In this work, we introduce two methods to handle nearly isometric shape correspondence problem. The first method, Sparsity-Enforced Quadratic Assignment (SEQA), introduces a novel local pairwise descriptor and then develops a simple, effective iterative method to solve the resulting quadratic assignment through sparsity control. The pairwise descriptor is based on the stiffness and mass matrix of finite element approximation of the Laplace-Beltrami differential operator. The key idea of our iterative algorithm is to select pairs with good correspondence as anchor pairs based on a new criterion we introduced called local mapping distortion, and then solve a regularized quadratic assignment problem the neighborhoods of corresponding selected anchor pairs through sparsity control. Various pointwise global features with reference to these anchor pairs can be used to improve the dense shape correspondence further. Inspired by the powerful ability of local mapping distortion on selecting accurate anchor pairs, we further introduce the second method, Dual Iterative Refinement (DIR). It is a simple and efficient algorithm which combines dual features, spatial and spectral, or local and global, in a complementary and optimal way. DIR first uses local spatial feature, local mapping distortion to obtain anchor pairs which are used to determine an appropriate dimension of the spectral space and the corresponding functional map. Then the functional map is used to update a new correspondence. Both methods allow us to deal with open surfaces, partial matching, and topological perturbations robustly. We use various experiments to show the efficiency, quality, and versatility of our methods on large data sets, patches, and point clouds (without global meshes).

In this dissertation we propose a ray-based finite element method (ray-FEM) for the high-frequency Helmholtz equation in smooth media, whose basis are learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of $\mathcal{O}(\omega^{-\frac{1}{2}})$, where $\omega$ is the frequency parameter in the Helmholtz equation.

The local basis are motivated by the geometric optics ansatz and are composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively.

In addition, a fast sweeping-type preconditioner is used to solve the resulting linear system. We present numerical examples in 2D to show both efficiency and convergence of our method as the frequency becomes larger and larger. In particular, we show empirically that the overall complexity is $\mathcal{O}(\omega^2)$ up to a poly-logarithmic factor.

Dynamic materials including self-healing polymers and vitrimers have huge potential for improving current material’s performance, lifetime and processability. In this thesis, we have synthesized a self-healing system by incorporating zinc amide interaction and a vitrimer system via transesterification of boronic esters. Chapter 1 provides a brief introduction to the background of self-healing polymers and vitrimers. Chapter 2 details the design and development of a zinc amide self-healing polymer system. Chapter 3 discusses attempts to synthesize a catalyst free, tunable boronic ester vitrimer.

For acoustic scattering we consider the inverse problem for a complex scatterer, which might be composed of both inhomogeneous anisotropic mediums and impenetrable obstacles. It is shown that the fixed-frequency scattering amplitude uniquely determines the scattering support of a complex scatterer, disregarding its contents. We also study a special kind of transmission problem.

For electromagnetic scattering, we develop a direct scheme of imaging multiple electromagnetic scatterers. The multiple scatterers may consists of both inhomogeneities and impenetrable perfect electromagnetic conductive obstacles. Moreover, there might be extended components presented simultaneously in the multiple scatterers. The proposed reconstruction scheme is based on an imaging functional which can be directly calculated without any inversion involved. We also provide a heuristic principle study for this imaging scheme. We also provide extensive numerical study to illustrate the effectiveness and efficiency of the scheme.

Markov chain Monte Carlo (MCMC) methods have been widely used in Bayesian inference involving intractable probabilistic models. However, simple MCMC algorithms (e.g., random walk Metropolis and Gibbs sampling) are notorious for their lack of computational efficiency in complex, high-dimensional models and poor scaling to large data sets. In recent years, many advanced MCMC methods (e.g., Hamiltonian Monte Carlo and Riemannian Manifold Hamiltonian Monte Carlo) have been proposed that utilize geometrical and statistical quantities from the model in order to explore the target distribution more effectively. The gain in the efficacy of exploration, however, often comes at a significant computational cost which hinders their application to problems with large data sets or complex likelihoods.

In practice, it remains challenging to design scalable MCMC algorithms that can balance computational complexity and exploration efficacy well. To address this issue, some recent algorithms rely on stochastic gradient methods by approximating full data gradients using mini-batches of data. In contrast, this thesis focuses on accelerating the computation of MCMC samplers based on various surrogate methods via exploring the regularity of the target distribution.

We start with a precomputing strategy that can be used to build efficient surrogates in relatively low-dimension parameter spaces. We then propose a random network surrogate architecture which can effectively capture the collective properties of large data sets or complex models with scalability, flexibility and efficiency. Finally, we provide a variational perspective for our random network surrogate methods and propose an approximate inference framework that combines the advantages of both variational Bayes and Markov chain Monte Carlo methods. The properties and efficiency of our proposed methods are demonstrated on a variety of synthetic and real-world data problems.