We develop a numerical method for the decomposition of multivariate functions based on recursively

applying biorthogonal decompositions in function spaces. The result is an approximation of

the multivariate function by sums of products of univariate functions. Decompositions of this type

can conveniently be visualized by binary trees and in some sense are a functional analog of the decompositions

in tensor numerical methods that are obtained through sequences of matrix reshaping

and singular value decomposition. The underlying theory of recursive biorthogonal decomposition

in function spaces is developed and computational aspects are discussed. This decomposition is

generalized to handle time dependence in such a way which allows for the decomposition and propagation

of solutions to nonlinear time dependent partial differential equations. In this way we obtain

a numerical solution for time dependent problems which remains on a low parametric manifold of

constant rank for all time. We also discuss the addition and removal of time dependent modes during

propagation to allow for robust adaptive solvers. Applications to prototype linear hyperbolic

problems are presented and discussed.

The Mori-Zwanzig (MZ) formulation is a technique from irreversible statistical mechanics that allows the development of formally exact evolution equation for the quantities of interest such as macroscopic observables in high-dimensional dynamical systems. Although being widely used in physics and applied mathematics as a tool of dimension reduction, the analytical properties of the equation are still unknown, which makes the quantification and approximation of the MZ equation arduous tasks. In this dissertation, we address this problem from both theoretical and computational points of view. For the first time, we study the MZ equation, especially the memory integral term, using the theory of strongly continuous semigroups, and establish an estimation theory which works for classical and stochastic dynamical systems. In particular, some recent results from the H\"ormader analysis of hypoelliptic equations are applied to get exponential decay estimates of the MZ memory kernel. We also develop a series expansion technique to approximate the MZ equation, and provide associated combinatorial algorithms to calculate the expansion coefficients from first principles. The new approximation methods are tested on various linear and nonlinear dynamical systems, with convergence results obtained both theoretically and numerically. Further developments of the Mori-Zwanzig formulation based on the mathematical framework provided in this work can be expected, which can be used in general dimension reduction problems from physics and mathematics.

High-throughput screening of compounds for desirable electronic properties can allow for accelerated discovery and design of materials. Density functional theory (DFT) is the popular approach used for these quantum chemical calculations, but it can be computationally expensive on a large scale. Recently, machine learning methods have gained traction as a supplementation to DFT, with well-trained models achieving similar accuracy as DFT itself. However, training a machine learning model to be accurate and generalizable to unseen materials requires a large amount of training data. This work proposes a method to minimize the need for novel data creation for training by using transfer learning and publicly-available databases, allowing for both data-efficient and accurate machine learning to mitigate the computational cost of DFT.

The numerical simulation of high-dimensional partial differential equations (PDEs) is a challenging and important problem in science and engineering. Classical methods based on tensor product representations are not viable in high-dimensions, as the number of degrees of freedom grows exponentially fast with the problem dimension. In this dissertation we present low-rank tensor methods for approximating high-dimensional PDEs, which have a number of degrees of freedom and computational cost that grow linearly with the problem dimension. These methods are based on projecting a given PDE onto a low-rank tensor manifold and then constructing an approximate PDE solution as a path on the manifold. In order to control the accuracy of the low-rank tensor approximation we present a rank-adaptive algorithm that can add or remove tensor modes adaptively from the PDE solution during time integration. We also present a tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is to determine a coordinate transformation of a given functions domain so that the function in the new coordinate system has smaller tensor rank. We demonstrate each of the presented low-rank tensor methods by providing several numerical applications to multivariate functions and PDEs.

We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. In particular, we develop a mathematical framework for the analysis of these algorithms which encompasses both explicit and implicit time stepping. With this framework we prove convergence of a wide range of step-truncation methods, including one-step and multi-step methods. These methods rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a linear advection problem, a clasas of Fokker-Planck equations, the Allen-Cahn equation, the nonlinear Schrodinger, and a Burgers' equation with uncertain initial condition.