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Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method…

The finite element method (FEM) is commonly used in computational cardiac simulations. For this method, a mesh is constructed to represent the geometry and, subsequently, to approximate the solution. To accurately capture curved geometrical…

This paper concerns the numerical approximation of the Euler equations for multicomponent flows. A numerical method is proposed to reduce spurious oscillations that classically occur around material interfaces. It is based on the "Explicit…

In this paper, we introduce the neural empirical interpolation method (NEIM), a neural network-based alternative to the discrete empirical interpolation method for reducing the time complexity of computing the nonlinear term in a reduced…

We propose, analyze, and numerically validate a correction adaptive two-grid finite element method (CAT-GFEM) for nonselfadjoint or indefinite elliptic problems. In contrast to the adaptive two-grid finite element method (ATGFEM) of Li and…

Equilibrium reconstruction, which infers internal magnetic fields, plasmas current, and pressure distributions in tokamaks using diagnostic and coil current data, is crucial for controlled magnetic confinement nuclear fusion research.…

We explore the possibilities of applying structure-preserving numerical methods to a plasma hybrid model with kinetic ions and mass-less fluid electrons satisfying the quasi-neutrality relation. The numerical schemes are derived by finite…

The Quantum Angle Generator (QAG) is a new full Quantum Machine Learning model designed to generate accurate images on current Noise Intermediate Scale (NISQ) Quantum devices. Variational quantum circuits form the core of the QAG model, and…

Spatiotemporally chaotic systems, such as the solutions of some nonlinear partial differential equations, are dynamical systems that evolve toward a lower dimensional manifold. This manifold has an intricate geometry with heterogeneous…

The Shifted Boundary Method (SBM) trades some part of the burden of body-fitted meshing for increased algebraic complexity. While the resulting linear systems retain the standard $\mathcal{O}(h^{-2})$ conditioning of second-order operators,…

Recently, Masked Image Modeling (MIM) achieves great success in self-supervised visual recognition. However, as a reconstruction-based framework, it is still an open question to understand how MIM works, since MIM appears very different…

Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus,…

Let $(M, g)$ be an $n$-dimensional complete Riemannian manifold with $Ric(M)\geq-(n-1)Q$, where $Q\geq0$ is a constant. We obtain an interior gradient bound for minimal graphs in $M\times R$ under some technical assumptions. For details,…

We develop and analyze a new algorithm for empirical risk minimization, which is the key paradigm for training supervised machine learning models. Our method---SAGD---is based on a probabilistic interpolation of SAGA and gradient descent…

In this paper, we present a mathematical analysis of time-dependent $N$-body electronic systems and establish mixed regularity for the corresponding wavefunctions. Based on this, we develop sparse grid approximations to reduce computational…

Noisy quantum devices demand error-mitigation techniques to be accurate yet simple and efficient in terms of number of shots and processing time. Many established approaches (e.g., extrapolation and quasi-probability cancellation) impose…

We focus on the problem of minimizing the sum of smooth component functions (where the sum is strongly convex) and a non-smooth convex function, which arises in regularized empirical risk minimization in machine learning and distributed…

In numerical simulation, structured mesh generation often requires a lot of time and manpower investment. The general scheme for structured quad mesh generation is to find a mapping between the computational domain and the physical domain.…

Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can…

Simulating nonlinear reaction-diffusion dynamics on complex, non-Euclidean manifolds remains a fundamental challenge in computational morphogenesis, constrained by high-fidelity mesh generation costs and symplectic drift in discrete…