Andrew J. Wathen

Q. T. Le Gia, Ian H. Sloan, Andrew J. Wathen

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. Making use of a recently derived inf-sup condition [13] and the Brezzi stability and convergence theorem for this approximation scheme, we show that the linear system can be optimally preconditioned with a suitable block-diagonal preconditioner. Numerical experiments with a non-uniform distribution of data points support the theoretical conclusions.

Anthony Goddard, Andrew Wathen

McDonald, Pestana and Wathen (SIAM J. Sci. Comput. 40(2), pp. A2012-A1033, 2018) present a method for preconditioning of time-dependent PDEs via approximation by a nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners for the non-symmetric block Toeplitz matrices which arise from an all-at-once formulation. They suggest that such an approach might be efficiently implemented in parallel. In this short article, we present parallel numerical results for their preconditioner which exhibit strong scaling. We also extend their preconditioner via a Neumann series approach, which also allows for efficient parallel execution. Our simple implementation (in C++ and MPI) is available at the Git repository PARALAAOMPI. https://github.com/anthonyjamesgoddard/PARALAAOMPI

Shengxin Zhu, Andrew J Wathen

Algorithms for Gaussian process, marginal likelihood methods or restricted maximum likelihood methods often require derivatives of log determinant terms. These log determinants are usually parametric with variance parameters of the underlying statistical models. This paper demonstrates that, when the underlying matrix is sparse, how to take the advantage of sparse inversion---selected inversion which share the same sparsity as the original matrix---to accelerate evaluating the derivative of log determinant.