• Abstract

Since the memory effect is taken into account, the singularly perturbed subdiffusion equation can better describe the diffusion phenomenon with small diffusion coefficients. However, near the boundary configured with non-smooth boundary values, the solution of the singularly perturbed subdiffusion equation has a boundary layer of thickness $\mathcal{O}(ε),$ which brings great challenges to the construction of the efficient numerical schemes. By decomposing the Caputo fractional derivative, the singularly perturbed subdiffusion equation is formally transformed into a class of steadystate diffusive-reaction equation. By means of a kind of tailored finite point method (TFPM) scheme for solving steady-state diffusion-reaction equations and the $\mathcal{L}1$ formula for discretizing the Caputo fractional derivative, we construct a new $\mathcal{L}1$-TFPM scheme for solving singularly perturbed subdiffusion equations. Our proposed numerical scheme satisfies the discrete extremum principle and is unconditionally numerically stable. Besides, we prove that the new TFPM scheme can obtain reliable numerical solutions as $h ≪ ε$ and $ε ≪ h.$ However, there will be a large error loss due to the resonance effect as $h ∼ ε.$ Numerical experimental results can demonstrate the validity of the numerical scheme.

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This paper is concerned with the numerical analyses of finite element methods for the nonlinear backward stochastic Stokes equations (BSSEs) where the forcing term is coupled with $z.$ Under several developed analysis techniques, the error estimates of the proposed semi-discrete and fully discrete schemes, as well as their boundedness, are rigorously presented and established. Optimal convergence rates of the fully discrete scheme are obtained not only for the velocity $u$ and auxiliary stochastic process $z$ but also for the pressure $p.$ For the efficiency of solving BSSEs, the proposed numerical scheme is parallelly designed in stochastic space. Numerical results are finally provided and tested in parallel to validate the theoretical results.

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The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical method for mean curvature flows by using the Onsager principle as an approximation tool. We first show that the mean curvature flow can be derived naturally from the Onsager variational principle. Then we consider a piecewise linear approximation of the curve and derive a discrete geometric flow. The discrete flow is described by a system of ordinary differential equations for the nodes of the discrete curve. We prove that the discrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system can be solved by the improved Euler scheme and this leads to an efficient fully discrete scheme. We first consider the method for a simple mean curvature flow and then extend it to the volume preserving mean curvature flow and also a wetting problem on substrates. Numerical examples show that the method has optimal convergence rate and works well for all the three problems.

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This note discusses a long debated question whether there is randomness in quantum mechanics or not? Einstein’s view on the question is “God does not throw dice”. Our starting point for the discussion is the classification of products in the economic system, called ProductRank, which seems an analog of the “principal component analysis” in statistics. But the former is much more elaborate than the latter. Interestingly, we find an intrinsic common point among economic system, statistics and quantum mechanics, which then leads to a successful classification of the products in economy, as well as a mathematical view of “wave probability” in quantum mechanics. An application to the algorithm for eigenpair is included.

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In this work, we develop and analyze a family of up to fourth-order, unconditionally energy-stable, single-step schemes for solving gradient flows with global Lipschitz continuity. To address the exponential damping/growth behavior observed in Lawson’s integrating factor Runge-Kutta approach, we propose a novel strategy to maintain the original system’s steady state, leading to the construction of an exponential Runge-Kutta (ERK) framework. By integrating the linear stabilization technique, we provide a unified framework for examining the energy stability of the ERK method. Moreover, we show that certain specific ERK schemes achieve unconditional energy stability when a sufficiently large stabilization parameter is utilized. As a case study, using the no-slope-selection thin film growth equation, we conduct an optimal rate convergence analysis and error estimate for a particular three-stage, third-order ERK scheme coupled with Fourier pseudo-spectral discretization. This is accomplished through rigorous eigenvalue estimation and nonlinear analysis. Numerical experiments are presented to confirm the high-order accuracy and energy stability of the proposed schemes.

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This paper presents an innovative approach to computational acoustic imaging of biperiodic surfaces, exploiting the capabilities of an acoustic superlens to overcome the diffraction limit. We address the challenge of imaging physical entities in complex environments by considering the partial differential equations that govern the physics and solving the corresponding inverse problem. We focus on imaging infinite rough surfaces, specifically 2D diffraction gratings, and propose a method that leverages the transformed field expansion. We derive a reconstruction formula connecting the Fourier coefficients of the surface and the measured field, demonstrating the potential for unlimited resolution under ideal conditions. We also introduce an approximate discrepancy principle to determine the cut-off frequency for the truncated Fourier series expansion in surface profile reconstruction. Furthermore, we elucidate the resolution enhancement effect of the superlens by deriving the discrete Fourier transform of white Gaussian noise. Our numerical experiments confirm the effectiveness of the proposed method, demonstrating high subwavelength resolution even under slightly non-ideal conditions. This study extends the current understanding of superlens-based imaging and provides a robust framework for future research.

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The maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations. To propose MBP-preserving schemes with high spatial accuracy, in the first part of this series, we developed a class of time semidiscrete stochastic Runge-Kutta (SRK) methods for semilinear parabolic equations, and constructed the first- and second-order fully discrete MBP-preserving SRK schemes. In this paper, to develop higher order fully discrete MBP-preserving SRK schemes with spectral accuracy in space, we use the Sinc quadrature rule to approximate the conditional expectations in the time semidiscrete SRK methods and propose a class of fully discrete MBP-preserving SRK schemes with up to fourth-order accuracy in time for semilinear equations. Based on the property of the Sinc quadrature rule, we theoretically prove that the proposed fully discrete SRK schemes preserve the MBP and can achieve an exponential order convergence rate in space. In addition, we reveal that the conditional expectation with respect to the Bronwian motion in the time semidiscrete SRK method is essentially equivalent to the exponential Laplacian operator under the periodic boundary condition. Ample numerical experiments are also performed to demonstrate our theoretical results and to show the exponential order convergence rate in space of the proposed schemes.

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Federated learning (FL) has been becoming a popular interdisciplinary research area in both applied mathematics and information sciences. Mathematically, FL aims to collaboratively optimize aggregate objective functions over distributed datasets while satisfying a variety of privacy and system constraints. Different from conventional distributed optimization methods, FL needs to address several specific issues (e.g. non-i.i.d. data and differential private noises), which pose a set of new challenges in the problem formulation, algorithm design, and convergence analysis. In this paper, we will systematically review existing FL optimization research including their assumptions, formulations, methods, and theoretical results. Potential future directions are also discussed.

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$ℓ_1$ based sparse regularization plays a central role in compressive sensing and image processing. In this paper, we propose $ℓ_1$DecNet, as an unfolded network derived from a variational decomposition model, which incorporates $ℓ_1$ related sparse regularizations and is solved by a non-standard scaled alternating direction method of multipliers. $ℓ_1$DecNet effectively separates a spatially sparse feature and a learned spatially dense feature from an input image, and thus helps the subsequent spatially sparse feature related operations. Based on this, we develop $ℓ_1$DecNet+, a learnable architecture framework consisting of our $ℓ_1$DecNet and a segmentation module which operates over extracted sparse features instead of original images. This architecture combines well the benefits of mathematical modeling and data-driven approaches. To our best knowledge, this is the first study to incorporate mathematical image prior into feature extraction in segmentation network structures. Moreover, our $ℓ_1$DecNet+ framework can be easily extended to 3D case. We evaluate the effectiveness of $ℓ_1$DecNet+ on two commonly encountered sparse segmentation tasks: retinal vessel segmentation in medical image processing and pavement crack detection in industrial abnormality identification. Experimental results on different datasets demonstrate that, our $ℓ_1$DecNet+ architecture with various lightweight segmentation modules can achieve equal or better performance than their enlarged versions respectively. This leads to especially practical advantages on resource-limited devices.

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