Related papers: An efficient and fast sparse grid algorithm for hi…
The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of…
We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $\Theta(ds \log…
Creating low dimensional representations of a high dimensional data set is an important component in many machine learning applications. How to cluster data using their low dimensional embedded space is still a challenging problem in…
The sparse pseudo-input Gaussian process (SPGP) is a new approximation method for speeding up GP regression in the case of a large number of data points N. The approximation is controlled by the gradient optimization of a small set of M…
In this paper, we study the (decentralized) distributed optimization problem with high-dimensional sparse structure. Building upon the FedDA algorithm, we propose a (Decentralized) FedDA-GT algorithm, which combines the \textbf{gradient…
This paper proposes a mode multigrid (MMG) method, and applies it to accelerate the convergence of the steady state flow on unstructured grids. The dynamic mode decomposition (DMD) technique is used to analyze the convergence process of…
We propose a novel, efficient approach for distributed sparse learning in high-dimensions, where observations are randomly partitioned across machines. Computationally, at each round our method only requires the master machine to solve a…
Generalized inverses play a fundamental role in numerical linear algebra, particularly when matrices are rectangular, singular, or rank deficient. Even when the input matrix is sparse, generalized inverses such as the M-P pseudoinverse are…
This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci.…
We solve Poisson's equation using new multigrid algorithms that converge rapidly. The novel feature of the 2D and 3D algorithms are the use of extra diagonal grids in the multigrid hierarchy for a much richer and effective communication…
In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions of the…
Consider an $s$-dimensional function being evaluated at $n$ points of a low discrepancy sequence (LDS), where the objective is to approximate the one-dimensional functions that result from integrating out $(s-1)$ variables. Here, the…
Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The…
This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-dimensional stochastic…
Standard gradient-based iteration algorithms for optimization, such as gradient descent and its various proximal-based extensions to nonsmooth problems, are known to converge slowly for ill-conditioned problems, sometimes requiring many…
The \emph{deterministic} sparse grid method, also known as Smolyak's method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about…
Reducing the memory footprint of neural networks is a crucial prerequisite for deploying them in small and low-cost embedded devices. Network parameters can often be reduced significantly through pruning. We discuss how to best represent…
Distributed optimization has been widely used as one of the most efficient approaches for model training with massive samples. However, large-scale learning problems with both massive samples and high-dimensional features widely exist in…
This article presents two novel adaptive-sparse polynomial dimensional decomposition (PDD) methods for solving high-dimensional uncertainty quantification problems in computational science and engineering. The methods entail global…
In simulation technology, computationally expensive objective functions are often replaced by cheap surrogates, which can be obtained by interpolation. Full grid interpolation methods suffer from the so-called curse of dimensionality,…