Related papers: A Sketched Finite Element Method for Elliptic Mode…
A finite element method for elliptic problems with discontinuous coefficients is presented. The discontinuity is assumed to take place along a closed smooth curve. The proposed method allows to deal with meshes that are not adapted to the…
Sketching is a probabilistic data compression technique that has been largely developed in the computer science community. Numerical operations on big datasets can be intolerably slow; sketching algorithms address this issue by generating a…
This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon…
New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…
We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the…
We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational…
We consider distributed convex optimization problems that involve a separable objective function and nontrivial functional constraints, such as Linear Matrix Inequalities (LMIs). We propose a decentralized and computationally inexpensive…
We consider the problem of recovering an $n_1 \times n_2$ low-rank matrix with $k$-sparse singular vectors from a small number of linear measurements (sketch). We propose a sketching scheme and an algorithm that can recover the singular…
We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional…
Kernel methods are learning algorithms that enjoy solid theoretical foundations while suffering from important computational limitations. Sketching, which consists in looking for solutions among a subspace of reduced dimension, is a well…
In this work we propose an efficient and accurate multi-scale optical simulation algorithm by applying a numerical version of slowly varying envelope approximation in FEM. Specifically, we employ the fast iterative method to quickly compute…
Multiple scale homogenization problems are reduced to single scale problems in higher dimension. It is shown that sparse tensor product Finite Element Methods (FEM) allow the numerical solution in complexity independent of the dimension and…
In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new…
In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization,…
This paper discusses predictive inference and feature selection for generalized linear models with scarce but high-dimensional data. We argue that in many cases one can benefit from a decision theoretically justified two-stage approach:…
A general method for accelerating fixed point schemes for problems related to partial differential equations is presented in this article. The speedup is obtained by training a reduced-order model on-the-fly, removing the need to do an…
In the present work we introduce a novel refinement algorithm for two-dimensional elliptic partial differential equations discretized with Virtual Element Method (VEM). The algorithm improves the numerical solution accuracy and the mesh…
In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead…
We present a new method for stochastic shape optimisation of engineering structures. The method generalises an existing deterministic scheme, in which the structure is represented and evolved by a level-set method coupled with mathematical…
This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly…