Related papers: Tensor-based techniques for fast discretization an…
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and…
Stratified fluids composed of a sequence of alternate layers show interesting macroscopic properties, which may be quite different from those of the individual constituent fluids. On a macroscopic scale, such systems can be considered a…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
We analyze classical dimer models on the square and triangular lattice using a tensor network representation of the dimers. The correlation functions are numerically calculated using the recently developed "Tensor renormalization group"…
Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally,…
Although FFT-based methods are renowned for their numerical efficiency and stability, traditional discretizations fail to capture material interfaces that are not aligned with the grid, resulting in suboptimal accuracy. To address this…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
This paper is concerned with the construction, analysis and realization of a numerical method to approximate the solution of high dimensional elliptic partial differential equations. We propose a new combination of an Adaptive Wavelet…
Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that…
We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients…
This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The resulting effective deterministic model is given through a quasilocal discrete integral…
In this work we focus on the preconditioning of a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner…
We study the use of polyhedral discretizations for the solution of heat diffusion and elastodynamic problems in computer graphics. Polyhedral meshes are more natural for certain applications than pure triangular or quadrilateral meshes,…
Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to…
In the last decade, tensors have shown their potential as valuable tools for various tasks in numerical linear algebra. While most of the research has been focusing on how to compress a given tensor in order to maintain information as well…
Elliptic partial differential equations must be solved numerically for many problems in numerical relativity, such as initial data for every simulation of merging black holes and neutron stars. Existing elliptic solvers can take multiple…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting…
In this article, we introduce a fast and memory efficient solver for sparse matrices arising from the finite element discretization of elliptic partial differential equations (PDEs). We use a fast direct (but approximate) multifrontal…
A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients…