Related papers: Low-rank approximation for multiscale PDEs
Statistical learning additions to physically derived mathematical models are gaining traction in the literature. A recent approach has been to augment the underlying physics of the governing equations with data driven Bayesian statistical…
We revisit the use of Stochastic Gradient Descent (SGD) for solving convex optimization problems that serve as highly popular convex relaxations for many important low-rank matrix recovery problems such as \textit{matrix completion},…
Many problems in computational science and engineering involve partial differential equations and thus require the numerical solution of large, sparse (non)linear systems of equations. Multigrid is known to be one of the most efficient…
This work describes a domain embedding technique between two non-matching meshes used for generating realizations of spatially correlated random fields with applications to large-scale sampling-based uncertainty quantification. The goal is…
Inverse problems occur in a variety of parameter identification tasks in engineering. Such problems are challenging in practice, as they require repeated evaluation of computationally expensive forward models. We introduce a unifying…
The numerical solution of parameter identification inverse problems for kinetic equations can exhibit high computational and memory costs. In this paper, we propose a dynamical low-rank scheme for the reconstruction of the scattering…
Elliptic partial differential equations (PDEs) frequently arise in continuum descriptions of physical processes relevant to science and engineering. Multilevel preconditioners represent a family of scalable techniques for solving discrete…
We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We…
LP-type problems such as the Minimum Enclosing Ball (MEB), Linear Support Vector Machine (SVM), Linear Programming (LP), and Semidefinite Programming (SDP) are fundamental combinatorial optimization problems, with many important…
Weak constraint four-dimensional variational data assimilation is an important method for incorporating data (typically observations) into a model. The linearised system arising within the minimisation process can be formulated as a saddle…
This work considers the problem of computing the canonical polyadic decomposition (CPD) of large tensors. Prior works mostly leverage data sparsity to handle this problem, which is not suitable for handling dense tensors that often arise in…
In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart-Thomas finite element spaces. It requires to solve local problems in small…
We propose a multilevel Markov chain Monte Carlo (MCMC) method for the Bayesian inference of random field parameters in PDEs using high-resolution data. Compared to existing multilevel MCMC methods, we additionally consider level-dependent…
This paper presents novel adaptive space-time reduced-rank interference suppression least squares algorithms based on joint iterative optimization of parameter vectors. The proposed space-time reduced-rank scheme consists of a joint…
Memristor crossbars enable vector-matrix multiplication (VMM), and are promising for low-power applications. However, it can be difficult to write the memristor conductance values exactly. To improve the accuracy of VMM, we propose a scheme…
Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems…
We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered…
This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of…
We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…
The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational…