Analytical Methods for Structured Matrix Computations

The design of fast algorithms is not only about achieving faster speeds but also about retaining the ability to control the error and numerical stability. This is crucial to the reliability of computed numerical solutions. This dissertation studies topics related to structured matrix computations with an emphasis on their numerical analysis aspects and algorithms. The methods discussed here are all based on rich analytical results that are mathematically justified. In chapter 2, we present a series of comprehensive error analyses to an analytical matrix compression method and it serves as a theoretical explanation of the proxy point method. These results are also important instructions on optimizing the performance. In chapter 3, we propose a non-Hermitian eigensolver by combining HSS matrix techniques with a contour-integral based method. Moreover, probabilistic analysis enables further acceleration of the method in addition to manipulating the HSS representation algebraically. An application of the HSS matrix is discussed in chapter 4 where we design a structured preconditioner for linear systems generated by AIIM. We improve the numerical stability for the matrix-free HSS construction process and make some additional modifications tailored to this particular problem

JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities

We present a scalable strategy for development of mesh-free hybrid neuro-symbolic partial differential equation solvers based on existing mesh-based numerical discretization methods. Particularly, this strategy can be used to efficiently train neural network surrogate models of partial differential equations by (i) leveraging the accuracy and convergence properties of advanced numerical methods, solvers, and preconditioners, as well as (ii) better scalability to higher order PDEs by strictly limiting optimization to first order automatic differentiation. The presented neural bootstrapping method (hereby dubbed NBM) is based on evaluation of the finite discretization residuals of the PDE system obtained on implicit Cartesian cells centered on a set of random collocation points with respect to trainable parameters of the neural network. Importantly, the conservation laws and symmetries present in the bootstrapped finite discretization equations inform the neural network about solution regularities within local neighborhoods of training points. We apply NBM to the important class of elliptic problems with jump conditions across irregular interfaces in three spatial dimensions. We show the method is convergent such that model accuracy improves by increasing number of collocation points in the domain and predonditioning the residuals. We show NBM is competitive in terms of memory and training speed with other PINN-type frameworks. The algorithms presented here are implemented using \texttt{JAX} in a software package named \texttt{JAX-DIPS} (https://github.com/JAX-DIPS/JAX-DIPS), standing for differentiable interfacial PDE solver. We open sourced \texttt{JAX-DIPS} to facilitate research into use of differentiable algorithms for developing hybrid PDE solvers