Uniformly convergent adaptive methods for a class of parametric operator equations

Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations

This paper is concerned with the design and implementation of efficient solution algorithms for elliptic PDE problems with correlated random data. The energy orthogonality that is built into stochastic Galerkin approximations is cleverly exploited to give an innovative energy error estimation strategy that utilizes the tensor product structure of the approximation space. An associated error est...

Adaptive frame methods for elliptic operator equations

Stephan Dahlke a, Massimo Fornasier b and Thorsten Raasch a a Philipps-Universität Marburg, FB 12 Mathematik und Informatik, Hans-Meerwein Straße, Lahnberge, 35032 Marburg, Germany E-mail: {dahlke;raasch}@mathematik.uni-marburg.de b Università “La Sapienza” in Roma, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via Antonio Scarpa, 16/B, 00161 Roma, Italy E-mail: mfornasi...

A Uniformly Convergent Adaptive Particle Filter

Particle filters are Monte Carlo methods that aim to approximate the optimal filter of a partially observed Markov chain. In this paper, we study the case in which the transition kernel of the Markov chain depends on unknown parameters: we construct a particle filter for the simultaneous estimation of the parameter and the partially observed Markov chain (adaptive estimation) and we prove the c...

Adaptive wavelet methods for elliptic operator equations: Convergence rates

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so cal...

Optimal adaptive wavelet methods - for linear operator equations