On Multiscale and Statistical Numerical Methods for PDEs and Inverse Problems

This thesis focuses on numerical methods for scientific computing and scientific machine learning, specifically on solving partial differential equations and inverse problems. The design of numerical algorithms usually encompasses a spectrum that ranges from specialization to generality. Classical approaches, such as finite element methods, and contemporary scientific machine learning approaches, like neural nets, can be viewed as lying at relatively opposite ends of this spectrum. Throughout this thesis, we tackle mathematical challenges associated with both ends by advancing rigorous multiscale and statistical numerical methods. Regarding the multiscale numerical methods, we present an exponentially convergent multiscale finite element method for solving high-frequency Helmholtz's equation with rough coefficients. To achieve this, we first identify the local low-complexity structure of Helmholtz's equations when the resolution is smaller than the wavelength. Then, we construct local basis functions by solving local spectral problems and couple them globally through non-overlapped domain decomposition and Galerkin's method. This results in a numerical method that achieves nearly exponentially convergent accuracy regarding the number of local basis functions, even when the solution is highly non-smooth. We also analyze the role of a subsampled lengthscale in variational multiscale methods, characterizing the tradeoff between accuracy and efficiency in the numerical upscaling of heterogeneous PDEs and scattered data approximation. As for the statistical numerical methods, we discuss using Gaussian processes and kernel methods to solve nonlinear PDEs and inverse problems. This framework incorporates the flavor of scientific machine learning automation and extends classical meshless solvers. It transforms general PDE problems into quadratic optimization with nonlinear constraints. We present the theoretical underpinning of the methodology. For the sc