Related papers: Multilevel tensor approximation of PDEs with rando…
A distributed adaptive algorithm is proposed to solve a node-specific parameter estimation problem where nodes are interested in estimating parameters of local interest, parameters of common interest to a subset of nodes and parameters of…
Learning generative probabilistic models is a core problem in machine learning, which presents significant challenges due to the curse of dimensionality. This paper proposes a joint dimensionality reduction and non-parametric density…
In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear time-dependent partial differential equations (PDEs). In our recent work, we introduced a bi-level regularization…
We propose a hierarchical tensor-network approach for approximating high-dimensional probability density via empirical distribution. This leverages randomized singular value decomposition (SVD) techniques and involves solving linear…
We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm…
This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain $D$ such that…
In this work, we propose a new randomized algorithm for computing a low-rank approximation to a given matrix. Taking an approach different from existing literature, our method first involves a specific biased sampling, with an element being…
In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within…
A generalized finite element method is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter $\varepsilon$, based on locally approximating the solution on each subdomain by solution of a…
Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
This paper is devoted to the design and analysis of a numerical algorithm for approximating solutions of a degenerate cross-diffusion system, which models particular instances of taxis-type migration processes under local sensing…
Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion…
We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
Multilevel sampling methods, such as multilevel and multifidelity Monte Carlo, multilevel stochastic collocation, or delayed acceptance Markov chain Monte Carlo, have become standard uncertainty quantification (UQ) tools for a wide class of…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in…
Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media,…