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In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix…
In the current era of vast data and transparent machine learning, it is essential for techniques to operate at a large scale while providing a clear mathematical comprehension of the internal workings of the method. Although there already…
Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent…
This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices,…
Generalized or extended finite element methods (GFEM/XFEM) are in general badly conditioned and have numerous additional degrees of freedom (DOF) compared with the FEM because of introduction of enriched functions. In this paper, we develop…
Value-function (VF) approximation is a central problem in Reinforcement Learning (RL). Classical non-parametric VF estimation suffers from the curse of dimensionality. As a result, parsimonious parametric models have been adopted to…
Non-negative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of non-negative data. Although it has successfully been applied in several applications, it does not always result in…
In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive…
We present a new direct logarithmically optimal in theory and fast in practice algorithm to implement the high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. The key points…
The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the…
Finite element analysis of solid mechanics is a foundational tool of modern engineering, with low-order finite element methods and assembled sparse matrices representing the industry standard for implicit analysis. We use performance models…
We develop novel stabilized cut discontinuous Galerkin (CutDG) methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in $\mathbb{R}^d$ where the domain boundary can cut…
In this work, we develop an optimization framework for problems whose solutions are well-approximated by Hierarchical Tucker (HT) tensors, an efficient structured tensor format based on recursive subspace factorizations. By exploiting the…
We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three…
In this work, the joint use of a mixed penalty-interior point method and direct search is proposed, to address {nonlinear} constrained derivative-free optimization problems. A merit function is considered, wherein the set of nonlinear…
We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a…
We develop a cut finite element method (CutFEM) for convection-diffusion problems posed on mixed-dimensional domains, i.e., unions of manifolds of different dimensions arranged in a hierarchical structure where lower-dimensional components…
This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods…
This paper investigates the efficient solution of penalized quadratic regressions in high-dimensional settings. A novel and efficient algorithm for ridge-penalized quadratic regression is proposed, leveraging the matrix structures of the…
This is a study of certain finite element methods designed for convection-dominated, time-dependent partial differential equations. Specifically, we analyze high order space-time tensor product finite element discretizations, used in a…