In this paper new innovative fourth order compact schemes for Robin and Neumann boundary conditions have been developed for boundary value problems of elliptic PDEs in two and three dimensions. Different from traditional finite difference operator approach, which may not work for flux type of boundary conditions, carefully designed undetermined coefficient methods are utilized in developing high order compact (HOC) schemes. The new methods not only can be utilized to design HOC schemes for flux type of boundary conditions but can also be applied to general elliptic PDEs including Poisson, Helmholtz, diffusion-advection, and anisotropic equations with linear boundary conditions. In the new developed HOC methods, the coefficient matrices are generally M-matrices, which guarantee the discrete maximum principle for well-posed problems, so the convergence of the HOC methods. The developed HOC methods are versatile and can cover most of high order compact schemes in the literature. The HOC methods for Robin boundary conditions and for anisotropic diffusion and advection equations with Robin or even Dirichlet boundary conditions are likely the first ones that have ever been developed. With the help of pseudo-inverse, or SVD solutions, we have also observed that the developed HOC methods usually have smaller error constants compared with traditional HOC methods when applicable. Non-trivial examples with large wave numbers and oscillatory solutions are presented to confirm the performance of the new HOC methods.
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We introduce a flexible and scalable class of Bayesian geostatistical models for discrete data, based on the class of nearest neighbor mixture transition distribution processes (NNMP), referred to as discrete NNMP. The proposed class characterizes spatial variability by a weighted combination of first-order conditional probability mass functions (pmfs) for each one of a given number of neighbors. The approach supports flexible modeling for multivariate dependence through specification of general bivariate discrete distributions that define the conditional pmfs. Moreover, the discrete NNMP allows for construction of models given a pre-specified family of marginal distributions that can vary in space, facilitating covariate inclusion. In particular, we develop a modeling and inferential framework for copula-based NNMPs that can attain flexible dependence structures, motivating the use of bivariate copula families for spatial processes. Compared to the traditional class of spatial generalized linear mixed models, where spatial dependence is introduced through a transformation of response means, our process-based modeling approach provides both computational and inferential advantages. We illustrate the benefits with synthetic data examples and an analysis of North American Breeding Bird Survey data.
The level set estimation problem seeks to find all points in a domain ${\cal X}$ where the value of an unknown function $f:{\cal X}\rightarrow \mathbb{R}$ exceeds a threshold $\alpha$. The estimation is based on noisy function evaluations that may be acquired at sequentially and adaptively chosen locations in ${\cal X}$. The threshold value $\alpha$ can either be \emph{explicit} and provided a priori, or \emph{implicit} and defined relative to the optimal function value, i.e. $\alpha = (1-\epsilon)f(x_\ast)$ for a given $\epsilon > 0$ where $f(x_\ast)$ is the maximal function value and is unknown. In this work we provide a new approach to the level set estimation problem by relating it to recent adaptive experimental design methods for linear bandits in the Reproducing Kernel Hilbert Space (RKHS) setting. We assume that $f$ can be approximated by a function in the RKHS up to an unknown misspecification and provide novel algorithms for both the implicit and explicit cases in this setting with strong theoretical guarantees. Moreover, in the linear (kernel) setting, we show that our bounds are nearly optimal, namely, our upper bounds match existing lower bounds for threshold linear bandits. To our knowledge this work provides the first instance-dependent, non-asymptotic upper bounds on sample complexity of level-set estimation that match information theoretic lower bounds.
It is nontrivial to store rapidly growing big data nowadays, which demands high-performance lossless compression techniques. Likelihood-based generative models have witnessed their success on lossless compression, where flow based models are desirable in allowing exact data likelihood optimisation with bijective mappings. However, common continuous flows are in contradiction with the discreteness of coding schemes, which requires either 1) imposing strict constraints on flow models that degrades the performance or 2) coding numerous bijective mapping errors which reduces the efficiency. In this paper, we investigate volume preserving flows for lossless compression and show that a bijective mapping without error is possible. We propose Numerical Invertible Volume Preserving Flow (iVPF) which is derived from the general volume preserving flows. By introducing novel computation algorithms on flow models, an exact bijective mapping is achieved without any numerical error. We also propose a lossless compression algorithm based on iVPF. Experiments on various datasets show that the algorithm based on iVPF achieves state-of-the-art compression ratio over lightweight compression algorithms.
We consider a statistical inverse learning problem, where the task is to estimate a function $f$ based on noisy point evaluations of $Af$, where $A$ is a linear operator. The function $Af$ is evaluated at i.i.d. random design points $u_n$, $n=1,...,N$ generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and $p$-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.
Due to the explosion in the size of the training datasets, distributed learning has received growing interest in recent years. One of the major bottlenecks is the large communication cost between the central server and the local workers. While error feedback compression has been proven to be successful in reducing communication costs with stochastic gradient descent (SGD), there are much fewer attempts in building communication-efficient adaptive gradient methods with provable guarantees, which are widely used in training large-scale machine learning models. In this paper, we propose a new communication-compressed AMSGrad for distributed nonconvex optimization problem, which is provably efficient. Our proposed distributed learning framework features an effective gradient compression strategy and a worker-side model update design. We prove that the proposed communication-efficient distributed adaptive gradient method converges to the first-order stationary point with the same iteration complexity as uncompressed vanilla AMSGrad in the stochastic nonconvex optimization setting. Experiments on various benchmarks back up our theory.
We consider the problem of estimating a $d$-dimensional discrete distribution from its samples observed under a $b$-bit communication constraint. In contrast to most previous results that largely focus on the global minimax error, we study the local behavior of the estimation error and provide \emph{pointwise} bounds that depend on the target distribution $p$. In particular, we show that the $\ell_2$ error decays with $O\left(\frac{\lVert p\rVert_{1/2}}{n2^b}\vee \frac{1}{n}\right)$ (In this paper, we use $a\vee b$ and $a \wedge b$ to denote $\max(a, b)$ and $\min(a,b)$ respectively.) when $n$ is sufficiently large, hence it is governed by the \emph{half-norm} of $p$ instead of the ambient dimension $d$. For the achievability result, we propose a two-round sequentially interactive estimation scheme that achieves this error rate uniformly over all $p$. Our scheme is based on a novel local refinement idea, where we first use a standard global minimax scheme to localize $p$ and then use the remaining samples to locally refine our estimate. We also develop a new local minimax lower bound with (almost) matching $\ell_2$ error, showing that any interactive scheme must admit a $\Omega\left( \frac{\lVert p \rVert_{{(1+\delta)}/{2}}}{n2^b}\right)$ $\ell_2$ error for any $\delta > 0$. The lower bound is derived by first finding the best parametric sub-model containing $p$, and then upper bounding the quantized Fisher information under this model. Our upper and lower bounds together indicate that the $\mathcal{H}_{1/2}(p) = \log(\lVert p \rVert_{{1}/{2}})$ bits of communication is both sufficient and necessary to achieve the optimal (centralized) performance, where $\mathcal{H}_{{1}/{2}}(p)$ is the R\'enyi entropy of order $2$. Therefore, under the $\ell_2$ loss, the correct measure of the local communication complexity at $p$ is its R\'enyi entropy.
The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs $G$, $H$, and lists $L(v)\subseteq V(H)$ for every $v\in V(G)$, a {\em list homomorphism} is a function $f:V(G)\to V(H)$ that preserves the edges (i.e., $uv\in E(G)$ implies $f(u)f(v)\in E(H)$) and respects the lists (i.e., $f(v)\in L(v))$. Standard techniques show that if $G$ is given with a tree decomposition of width $t$, then the number of list homomorphisms can be counted in time $|V(H)|^t\cdot n^{\mathcal{O}(1)}$. Our main result is determining, for every fixed graph $H$, how much the base $|V(H)|$ in the running time can be improved. For a connected graph $H$ we define $\operatorname{irr}(H)$ the following way: if $H$ has a loop or is nonbipartite, then $\operatorname{irr}(H)$ is the maximum size of a set $S\subseteq V(H)$ where any two vertices have different neighborhoods; if $H$ is bipartite, then $\operatorname{irr}(H)$ is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected $H$, we define $\operatorname{irr}(H)$ as the maximum of $\operatorname{irr}(C)$ over every connected component $C$ of $H$. We show that, for every fixed graph $H$, the number of list homomorphisms from $(G,L)$ to $H$ * can be counted in time $\operatorname{irr}(H)^t\cdot n^{\mathcal{O}(1)}$ if a tree decomposition of $G$ having width at most $t$ is given in the input, and * cannot be counted in time $(\operatorname{irr}(H)-\epsilon)^t\cdot n^{\mathcal{O}(1)}$ for any $\epsilon>0$, even if a tree decomposition of $G$ having width at most $t$ is given in the input, unless the #SETH fails. Thereby we give a precise and complete complexity classification featuring matching upper and lower bounds for all target graphs with or without loops.
In numerical simulations of complex flows with discontinuities, it is necessary to use nonlinear schemes. The spectrum of the scheme used have a significant impact on the resolution and stability of the computation. Based on the approximate dispersion relation method, we combine the corresponding spectral property with the dispersion relation preservation proposed by De and Eswaran (J. Comput. Phys. 218 (2006) 398-416) and propose a quasi-linear dispersion relation preservation (QL-DRP) analysis method, through which the group velocity of the nonlinear scheme can be determined. In particular, we derive the group velocity property when a high-order Runge-Kutta scheme is used and compare the performance of different time schemes with QL-DRP. The rationality of the QL-DRP method is verified by a numerical simulation and the discrete Fourier transform method. To further evaluate the performance of a nonlinear scheme in finding the group velocity, new hyperbolic equations are designed. The validity of QL-DRP and the group velocity preservation of several schemes are investigated using two examples of the equation for one-dimensional wave propagation and the new hyperbolic equations. The results show that the QL-DRP method integrated with high-order time schemes can determine the group velocity for nonlinear schemes and evaluate their performance reasonably and efficiently.
The list-decodable code has been an active topic in theoretical computer science.There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only on the radius. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the sizes of list-decodable codes. Our results give exponential improvements on the recent generalized Singleton upper bound in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are large. A generalized Singleton upper bound for average-radius list-decodable codes is also given from our general covering code upper bound. Even for the list size $L=1$ case our covering code upper bounds give highly non-trivial upper bounds on the sizes of codes with the given minimum distance. The asymptotic forms of covering code bounds in the Hamming metric setting lead to an asymptotic bound for list-decodable binary codes, which is similar to and weaker than the classical McEliece-Rudemich-Rumsey-Welch bound. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes. We apply our general covering code upper bounds for list-decodable rank-metric codes, list-decodable subspace codes, list-decodable insertion codes and list-decodable deletion codes. Some new better results about non-list-decodability of rank-metric codes and subspace codes are obtained.
Let $\mathcal{M}$ be a smooth $d$-dimensional submanifold of $\mathbb{R}^N$ with boundary that's equipped with the Euclidean (chordal) metric, and choose $m \leq N$. In this paper we consider the probability that a random matrix $A \in \mathbb{R}^{m \times N}$ will serve as a bi-Lipschitz function $A: \mathcal{M} \rightarrow \mathbb{R}^m$ with bi-Lipschitz constants close to one for three different types of distributions on the $m \times N$ matrices $A$, including two whose realizations are guaranteed to have fast matrix-vector multiplies. In doing so we generalize prior randomized metric space embedding results of this type for submanifolds of $\mathbb{R}^N$ by allowing for the presence of boundary while also retaining, and in some cases improving, prior lower bounds on the achievable embedding dimensions $m$ for which one can expect small distortion with high probability. In particular, motivated by recent modewise embedding constructions for tensor data, herein we present a new class of highly structured distributions on matrices which outperform prior structured matrix distributions for embedding sufficiently low-dimensional submanifolds of $\mathbb{R}^N$ (with $d \lesssim \sqrt{N}$) with respect to both achievable embedding dimension, and computationally efficient realizations. As a consequence we are able to present, for example, a general new class of Johnson-Lindenstrauss embedding matrices for $\mathcal{O}(\log^c N)$-dimensional submanifolds of $\mathbb{R}^N$ which enjoy $\mathcal{O}(N \log (\log N))$-time matrix vector multiplications.
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