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Fall 2022
The Helmholtz equation is a fundamental wave propagation model in the timeharmonic setting, which appears in many applications such as electromagnetics, geophysics, and ocean acoustics. It is challenging and computationally expensive to solve due to (1) its highly oscillating solution and (2)...

The Strong Restricted Isometry Property of SubGaussian Matrices and the Erasure Robustness Property of Gaussian Random Frames
DownloadSpring 2016
In this thesis we will study the robustness property of subgaussian random matrices. We first show that the nearly isometry property will still hold with high probability if we erase a certain portion of rows from a subgaussian matrix, and we will estimate the erasure ratio with a given small...

Fall 2017
One main goal of this thesis is to bring forth a systematic and simple construction of a multiwavelet basis on a bounded interval. The construction that we present possesses orthogonality in the derivatives of the multiwavelet basis among all scale levels. Since we are mainly interested in Riesz...

Fall 2015
This thesis concentrates on the construction of directional tensor product complex tight framelets. It uses a complex tight framelet filter bank in one dimension and the tensor product of the onedimensional filter bank to obtain highdimensional filter bank. It has a number of advantages over...

Fall 2022
Interface problems arise in many applications such as modeling of underground waste disposal, oil reservoirs, composite materials, and many others. The coefficient $a$, the source term $f$, the solution $u$ and the flux $a\nabla u\cdot \vec{n}$ are possibly discontinuous across the interface...

Spring 2021
Generalizing wavelets by adding desired redundancy and flexibility, framelets (a.k.a. wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale...