Errors Theory using Dirichlet Forms, Linear Partial Differential Equations and Wavelets

TL;DR

This paper applies Dirichlet Forms error theory to linear PDEs, utilizing wavelet bases to analyze how uncertainties in terminal conditions propagate through solutions, simplifying computations in certain LPDE classes.

Contribution

It introduces a novel combination of error theory with wavelet decomposition for LPDEs, providing new insights into uncertainty transmission and computational simplification.

Findings

Wavelet bases effectively decompose LPDE solutions.

Uncertainty propagation can be analyzed via Dirichlet Forms.

Simplified computational methods for specific LPDE classes.

Abstract

We present an application of error theory using Dirichlet Forms in linear partial differential equations (LPDE). We study the transmission of an uncertainty on the terminal condition to the solution of the LPDE thanks to the decomposition of the solution on a wavelets basis. We analyze the basic properties and a particular class of LPDE where the wavelets bases show their powerful, the combination of error theory and wavelets basis justifies some hypotheses, helpful to simplify the computation.