A study on deflation techniques and POD methods for the acceleration of Krylov subspace methods
Jenny Tjan

Site of the project: TU Delft (collaboration with Padua University, see picture below)

Supervisors TU Delft: Kees Vuik and Gabriela Diaz Cortes

start of the project: November 2017

In February 2018 the Interim Thesis has appeared and a presentation has been given.

The Master project has been finished in August 2018 by the completion of the Masters Thesis and a final presentation has been given.

For working address etc. we refer to our alumnipage.

Summary of the master project:
After discretization of the problem equations describing e.g. reservoir simulation or groundwater flow problems, we arrive to systems of linear equations. If the problem is large or ill conditioned, i.e. the matrix of the system has high condition number, solution of the problem is time consuming. Therefore, it is necessary to find a way to reduce this time. The first approach is to use iterative solver, but sometimes, this approach is not enough. Then, preconditioning techniques have to be used.

Proper Orthogonal Decomposition (POD) based on information obtained from the system has been found as a good approach to obtain an acceleration of iterative methods. Deflation methods have also been studied with the same purpose, showing as well, a good performance.

For an optimal performance of the deflation techniques, it is necessary to find good deflation vectors. If a good selection of these vectors is made, only a a small increase in the required computing time per iteration and an important decrease in the number of iterations is achieved. The capture of a series of snapshots, solutions of the system with slightly different characteristics, are used to construct a POD basis.

In this project, we want to explore the similarities and differences between the use of POD basis preconditioner and as deflation vectors. The topics to cover during this MSc thesis are:
• Implementation of POD basis vectors together with deflation matrices as preconditioners.
• Implementation of POD basis vectors as deflation vectors.
• Comparison of both approaches.
The studied problem will be a linear system with large number of unknowns and high condition number (resulting from reservoir simulation or groundwater flow problems). The POD basis will be obtained from snapshots, solutions of the system at various time steps.

Reference:

G.B. Diaz Cortes and C. Vuik and J.D. Jansen (pdf, bibtex)
On POD-based Deflation Vectors for DPCG applied to porous media problems
Journal of Computational and Applied Mathematics, 330, pp. 193-213, 2018

## Contact information: Kees Vuik

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