arXiv subjects

Zhaohui Fu

Publications and source records attributed to Zhaohui Fu.

7 recordsLinked to original sources

arXiv2026

Optimal Sobolev Approximation by Deterministic and Random Shallow Sigmoidal Networks

Shallow networks with prescribed or randomly sampled hidden parameters are widely used as numerical trial spaces, yet their optimal Sobolev approximation power with standard smooth sigmoidal activations in general dimension remains unresolved. We establish the corresponding optimal rates for a class of smooth sigmoidal activations with Schwartz-class derivative decay, including $\tanh$, the logistic sigmoid, and the error function $erf$. We first construct deterministic direction--offset dictionaries with $M$ features such that every $u\in H^k(\Omega)$ can be approximated with error of order $M^{-(k-m)/d}$ in $H^m(\Omega)$ for all $0\le m\le k$. This rate is optimal in the sense of Kolmogorov widths for Sobolev balls. We further prove that dictionaries obtained by independent parameter sampling from any prescribed density bounded away from zero attain the same approximation exponent with high probability, up to logarithmic oversampling. The analysis develops a sigmoidal ridge representation and combines it with deterministic or probabilistic quadrature in direction--offset space while retaining polynomial control of the output coefficients. Numerical experiments across a broad range of dimensions, target regularities, and Sobolev error norms recover the predicted algebraic rates for both deterministic and random feature dictionaries.

math.NA↗

arXiv2026

A Discrete-Time Random Feature Method for Nonlinear Evolution Equations with Implicit-Explicit Runge--Kutta Time Stepping

We study a discrete-time random feature method for nonlinear, time-dependent partial differential equations. In contrast to continuous-time formulations that treat time as an additional input variable, the method advances the solution step by step, with each time level computed from previously available states. The spatial solution at each step is represented in the random feature trial space, and the time discretization is given by an implicit-explicit Runge--Kutta (IMEX-RK, 4 stages, third-order) scheme. After splitting the operator into linear and nonlinear parts, each stage admits a linear least-squares formulation, which avoids nonlinear least-squares solves. We also derive a global error estimate for the fully discrete method, separating the contributions of the stage-wise RFM approximation, perturbations in the least-squares coefficients, and the temporal discretization. Numerical experiments for the Allen--Cahn, Burgers, Korteweg--De Vries, and Cahn--Hilliard equations show relative $L^2$-errors of order $10^{-6}$ and convergence rates consistent with the third-order IMEX scheme. A comparison with an IMEX-PINN variant shows that the proposed method achieves higher accuracy at substantially lower computational cost.

math.NA↗

arXiv2025

Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity

We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution.

math.NA↗

arXiv2024

Higher-Order Energy-Decreasing Exponential Time Differencing Runge-Kutta methods for Gradient Flows

In this paper, we develop a general framework for constructing higher-order, unconditionally energy-stable exponential time differencing Runge-Kutta methods applicable to a range of gradient flows. Specifically, we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation. Our analysis reveals that commonly used third-order and fourth-order ETDRK schemes fail to meet these conditions. To address this, we introduce new third-order ETDRK schemes, designed with appropriate stabilization, which satisfy these conditions and thus guarantee the unconditional energy decaying property. We conduct extensive numerical experiments with these new schemes to verify their accuracy, stability, behavior under large time steps, long-term evolution, and adaptive time stepping strategy across various gradient flows. This study is the first to examine the unconditional energy stability of high-order ETDRK methods, and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.

math.NA↗

arXiv2022

Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

math.NA↗

arXiv2022

Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equation

It is difficult to design high order numerical schemes which could preserve both the maximum bound property (MBP) and energy dissipation law for certain phase field equations. Strong stability preserving (SSP) Runge-Kutta methods have been developed for numerical solution of hyperbolic partial differential equations in the past few decades, where strong stability means the non-increasing of the maximum bound of the underlying solutions. However, existing framework of SSP RK methods can not handle nonlinear stabilities like energy dissipation law. The aim of this work is to extend this SSP theory to deal with the nonlinear phase field equation of the Allen-Cahn type which typically satisfies both maximum bound preserving (MBP) and energy dissipation law. More precisely, for Runge-Kutta time discretizations, we first derive a general necessary and sufficient condition under which MBP is satisfied; and we further provide a necessary condition under which the MBP scheme satisfies energy dissipation.

math.NA↗

arXiv2021

Equivalent formulations of the oxygen depletion problem, other implicit free boundary value problems, and implications for numerical approximation

The Oxygen Depletion problem is an implicit free boundary value problem. The dynamics allow topological changes in the free boundary. We show several mathematical formulations of this model from the literature and give a new formulation based on a gradient flow with constraint. All formulations are shown to be equivalent. We explore the possibilities for the numerical approximation of the problem that arise from the different formulations. We show a convergence result for an approximation based on the gradient flow with constraint formulation that applies to the general dynamics including topological changes. More general (vector, higher order) implicit free boundary value problems are discussed. Several open problems are described.

math.AP↗