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In conventional machine learning (ML) approaches applied to electroencephalography (EEG), this is often a limited focus, isolating specific brain activities occurring across disparate temporal scales (from transient spikes in milliseconds…
Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation…
We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable…
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…
High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order…
Gaussian curvature is an important geometric property of surfaces, which has been used broadly in mathematical modeling. Due to the full nonlinearity of the Gaussian curvature, efficient numerical methods for models based on it are uncommon…
We propose a novel framework to solve PDEs on moving manifolds, where the evolving surface is represented by a moving point cloud. This has the advantage of avoiding the need to discretize the bulk volume around the surface, while also…
For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for…
The problem of reconstructing brain activity from electric potential measurements performed on the surface of a human head is not an easy task: not just because the solution of the related inverse problem is fundamentally ill-posed (not…
Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they show up frequently perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is…
We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the…
We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers…
This book aims to provide a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. Such a class of equations often arises in analysis, probability theory,…
Surface integral equations (SIEs)-based boundary element methods are widely used for analyzing electromagnetic scattering scenarii. However, after discretization of SIEs, the spectrum and eigenvectors of the boundary element matrices are…
Isogeometric Analysis (IGA) is a computational technique for the numerical approximation of partial differential equations (PDEs). This technique is based on the use of spline-type basis functions, that are able to hold a global smoothness…
The electrocardiogram (ECG) is a well-known technique used to diagnose cardiac diseases. To acquire the spatial signal characteristics from the thorax, multiple electrodes are commonly used. Displacements of electrodes affect the signal…