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Low rank methods for PDE-constrained optimization

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UNQW03 - Reducing dimensions and cost for UQ in complex systems

Optimization subject to PDE constraints is crucial in many applications . Numerical analysis has contributed a great deal to allow for the efficient solution of these problems and our focus in this talk will be on the solution of the large scale linear systems that represent the first order optimality conditions. We illustrate that these systems, while being of very large dimension, usually contain a lot of mathematical structure. In particular, we focus on low-rank methods that utilize the Kronecker product structure of the system matrices. These methods allow the solution of a time-dependent problem with the storage requirements of a small multiple of the steady problem. Furthermore, this technique can be used to tackle the added dimensionality when we consider optimization problems subject to PDEs with uncertain coefficients. The stochastic Galerkin FEM technique leads to a vast dimensional system that would be infeasible on any computer but using low-rank techniques this can be solved on a standard laptop computer.

This talk is part of the Isaac Newton Institute Seminar Series series.

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