<div>On rectangular meshes, the simplest spectral element method for elliptic equations is the classical Lagrangian <i>Q</i>^k finite element method with only (<i>k</i>+1)-point Gauss-Lobatto quadrature, which can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is (<i>k</i> + 2)-th order accurate for <i>k</i> ≥ 2, whereas <i>Q</i>^<i>k</i> spectral element method is usually considered as a (<i>k</i> + 1)-th order accurate scheme in <i>L²</i>-norm. This result can be extended to linear wave, parabolic and linear Schrödinger equations.</div><div><br></div><div><div>Additionally, the <i>Q^k</i> finite element method for elliptic problems can also be viewed as a finite difference scheme on all Gauss-Lobatto points if the variable coefficients are replaced by their piecewise <i>Q^k </i>Lagrange interpolants at the Gauss Lobatto points in each rectangular cell, which is also proven to be (<i>k</i> + 2)-th order accurate.</div></div><div><br></div><div><div>Moreover, the monotonicity and discrete maximum principle can be proven for the fourth order accurate Q2 scheme for solving a variable coefficient Poisson equation, which is the first monotone and high order accurate scheme for a variable coefficient elliptic operator.</div></div><div><br></div><div><div>Last but not the least, we proved that certain high order accurate compact finite difference methods for convection diffusion problems satisfy weak monotonicity. Then a simple limiter can be designed to enforce the bound-preserving property when solving convection diffusion equations without losing conservation and high order accuracy.</div><div><br></div></div>

In this thesis, we first propose a new scalar auxiliary variable (SAV) approach for general dissipative nonlinear systems. This new approach is half computational cost of the original SAV approach, can be extended to high order unconditionally energy stable backward differentiation formula (BDF) schemes and not restricted to the gradient flow structure. Rigorous error estimates for this new SAV approach are conducted for the Allen-Cahn and Cahn-Hilliard type equations from the BDF1 to the BDF5 schemes in a unified form. As an application of this new approach, we construct high order unconditionally stable, fully discrete schemes for the incompressible Navier-Stokes equation with periodic boundary condition. The corresponding error estimates for the fully discrete schemes are also reported. Secondly, by combining the new SAV approach with functional transformation, we propose a new method to construct high-order, linear, positivity/bound preserving and unconditionally energy stable schemes for general dissipative systems whose solutions are positivity/bound preserving. We apply this new method to second order equations: the Allen-Cahn equation with logarithm potential, the Poisson-Nernst-Planck equation and the Keller-Segel equations and fourth order equations: the thin film equation and the Cahn-Hilliard equation with logarithm potential. Ample numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

<p dir="ltr">This work introduces an innovative algorithm for co-simulating time-dependent Drift Diffusion (DD) equations with Maxwell\textquotesingle s equations to characterize semiconductor devices. Traditionally, the DD equations, derived from the Boltzmann transport equations, are used alongside Poisson\textquotesingle s equation to model electronic carriers in semiconductors. While DD equations coupled with Poisson\textquotesingle s equation underpin commercial TCAD software for micron-scale device simulation, they are limited by electrostatic assumptions and fail to capture time dependent high-frequency effects. Maxwell\textquotesingle s equations are fundamental to classical electrodynamics, enabling the prediction of electrical performance across frequency range crucial to advanced device fabrication and design. However, their integration with DD equations has not been studied thoroughly. The proposed method advances current simulation techniques by introducing a new broadband patch-based method to solve time-domain 3-D Maxwell\textquotesingle s equations and integrating it with the solution of DD equations. This technique is free of the low-frequency breakdown issues prevalent in conventional full-wave simulations. Meanwhile, it enables large-scale simulations with reduced computational complexity. This work extends the simulation to encompass the complete device, including metal contacts and interconnects. Thus, it captures the entire electromagnetic behavior, which is especially critical in electrically larger systems and high-frequency scenarios. The electromagnetic interactions of the device with its contacts and interconnects are investigated, providing insights into performance at the chip level. Validation through numerical experiments and comparison with results from commercial TCAD tools confirm the effectiveness of the proposed method. </p>

<p dir="ltr">Matrix decompositions play a pivotal role in matrix computation and applications. While general dense matrix-vector multiplications and linear equation solvers are prohibitively expensive, matrix decompositions offer fast alternatives for matrices meeting specific properties. This dissertation delves into my contributions to two fast matrix multiplication algorithms and one fast linear equation solver algorithm tailored for certain matrices and applications, all based on efficient matrix decompositions. Fast dimensionality reduction methods in spectral clustering, based on efficient eigen-decompositions, are also explored.</p><p dir="ltr">The first matrix decomposition introduced is the "kernel-independent" interpolative decomposition butterfly factorization (IDBF), acting as a data-sparse approximation for matrices adhering to a complementary low-rank property. Constructible in $O(N\log N)$ operations for an $N \times N$ matrix via hierarchical interpolative decompositions (IDs), the IDBF results in a product of $O(\log N)$ sparse matrices, each with $O(N)$ non-zero entries. This factorization facilitates rapid matrix-vector multiplication in $O(N \log N)$ operations, making it a versatile framework applicable to various scenarios like special function transformation, Fourier integral operators, and high-frequency wave computation.</p><p dir="ltr">The second matrix decomposition accelerates matrix-vector multiplication for computing multi-dimensional Jacobi polynomial transforms. Leveraging the observation that solutions to Jacobi's differential equation can be represented through non-oscillatory phase and amplitude functions, the corresponding matrix is expressed as the Hadamard product of a numerically low-rank matrix and a multi-dimensional discrete Fourier transform (DFT) matrix. This approach utilizes $r^d$ fast Fourier transforms (FFTs), where $r = O(\log n / \log \log n)$ and $d$ is the dimension, resulting in an almost optimal algorithm for computing the multidimensional Jacobi polynomial transform.</p><p dir="ltr">An efficient numerical method is developed based on a matrix decomposition, Hierarchical Interpolative Factorization, for solving modified Poisson-Boltzmann (MPB) equations. Addressing the computational bottleneck of evaluating Green's function in the MPB solver, the proposed method achieves linear scaling by combining selected inversion and hierarchical interpolative factorization. This innovation significantly reduces the computational cost associated with solving MPB equations, particularly in the evaluation of Green's function.</p><p dir="ltr"><br></p><p dir="ltr">Finally, eigen-decomposition methods, including the block Chebyshev-Davidson method and Orthogonalization-Free methods, are proposed for dimensionality reduction in spectral clustering. By leveraging well-known spectrum bounds of a Laplacian matrix, the Chebyshev-Davidson methods allow dimensionality reduction without the need for spectrum bounds estimation. And instead of the vanilla Chebyshev-Davidson method, it is better to use the block Chebyshev-Davidson method with an inner-outer restart technique to reduce total CPU time and a progressive polynomial filter to take advantage of suitable initial vectors when available, for example, in the streaming graph scenario. Theoretically, the Orthogonalization-Free method constructs a unitary isomorphic space to the eigenspace or a space weighting the eigenspace, solving optimization problems through Gradient Descent with Momentum Acceleration based on Conjugate Gradient and Line Search for optimal step sizes. Numerical results indicate that the eigenspace and the weighted eigenspace are equivalent in clustering performance, and scalable parallel versions of the block Chebyshev-Davidson method and OFM are developed to enhance efficiency in parallel computing.</p>

<p dir="ltr">This thesis delves into the development and integration of energy-dissipative methods, with applications spanning numerical analysis, optimization, and deep neural networks, primarily targeting gradient flows and porous medium equations. In the realm of optimization, we introduce the element-wise relaxed scalar auxiliary variable (E-RSAV) algorithm, showcasing its robustness and convergence through extensive numerical experiments. Complementing this, we design an Energy-Dissipative Evolutionary Deep Operator Neural Network (DeepONet) to numerically address a suite of partial differential equations. By employing a dual-subnetwork structure and utilizing the Scalar Auxiliary Variable (SAV) method, the network achieves impeccable approximations of operators while upholding the Energy Dissipation Law, even when training data comprises only the initial state. Lastly, we formulate first-order schemes tailored for Wasserstein gradient flows. Our schemes demonstrate remarkable properties, including mass conservation, unique solvability, positivity preservation, and unconditional energy dissipation. Collectively, the innovations presented here offer promising pathways for efficient and accurate numerical solutions in both gradient flows and Wasserstein gradient flows, bridging the gap between traditional optimization techniques and modern neural network methodologies.</p>

<p dir="ltr">This thesis presents advances in numerical methods for partial differential equations (PDEs) and optimization problems, with a focus on improving efficiency, stability, and accuracy across various applications. We begin by addressing 3D Poisson-type equations, developing a GPU-accelerated spectral-element method that utilizes the tensor product structure to achieve extremely fast performance. This approach enables solving problems with over one billion degrees of freedom in less than one second on modern GPUs, with applications to Schrödinger and Cahn<i>–</i>Hilliard equations demonstrated. Next, we focus on parabolic PDEs, specifically the Cahn<i>–</i>Hilliard equation with dynamical boundary conditions. We propose an efficient energy-stable numerical scheme using a unified framework to handle both Allen<i>–</i>Cahn and Cahn<i>–</i>Hilliard type boundary conditions. The scheme employs a scalar auxiliary variable (SAV) approach to achieve linear, second-order, and unconditionally energy stable properties. Shifting to a machine learning perspective for PDEs, we introduce an unsupervised learning-based numerical method for solving elliptic PDEs. This approach uses deep neural networks to approximate PDE solutions and employs least-squares functionals as loss functions, with a focus on first-order system least-squares formulations. In the realm of optimization, we present an efficient and robust SAV based algorithm for discrete gradient systems. This method modifies the standard SAV approach and incorporates relaxation and adaptive strategies to achieve fast convergence for minimization problems while maintaining unconditional energy stability. Finally, we address optimization in the context of machine learning by developing a structure-guided Gauss<i>–</i>Newton method for shallow ReLU neural network optimization. This approach exploits both the least-squares and neural network structures to create an efficient iterative solver, demonstrating superior performance on challenging function approximation problems. Throughout the thesis, we provide theoretical analysis, efficient numerical implementations, and extensive computational experiments to validate the proposed methods. </p>

The sequence-dependent curvature and flexibility of DNA is critical for many biochemically important processes. However, few experimental methods are available for directly probing these properties at the base-pair level. One promising way to predict these properties as a function of sequence is to model DNA with a set of base-pair parameters that describe the local stacking of the different possible base-pair step combinations. In this dissertation research, we develop and study a computational model for predicting the diffusion coefficients of short, relatively rigid DNA fragments from the sequence and the base-pair parameters. We focus on diffusion coefficients because various experimental methods have been developed to measure them. Moreover, these coefficients can also be computed numerically from the Stokes equations based on the three-dimensional shape of the macromolecule. By comparing the predicted diffusion coefficients with experimental measurements, we can potentially obtain refined estimates of various base-pair parameters for DNA. Our proposed model consists of three sub-models. First, we consider the geometric model of DNA, which is sequence-dependent and controlled by a set of base-pair parameters. We introduce a set of new base-pair parameters, which are convenient for computation and lead to a precise geometric interpretation. Initial estimates for these parameters are adapted from crystallographic data. With these parameters, we can translate a DNA sequence into a curved tube of uniform radius with hemispherical end caps, which approximates the effective hydrated surface of the molecule. Second, we consider the solvent model, which captures the hydrodynamic properties of DNA based on its geometric shape. We show that the Stokes equations are the leading-order, time-averaged equations in the particle body frame assuming that the Reynolds number is small. We propose an efficient boundary element method with a priori error estimates for the solution of the exterior Stokes equations. Lastly, we consider the diffusion model, which relates our computed results from the solvent model to relevant measurements from various experimental methods. We study the diffusive dynamics of rigid particles of arbitrary shape which often involves arbitrary cross- and self-coupling between translational and rotational degrees of freedom. We use scaling and perturbation analysis to characterize the dynamics at time scales relevant to different classic experimental methods and identify the corresponding diffusion coefficients. In the end, we give rigorous proofs for the convergence of our numerical scheme and show numerical evidence to support the validity of our proposed models by making comparisons with experimental data. / text

The main topic of the thesis are inverse scattering problems of electromagnetic waves from periodic structures. We study first the direct problem and its numerical resolution using volume integral equation methods with a focus on the case of strongly singular integral operators and discontinuous coefficients. In a second investigation of the direct problem we study conditions on the material parameters under which well-posedness is ensured for all positive wave numbers. Such conditions exclude the existence of guided waves. The considered inverse scattering problem is related to shape identification. To treat this class of inverse problems, we investigate the so-called Factorization method as a tool to identify periodic patterns from measured scattered waves. In this thesis, these measurements are always related to plane incident waves. The outline of the thesis is the following: The first chapter is the introduction where we give the state of the art and new results of the topics studied in the thesis. The main content consists of five chapters, divided into two parts. The first part deals with the scalar case where the TM electromagnetic polarization is considered. In the second chapter we present the volume integral equation method with new results on Garding inequalities, convergence theory and numerical validation. The third chapter is devoted to the analysis of the Factorization method for the inverse scalar problem as well as some numerical experiments. The second part is dedicated to the study of 3-D Maxwell's equations. The fourth and fifth chapters are respectively generalizations of the results of the second and third ones to the case of Maxwell's equations. The sixth chapter contains the analysis of uniqueness conditions for the direct scattering problem, that is, absence of guided modes.

In this thesis, we study the existence of pathwise random periodic solutions to both the semilinear stochastic differential equations with linear multiplicative noise and the semilinear stochastic partial differential equations with linear multiplicative noise in a Hilbert space. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases, and then perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0, T],L2Ω,Rd)) or C([0, T],L2(Ω x O)) and Schauder's fixed point theorem to show the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations.

Cette étude s’inscrit dans le domaine de la simulation électromagnétique des problèmes de grande taille tels que la diffraction d’ondes planes par de larges plateformes et le rayonnement d’antennes aéroportées. Elle consiste à développer une méthode combinant décomposition en sous-domaines et compression hiérarchique des équations intégrales de frontière. Pour cela, nous rappelons dans un premier temps les points importants de la méthode des équations intégrales de frontière et de leur compression hiérarchique par l’algorithme ACA (Adaptive Cross Approximation). Ensuite, nous présentons la formulation IE-DDM (Integral Equations – Domain Decomposition Method) obtenue à partir d’une représentation intégrale des sous-domaines. Les matrices résultant de la discrétisation de cette formulation sont stockées au format H-matrice (matricehiérarchique). Un solveur spécialement adapté à la résolution de la formulation IE-DDM et à sa représentation hiérarchique a été conçu. Cette étude met en évidence l’efficacité de la décomposition en sous-domaines en tant que préconditionneur des équations intégrales. De plus, la méthode développée est rapide pour la résolution des problèmes à incidences multiples ainsi que la résolution des problèmes basses fréquences / This thesis is about the electromagnetic simulation of large scale problems as the wave scattering from aircrafts and the airborne antennas radiation. It consists in the development of a method combining domain decomposition and hierarchical compression of the surface integral equations. First, we remind the principles of the boundary element method and the hierarchical representation of the surface integral equations with the Adaptive Cross Approximation algorithm. Then, we present the IE-DDM formulation obtained from a sub-domain integral representation. The matrices resulting of the discretization of the formulation are stored in the H-matrix format. A solver especially fitted with the hierarchical representation of the IE-DDM formulation has been developed. This study highlights the efficiency of the sub-domain decomposition as a preconditioner of the integral equations. Moreover, the method is fast for the resolution of multiple incidences and the resolution of low frequencies problems