Compressive Algorithms-Adaptive Solutions of PDEs and Variational Problems

Iterative algorithms for families of variational inequalities fixed points and equilibrium problems

Pdes with Compressed Solutions

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an L norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by m...

How to Best Sample a Solution Manifold?

Model reduction attempts to guarantee a desired “model quality”, e.g. given in terms of accuracy requirements, with as small a model size as possible. This article highlights some recent developments concerning this issue for the so called Reduced Basis Method (RBM) for models based on parameter dependent families of PDEs. In this context the key task is to sample the solution manifold at judic...

A New Adaptive Grid Method Based on Iterative Grid Redistribution

We introduce an iterative grid redistribution method based on the variational approach which enables us to gain more precise control of the grid distribution near the regions of large solution variations. The method is particularly effective for solving PDEs with singular solutions (e.g. blow up solutions).

Numerical solution of the variational PDEs arising in optimal control theory

An iterative method based on Picard’s approach to ODEs’ initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examp...