数值分析
This paper investigates properties of complex-valued eigenvalue trajectories for the interior transmission problem parametrized by the index of refraction for homogeneous media. Our theoretical analysis for the unit disk shows that the only…
The cascade remapping method, originally proposed by Nair et al. (2002) for atmospheric modeling, enables efficient and mass conservative semi Lagrangian (SL) transport through successive one dimensional remapping. While widely used in…
Rising demand in AI and automotive applications is accelerating 2.5D IC adoption, with multiple chiplets tightly placed to enable high-speed interconnects and heterogeneous integration. As chiplet counts grow, traditional placement tools,…
In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and…
We study the constructions of piecewise rational interpolation kernels that are supported on the interval $[-2,2]$, and present one novel rational cubic/linear and five quartic/linear interpolation kernels. All proposed kernels are…
This article explores the convergence properties of an $SLIR^\text{T}R^\text{P}D$ endemic model, incorporating Dirac and Radon measures, alongside distributed delays to represent latency and temporary immunity. A class of delays is defined…
We present a hybrid method for time-dependent particle transport that combines Monte Carlo (MC) estimation with a deterministic discrete ordinates (\(S_N\)) solve, augmented by quasi-Monte Carlo (QMC) sampling. For spatial discretizations,…
We present an improved multigrid preconditioner for the acoustic Helmholtz equation with enhanced scalability. Standard multigrid fails to converge for the Helmholtz equation, and the well-known complex shifted Laplacian method overcomes it…
In this paper, we present a unified framework for reduced basis approximations of parametrized partial differential equations defined on parameter-dependent domains. Our approach combines unfitted finite element methods with both classical…
This work primarily focuses on providing full implementation details for Worsey-Farin (WF) spline interpolation over tetrahedral elements. While this spline space is not new and the theory has been covered in other works, there is a lack of…
Many paintings from the 19th century have exhibited signs of fading and discoloration, often linked to cadmium yellow, a pigment widely used by artists during that time. In this work, we develop a mathematical model of the cadmium sulfide…
We address the inverse problem of identifying a time-dependent source coefficient in a one-dimensional heat equation with a fractional Laplacian subject to Dirichlet boundary conditions and an integral nonlocal data. An a priori estimate is…
In this paper, we establish universal approximation theorems for neural networks applied to general nonlinear ill-posed operator equations. In addition to the approximation error, the measurement error is also taken into account in our…
This work develops a non-intrusive, data-driven surrogate modeling framework based on Operator Inference (OpInf) for rapidly solving parameter-dependent matrix equations in many-query settings. Motivated by the requirements of the OpInf…
Sketching techniques have gained popularity in numerical linear algebra to accelerate the solution of least squares problems. The so-called $\varepsilon$-subspace embedding property of a sketching matrix $S$ has been largely used to…
Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning…
In this paper, we propose a new approaches for low rank approximation of quaternion tensors \cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using…
We propose an approach based on Artificial Neural Networks (ANNs) to evaluate geometric constants relevant to the analysis and design of numerical schemes for partial differential equations. These constants play a central role,…
In multi-objective optimization, multiple loss terms are weighted and added together to form a single objective. These weights are chosen to properly balance the competing losses according to some meta-goal. For example, in physics-informed…
Fractional operators (derivatives/integrals) are defined via the integration of the functions. When the function is produced by a spanning set of fractional power functions, traditional quadrature rules often need to be revised, failing to…