Alexey Chernov

Optimal convergence estimates for the trace of the polynomial

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Suli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

L^{2}

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solutions fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

-projection operator on a simplex

In the present paper we initiate the study of hp Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size h and in the polynomial degree p in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop a novel convergence theory for the multilevel sample variance estimators in the framework of the multilevel Monte Carlo methods. We prove that, dependent on the regularity of the quantity of interest, the multilevel sample variance estimator may achieve the same asymptotic cost/error relation as the multilevel sample mean, which is superior to the standard Monte Carlo method. Weaker regularity assumptions result in reduced convergence rates, quantified in our analysis. The general convergence theory is applied to a class of scalar elliptic obstacle problems with rough random obstacle profiles, which is a simple model of contact between a deformable body with a rough uncertain substrate. Numerical experiments confirm theoretical convergence proofs.

Basic principles of hp Virtual Elements on quasiuniform meshes

We develop a complete convergence theory for the Maximum Entropy method based on moment matching for a sequence of approximate statistical moments estimated by the Multilevel Monte Carlo method. Under appropriate regularity assumptions on the target probability density function, the proposed method is superior to the Maximum Entropy method with moments estimated by the Monte Carlo method. New theoretical results are illustrated in numerical examples.

Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems

We introduce tools for a unified analysis and a comparison of impedance transmission conditions (ITCs) for thin conducting sheets within the time-harmonic eddy current model in two dimensions. The first criterion is the robustness with respect to the frequency or skin depth, which means whether they give meaningful results for small and for large frequencies or conductivities. As a second tool we study the accuracy for a range of sheet thicknesses and frequencies for a relevant example and finally analyze their asymptotic order in different asymptotic regimes. For the latter we write all the ITCs in a common form and show how they can be realized within the finite element method. Two new conditions, which we call ITC-2-0 and ITC-2-1, are introduced in this article, which appear in a symmetric form. They are derived by asymptotic expansions in the asymptotic regime of constant ratio between skin depth and thickness like those in [K. Schmidt and A. Chernov, Technical report 1102, Institut for Numerical Simu...

Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method

We are aiming at sharp and explicit-in-dimension estimations of the cardinality of s -dimensional hyperbolic crosses where s may be large, and applications in high-dimensional approximations of functions having mixed smoothness. In particular, we provide new tight and explicit-in-dimension upper and lower bounds for the cardinality of hyperbolic crosses. We apply them to obtain explicit upper and lower bounds for e -dimensions-the inverses of the well known Kolmogorov N -widths-in the space L 2 ( T s ) of modified Korobov classes U r , a ( T s ) on the s -torus T s : = - π , π s . The functions in this class have mixed smoothness of order r and depend on an additional parameter a which is responsible for the shape of the hyperbolic cross and controls the bound of the smoothness component of the unit ball of K r , a ( T s ) as a subset in L 2 ( T s ) . We give also a classification of tractability for the problem of e -dimensions of U r , a ( T s ) . This theory is extended to high-dimensional approximations of non-periodic functions in the weighted space L 2 ( - 1 , 1 s , w ) with the tensor product Jacobi weight w by tensor products of Jacobi polynomials with powers in hyperbolic crosses.

A Unified Analysis of Transmission Conditions for Thin Conducting Sheets in the Time-Harmonic Eddy Current Model

Spherical splines are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical result.

New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness

Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite-element method can be avoided using impedance transmission conditions (ITCs). Those ITCs shall provide an accurate approximation for small sheet thicknesses d, where the accuracy is best possible independent of the conductivity or the frequency being small or large-this we will call robustness. We investigate the accuracy and robustness of popular and recently developed ITCs, and propose robust ITCs, which are accurate up to O(d2).

PSEUDODIFFERENTIAL EQUATIONS ON THE SPHERE WITH SPHERICAL SPLINES

We consider two-body contact problems in elastoplasticity (plasticity with isotropic hardening) with and without friction and present solution procedures based on the coupling of finite elements and boundary elements. One solution method consists in rewriting the problem with penalty terms taking care of the frictional contact conditions [4], see also [8]. Then, its discretized version is solved by applying the radial return algorithm for both friction and plastification. We perform a segment-to-segment contact discretization which allows also to treat friction. Another solution procedure uses mortar projections [2] together with a Dirichlet-toNeumann (DtN) algorithm for the frictional contact part [6]; here we still use radial return for the plasticity part. Furthermore, extending the approach in [7] we can rewrite the contact problems with friction (given as variational inequalities without regularization) as saddle point problems and directly apply Uzawa’s algorithm. Comments are given for adaptive procedures [5]. Numerical benchmarks are given for small deformations and demonstrate the wide applicability of the given methods.