Related papers: Layer-adapted meshes for singularly perturbed prob…
Stochastic differential equations (SDEs) are increasingly used in longitudinal data analysis, compartmental models, growth modelling, and other applications in a number of disciplines. Parameter estimation, however, currently requires…
This study presents constructions of the space-time Conservation Element and Solution Element (CESE) methods to accommodate adaptive unstructured quadrilateral meshes. Subsequently, a novel algorithm is devised to effectively manage the…
We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and…
A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter.…
To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of…
We consider the approximation of singularly perturbed linear second-order boundary value problems by $hp$-finite element methods. In particular, we include the case where the associated differential operator may not be coercive. Within this…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter.…
We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality…
We consider a variant of the conventional MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of mesh elements and on their interfaces. A convergence analysis of the approach is presented. Residue-type a…
An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve…
We address a three-tier data-driven approach to solve the inverse problem in complex systems modelling from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and…
In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods \textsf{AMFEM1} and \textsf{AMFEM2} for the Hodge Laplacian problem in finite element exterior calculus. We prove that…
We consider a hybrid FEM-BEM method to compute approximations of full-space linear elliptic transmission problems. First, we derive a priori and a posteriori error estimates. Then, building on the latter, we present an adaptive algorithm…
Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the…
This article presents an introduction to MMPDElab, a package written in MATLAB for adaptive mesh movement and adaptive moving mesh P1 finite element solution of second-order partial different equations having continuous solutions in one,…
This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…