Research Experience

Multigrid preconditioned Conjugate Gradient method has been the standard solver for computer graphics research. For some problem, like the one to be solved here involving a large matrix with a large condition number, we might need to not only use (geometric) multigrid to precondition the conjugate gradient solver but also use smoother to precondition the multigrid preconditioner. Here we use block Jacobi smoother and it will become the most time-consuming part of the preconditioner because of matrix-vector multiplication and the factorization (or direct inversion if the block size is small) of the diagonal block matrices.

To speed up the solver, we can parallelize each part of MGPCG. The purpose of this research is to look at how the factorization of the sparse diagonal block matrices can be parallelized. The context or the problem to solve is Darcy's equation discretized using finite element method and the matrix coming out of the FEM discretization is sparse and ill-conditioned with very clear block structure along the diagonal (the blocks along the diagonal will also be sparse).
I have been doing prototyping work in Python to implement multiple iterative methods including CG, MG, MGPCG and to try out different ways to speed up the block Jacobi smoother. I will switch to C/C++ with parallelization framework in the near future.
This research is mentored by Prof. Maurice Fabien.

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Scalar Auxiliary Variable (SAV) method was proposed in recent years to solve the gradient flow problem which Cahn-Hilliard (CH) equations are part of. One problem of the original SAV is that it does not satisfy the exact energy dissipation function and a new method of R-SAV (SAV with relaxation) has been proposed to solve this issue. Here is R-SAV being used to solve Cahn-Hilliard and Navier-Stokes coupled equation (CHNS).
Incompressible Navier-Stokes equation is usually solved by projection method, and to ensure the stability of the solution when using SAV to solve CH and projection to solve NS in CHNS coupled problem, there is a scheme called M-SAV (multiple SAV). Prof. Jie Shen also proposed an improved method for the original SAV that the SAV could be easily extended to higher order scheme. This research is to draw inspiration from the above improvements of the SAV method to proposed a scheme to solve the CHNS coupled equations that is stable, satisfies the original energy dissipation function and can be extended to higher order scheme.
This research is mentored by Prof. Yukun Yue.

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Under construction

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Under construction

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