Analytical Techniques for a Numerical Solution of the Linear Volterra Integral Equation of the Second Kind

In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.This Research is Partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533, and by Junta de Andalcía Grant FQM359

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

Elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are considered. We study the $\ell^p$ summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Lo\`eve representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion

Frame Theory for Signal Processing in Psychoacoustics

This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field

Adaptive Parameter Optimization For An Elliptic-Parabolic System Using The Reduced-Basis Method With Hierarchical A-Posteriori Error Analysis

In this paper the authors study a non-linear elliptic-parabolic system, which is motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation. The goal is to determine several scalar parameters in the coupled model in an optimal manner by utilizing a reliable reduced-order approach based on the reduced basis (RB) method. However, the states are coupled through a strongly non-linear function, and this makes the evaluation of online-efficient error estimates difficult. First the well-posedness of the system is proved. Then a Galerkin finite element and RB discretization is described for the coupled system. To certify the RB scheme hierarchical a-posteriori error estimators are utilized in an adaptive trust-region optimization method. Numerical experiments illustrate good approximation properties and efficiencies by using only a relatively small number of reduced bases functions.Comment: 24 pages, 3 figure