Numerische Mathematik最新文献

We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy density (gamma (varvec{n})), where (varvec{n}in mathbb {S}^1) represents the outward unit normal vector. We begin with the anisotropic surface diffusion which possesses two well-known geometric structures—area conservation and energy dissipation—during the evolution of the closed curve. By introducing a novel surface energy matrix (varvec{G}_k(varvec{n})) depending on (gamma (varvec{n})) and the Cahn-Hoffman (varvec{xi })-vector as well as a nonnegative stabilizing function (k(varvec{n})), we obtain a new conservative geometric partial differential equation and its corresponding variational formulation for the anisotropic surface diffusion. Based on the new weak formulation, we propose a full discretization by adopting the parametric finite element method for spatial discretization and a semi-implicit temporal discretization with a proper and clever approximation for the outward normal vector. Under a mild and natural condition on (gamma (varvec{n})), we can prove that the proposed full discretization is structure-preserving, i.e. it preserves the area conservation and energy dissipation at the discretized level, and thus it is unconditionally energy stable. The proposed SP-PFEM is then extended to simulate the evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality (and thus no need to re-mesh during the evolution) of the proposed SP-PFEM for simulating anisotropic geometric flows.

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

The method of fundamental solutions has been mainly applied to wave scattering problems in bounded domains and to our knowledge there have not been works addressing density results for general shapes, or addressing the calculation of the complex resonance frequencies that occur in exterior problems. We prove density and convergence of the fundamental solutions approximation in the context of wave scattering problems, with and without a priori knowledge of the frequency, which is of particular importance to detect resonance frequencies for trapping domains. We also present several numerical results that illustrate the good performance of the method in the calculation of complex resonance frequencies for trapping and non trapping domains in 2D and 3D.

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.

Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in (mathbb {R}^{n+1}) ((n=1,2)) when the scatterer, assumed to be a compact subset of (mathbb {R}^ntimes {0}), is a d-set for some (din (n-1,n]), so that, in particular, the scatterer has Hausdorff dimension d. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in (mathbb {R}^2) by Cantor sets, and in (mathbb {R}^3) by Cantor dusts.

Abstract

This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods.

Iterative regularization exploits the implicit bias of optimization algorithms to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure with explicit computational steps able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the (ell _1) penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.

Abstract

Non-stationary signals are ubiquitous in real life. Many techniques have been proposed in the last decades which allow decomposing multi-component signals into simple oscillatory mono-components, like the groundbreaking Empirical Mode Decomposition technique and the Iterative Filtering method. When a signal contains mono-components that have rapid varying instantaneous frequencies like chirps or whistles, it becomes particularly hard for most techniques to properly factor out these components. The Adaptive Local Iterative Filtering technique has recently gained interest in many applied fields of research for being able to deal with non-stationary signals presenting amplitude and frequency modulation. In this work, we address the open question of how to guarantee a priori convergence of this technique, and propose two new algorithms. The first method, called Stable Adaptive Local Iterative Filtering, is a stabilized version of the Adaptive Local Iterative Filtering that we prove to be always convergent. The stability, however, comes at the cost of higher complexity in the calculations. The second technique, called Resampled Iterative Filtering, is a new generalization of the Iterative Filtering method. We prove that Resampled Iterative Filtering is guaranteed to converge a priori for any kind of signal. Furthermore, we show that in the discrete setting its calculations can be drastically accelerated by leveraging on the mathematical properties of the matrices involved. Finally, we present some artificial and real-life examples to show the power and performance of the proposed methods.Kindly check and confirm that the Article note is correctly identified.

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.

The minimum energy path (MEP) is the most probable transition path that connects two equilibrium states of a potential energy landscape. It has been widely used to study transition mechanisms as well as transition rates in the fields of chemistry, physics, and materials science. In this paper, we derive a novel result establishing the stability of MEPs under perturbations of the energy landscape. The result also represents a crucial step towards studying the convergence of various numerical approximations of MEPs, such as the nudged elastic band and string methods.