Sparse Tensor Product Finite Element Research Articles

Sparse tensor product high dimensional finite elements for two-scale mixed problems

We develop the essentially optimal sparse tensor product finite element method for solving two-scale mixed problems in both the primal and dual forms. We study the two-scale homogenized mixed problems which are obtained in the limit where the microscopic scale tends to zero. These limiting problems are posed in a high dimensional tensorized product domain. Using sparse tensor product finite elements, we solve these problems with essentially optimal complexity to obtain an approximation for the solution within a prescribed accuracy. This is achieved when the solutions of the high dimensional two-scale homogenized mixed problems possess sufficient regularity with respect to both the slow and the fast variable at the same time. We show that this regularity requirement holds when the two-scale coefficient and the forcing satisfy mild smoothness conditions. From the finite element solutions for the two-scale homogenized mixed problems, we construct numerical correctors for the solutions of the original two-scale mixed problems. We prove an error estimate for these correctors in terms of both the homogenization error and the finite element error. Numerical examples confirm the theoretical results.

Sparse tensor product finite element method for nonlinear multiscale variational inequalities of monotone type

Abstract We study an essentially optimal finite element (FE) method for locally periodic nonlinear multiscale variational inequalities of monotone type in a domain $D\subset{\mathbb{R}}^d$ that depend on a macroscopic and $n$ microscopic scales. The scales are separable. Using multiscale convergence we deduce a multiscale homogenized variational inequality in a tensorized domain in the high-dimensional space ${\mathbb R}^{(n+1)d}$. Given sufficient regularity on the solution the sparse tensor product FE method is developed for this problem, which attains an essentially equal (i.e., it differs by only a logarithmic factor) level of accuracy to that of the full tensor product FE method, but requires an essentially optimal number of degrees of freedom which is equal to that for solving a problem in ${{\mathbb{R}}}^d$ apart from a logarithmic factor. For two-scale problems we deduce a new homogenization error for the nonlinear monotone variational inequality. A numerical corrector is then constructed with an explicit error in terms of the homogenization and the FE errors. For general multiscale problems we deduce a numerical corrector from the FE solution of the multiscale homogenized problem, but without an explicit error as such a homogenization error is not available.

High dimensional finite elements for time-space multiscale parabolic equations

The paper develops the essentially optimal sparse tensor product finite element method for a parabolic equation in a domain in $\mathbb {R}^{d}$ which depends on a microscopic scale in space and a microscopic scale in time. We consider the critical self similar case which has the most interesting homogenization limit. We solve the high dimensional time-space multiscale homogenized equation, which provides the solution to the homogenized equation which describes the multiscale equation macroscopically, and the corrector which encodes the microscopic information. For obtaining an approximation within a prescribed accuracy, the method requires an essentially optimal number of degrees of freedom that is essentially equal to that for solving a macroscopic parabolic equation in a domain in $\mathbb {R}^{d}$ . A numerical corrector is deduced from the finite element solution. Numerical examples for one and two dimensional problems confirm the theoretical results. Although the theory is developed for problems with one spatial microscopic scale, we show numerically that the method is capable of solving problems with more than one spatial microscopic scale.

Bayesian inverse problems for recovering coefficients of two scale elliptic equations

We consider the Bayesian inverse homogenization problem of recovering the locally periodic two scale coefficient of a two scale elliptic equation, given limited noisy information on the solution. We consider both the uniform and the Gaussian prior probability measures. We use the two scale homogenized equation whose solution contains the solution of the homogenized equation which describes the macroscopic behaviour, and the corrector which encodes the microscopic behaviour. We approximate the posterior probability by a probability measure determined by the solution of the two scale homogenized equation. We show that the Hellinger distance of these measures converges to zero when the microscale converges to zero, and establish an explicit convergence rate when the solution of the two scale homogenized equation is sufficiently regular. Sampling the posterior measure by Markov Chain Monte Carlo (MCMC) method, instead of solving the two scale equation using fine mesh for each proposal with extremely high cost, we can solve the macroscopic two scale homogenized equation. Although this equation is posed in a high dimensional tensorized domain, it can be solved with essentially optimal complexity by the sparse tensor product finite element method, which reduces the computational complexity of the MCMC sampling method substantially. We show numerically that observations on the macrosopic behaviour alone are not sufficient to infer the microstructure. We need also observations on the corrector. Solving the two scale homogenized equation, we get both the solution to the homogenized equation and the corrector. Thus our method is particularly suitable for sampling the posterior measure of two scale coefficients.

High Dimensional Finite Elements for Multiscale Wave Equations

For locally periodic multiscale wave equations in $\mathbb{R}^d$ that depend on a macroscopic scale and n microscopic separated scales, we solve the high dimensional limiting multiscale homogenized problem that is posed in $(n+1)d$ dimensions and is obtained by multiscale convergence. We consider the full and sparse tensor product finite element methods, and analyze both the spatial semidiscrete and the fully (both temporal and spatial) discrete approximating problems. With sufficient regularity, the sparse tensor product approximation achieves a convergence rate essentially equal to that for the full tensor product approximation, but requires only an essentially equal number of degrees of freedom as for solving an equation in $\mathbb{R}^d$ for the same level of accuracy. For the initial condition $u(0,x)=0$, we construct a numerical corrector from the finite element solution. In the case of two scales, we derive an explicit homogenization error which, together with the finite element error, produces an explicit rate of convergence for the numerical corrector. Numerical examples for two- and three-scale problems in one or two dimensions confirm our analysis.

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

We study the mixed formulation of the stochastic Hodge-Laplace problem dened on a $n$-dimensional domain $D (n ≥ 1)$, with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the $m-th$ moment $(m ≥ 1$) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.

Sparse Finite Element Method for Periodic Multiscale Nonlinear Monotone Problems

A sparse tensor product finite element (FE) method is developed for the high-dimensional limiting problem obtained by applying the multiscale convergence to a multiscale nonlinear monotone problem in $\mathbb{R}^d$. The limiting problem is posed in a product space, so tensor product FE spaces are used for discretization. This sparse FE method requires essentially the same number of degrees of freedom to achieve essentially equal accuracy to that of a standard FE scheme for a partial differential equation in $\mathbb{R}^d$. It is proved that for the Euler–Lagrange equation of a two scale convex variational problem in a smooth and convex domain the solution to the high-dimensional limiting equation is smooth. An analytic homogenization error is then established, which together with the FE error provides an explicit error estimate for an approximation to the solution of the original multiscale problem. Without this regularity, such an approximation always exists when the meshsize and the microscale converge to 0 but without a rate of convergence.

Anisotropic Operator Symbols Arising From Multivariate Jump Processes

It is shown that infinitesimal generators $${\mathcal{A}}$$ of certain multivariate pure jump Lévy copula processes give rise to a class of anisotropic symbols that extends the well-known classes of pseudo differential operators of Hörmander-type. In addition, we provide minimal regularity convergence analysis for a sparse tensor product finite element approximation to solutions of the corresponding stationary Kolmogorov equations $${\mathcal{A}}u = f$$ . The computational complexity of the presented approximation scheme is essentially independent of the underlying state space dimension.

Sparse finite element approximation of high-dimensional transport-dominated diffusion problems

We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form , , where is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm , provided u belongs to the space of functions with square-integrable mixed (k+1)st derivatives: where , i=0,1, and . We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems , and hence for p ≥ 1 the 'error constant' exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than . In any case, in the absence of assumptions that relate L, p and d, the error is still bounded by , where for all L, p, d ≥ 2.

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.

High-Dimensional Finite Elements for Elliptic Problems with Multiple Scales

Elliptic homogenization problems in a domain $\Omega \subset \mathbb{R}^d$ with n + 1 separated scales are reduced to elliptic one-scale problems in dimension (n + 1)d. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work, and memory requirements comparable to those in a standard FEM for single-scale problems in $\Omega$, while it gives numerical approximations of the correct homogenized limit as well as of all first-order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results.