♦ A fast direct solver for elliptic PDEs on locally-perturbed domains.
Many problems in science and engineering can be formulated as integral equations with elliptic kernels. In particular, in optimal control and design problems, the domain geometry evolves and results in a sequence of discretized linear systems to be constructed and inverted. While the systems can be constructed and inverted independently, the computational cost is relatively high. In the case where the change in the domain geometry for each new problem is only local, i.e. the geometry remains the same except within a small subdomain, we are able to reduce the cost of inverting the new system by reusing the pre-computed fast direct solvers of the original system. The resulting solver only requires inexpensive matrix-vector multiplications, thus dramatically reducing the cost of inverting the new linear system.
Related Presentations are given at:
• SIAM Conference on Computational Science and Engineering 2017, Feb. 27 - Mar. 3 (Contributed Talk)
• CAAM Graduate Seminar Talk, Mar. 8
• 2017 Rice Oil and Gas HPC Conference, Mar. 15 - Mar. 16 (Poster)
• 2017 Modern Advances in Computational and Applied Mathematics Workshop, Jun. 9 - Jun. 10 (Poster)
Y. Zhang and A. Gillman,
"A fast direct solver for boundary value problems on locally perturbed geometries," Journal of Computational Physics. 2018
♦ Fast direct solvers for elliptic PDEs on evolving geometries.
Problems such as modeling blood vesicles in Stokes flow can be cast as solving elliptic PDEs defined on evolving gemetries: for each time step, the surface of the vesicles deform globally. While previous developments of fast direct solvers show that they perform well in terms of both robustness and efficiency when applied to elliptic PDEs, applying these solvers to problmes with evolving geometries are limited partially due to the high cost of constructing a new solver for each time step. The proposed solvers will reduce this cost by constructing a fast direct solver for the initial geometry and then repeatedly reusing the information stored in this initial solver to obtain new solvers on the subsequent geometries.