix series), comes another edge-of-your-seat social thriller perfect for fans of We Were Liars and Cruel Summer.
Dhonielle Clayton is the New York Times best-selling author of the Belles series, The Mirror: Shattered Midnight, and the coauthor of Blackout and the Tiny Pretty Things duology, now a Netflix original series. She hails from the Washington, DC, suburbs on the Maryland side. She taught secondary school for several years, and is a former elementary and middle school librarian. She is COO of the non-profit We Need Diverse Books, and president of Cake Creative, an IP story kitchen dedicated to diverse representation. She's an avid traveler, and always on the hunt for magic and mischief. Up next: The Marvellers, her middle grade fantasy debut. You can find her on social media @brownbookworm.
Sona Charaipotra is the author of Symptoms of a Heartbreak and How Maya Got Fierce, and coauthor of the Tiny Pretty Things duology, now a Netflix original series. She earned her master's in screenwriting from NYU and an MFA in creative writing from the New School. A working journalist, Sona has held editorial roles at People, TeenPeople, ABCNews.com, MSN, and most recently, the Barnes & Noble Teen Blog (RIP), and contributed to publications from the New York Times to TeenVogue. She is a former We Need Diverse Books board member, and co-founder of CAKE Literary, a boutique book packager focused on high-concept diverse titles. Find her on the web talking about books, Bollywood movies, and chai.
"A juicy, elegant, absolutely flawless thriller with a twist. It's the diverse Gossip Girl we've been waiting for!"—Tiffany D. Jackson, New York Times best-selling author of Grown and White Smoke
Publication Date: March 1st, 2022
Maximum Grade Level: 12
Young Adult Fiction / Social Themes / Friendship
Young Adult Fiction / Social Themes / Sexual Abuse


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Whisk the coconut cream, the water and the vanilla.
Simmer, but don't let it boil.
In a separate bowl, whisk the yolks and the honey.
Add the hot mixture to the bowl, slowly, tempering.
Bring it all to the saucepan on medium temperature and keep whisking until it reaches 170F.
Remove from the heat and put it in a cold bath for 10-15 minutes.
Move it to the fridge.
Blend the strawberries in a food processor.
Using two spoons, scoop them out into a bowl.
Add a couple mint leaves to decorate.


Tuesday, November 13 - The WannaBlog!
Students returned to the library during both their Social Studies and Language Arts periods today, to complete our activities related to the civil rights movement and the life and actions of Dr. Martin Luther King, Jr.
As a reminder, students who need to take the quiz will do so during class tomorrow. So those students should be studying at home, using the materials that have been mentioned in previous Daily Updates.


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Carter Jones Does It Again!
Image credit: VeloNews
Optum pro Carter Jones, coached by PCG master coach Dr. Stephen McGregor, notched three top-ten performances at the 2014 Tour of Utah that culminated with a sixth place on the final stage and seventh on GC. The Tour of Utah has come to be known as America's toughest stage race, and this year it lived up to its billing. With most of the race above 6,500 feet and including climbs with extensive grades between 10 and 20% topping out at 9,000 feet, it's a race for the toughest of stage racers.
Jones was the Optum team's GC hope, and he didn't disappoint, with an eighth place finish on the first critical stage (stage 4) that finished on the brutal Powder Mountain climb. On stage 6 to Snowbird ski resort, a break including former Tour de France champion Cadel Evans got away, and although Carter was fifth out of the main contenders, Evans jumped over him on GC. On the final stage that climbed Empire Pass and finished in Park City, Jones and Evans battled up the climb in an effort to reel in the leading four riders, including overall winner Tom Danielson and second place Chris Horner. They didn't make contact before the final descent. Evans put on a descending clinic to get the win, and Jones finished sixth, just ten seconds back and five seconds behind Danielson. In the end, Jones finished seventh overall, the top-placed non-World Tour rider.
Next up is the USA Pro Challenge, another opportunity for Carter to demonstrate his climbing prowess. Hats off to you, Carter!
Click here to read more about Carter's win: http://www.cyclingnews.com/races/tour-of-utah-2014/stage-7/results.
Posted by Peaks Coaching Group


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2 Upstate NY Affordable Housing Properties Secure $21M
By Jessica Fiur
Affordable HousingFinance & InvestmentNews
Riverside Charitable Corp. will acquire and rehabilitate Raymond Watkins Apartments in Sarasota Springs and John Guy Prindle Apartments in Ilion.
By D.C. Stribling
Raymond Watkin Apartments
Riverside Charitable Corp. (RCC) has obtained $20.9 million in financing to acquire and rehabilitate two affordable multifamily properties in New York State. One property is Raymond Watkins Apartments in Sarasota Springs, the other John Guy Prindle Apartments in Ilion.
Raymond Watkins Apartments is a 112-unit age-restricted and affordable Section 8 subsidized multifamily property at 57 Ballston Ave. The community offers a mix of one- and two-bedroom units, and some handicap-accessible dwelling units. The $13.95 million loan is for a term of 30 years.
John Guy Prindle Apartments also consists of a total of 112 age-restricted and affordable Section 8 subsidized multifamily units, and is at 80 East North St. Like Watkins, the property offers a mix of one- and two-bedroom units, and some handicap-accessible units. The $7 million loan is for 35-year term.
The borrower was awarded tax exempt bond volume cap and an allocation of 4 Percent Low Income Housing Tax Credits, and will recapitalize both assets as long-term investments with FHA insured financing. Both properties have historically operated with the benefit of a Section 8 Housing Assistance Payment (HAP) Contract, and will continue to do so. In conjunction with the application submissions, the borrower submitted a request for a new 20-year HAP Contract on both properties.
Rehab Work
The total rehab budget is about $3.3 million for Raymond Watkins Apartments and $3 million for John Guy Prindle Apartments. "Both properties are in in good condition and well-maintained considering their age," noted Tim Hoppin, director at Hunt Mortgage Group. "The renovation work will be completed as an 'in place' rehab, with no tenant displacement."
Hunt Mortage, utilizing HUD's 223(f) loan program, facilitated the loans. The rehab will encompass concrete work, parking lot overlay, landscaping embellishments, new signage, full roof replacement, new handicap vestibule door system at one entry, full kitchen cabinetry/countertop replacements, new entry and individual interior unit doors, painting of all common areas and unit interiors, among other improvements.
RCC was founded in 1988 as a 501(c)(3) Nonprofit Public Benefit Corporation as a provider of affordable housing supportive services. Gil Seton Jr. and SP Investment Fund LLC are doing the redevelopment work.
Hunt Mortgage Group