Discrete Approximation Scheme Research Articles

Mathematical programs with distributionally robust chance constraints: Statistical robustness, discretization and reformulation

In this paper, we consider mathematical programs with distributionally robust chance constraints (MPDRCC), where the ambiguity set is given by the general moment information. From the contaminated data-driven viewpoint, we first study the qualitative statistical robustness of MPDRCC. Then, motivated by the computational tractability, we investigate the discrete approximation of MPDRCC. The corresponding convergence results of the optimal value and the optimal solution set of the discrete approximation problem are established. After that, a reformulation of the discrete approximation problem is presented under standard assumptions, which is applied to solve MPDRCC approximately according to the above convergence results. Finally, two applications are reported, and some numerical results show that the statistical robustness assertion and the discrete approximation scheme are practical and effective.

Stability and discretization error analysis for the Cahn–Hilliard system via relative energy estimates

The stability of solutions to the Cahn–Hilliard equation with concentration dependent mobility with respect to perturbations is studied by means of relative energy estimates. As a by-product of this analysis, a weak-strong uniqueness principle is derived on the continuous level under realistic regularity assumptions on strong solutions. The stability estimates are further inherited almost verbatim by appropriate Galerkin approximations in space and time. This allows to derive sharp bounds for the discretization error in terms of certain projection errors and to establish order-optimal a priori error estimates for semi- and fully discrete approximation schemes. Numerical tests are presented for illustration of the theoretical results.

Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations

<abstract><p>In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $ H^1 $-Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $ L1 $ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.</p></abstract>

Fast simulation for Gaussian random fields on compact Riemannian manifolds

We develop a fully discrete approximation scheme to fast simulate centered Gaussian random fields (GRFs) over compact Riemannian manifolds. These GRFs can be either anisotropic or isotropic ones. The scheme is proposed by the truncation of a series expansion depending solely on the decay of the angular power spectrum, which is independent of the chosen space and time discretization. The pth moment convergence for the scheme is proved and the error bound with its associated rate of convergence is determined. Three simulation case studies are conducted by applying the Laplacian operator to find the required surface harmonics for a torus endowed with a Riemannian metric and by applying the ellipsoidal Beltrami and Laplacian operators to find the required surface ellipsoidal harmonics respectively for anisotropic and isotropic GRFs over ellipsoids endowed with two different Riemannian metrics.

Fully discrete approximation schemes for rate-independent crack evolution.

Over the past two decades, several distinct solution concepts for rate-independent evolutionary systems driven by non-convex energies have been suggested in an attempt to model properly jump discontinuities in time. Much attention has been paid in this context to the modelling of crack propagation. This paper studies two fully discrete (in time and space) approximation schemes for the rate-independent evolution of a single crack in a two-dimensional linear elastic material. The crack path is assumed to be known in advance, and the evolution of the crack tip along it relies on the Griffith theory. On the time-discrete level, the first scheme is based on local minimization, whereas the second scheme is a regularized version of the first one. The crucial feature of the schemes is their adaptive time-stepping nature, with finer time steps at those points where the evolution of the crack tip might develop a discontinuity. The set of discretization parameters includes the mesh size, crack increment, locality parameter and regularization parameter. In both cases, we explore the interplay between the discretization parameters and derive sufficient conditions on them ensuring the convergence of discrete interpolants to parametrized balanced viscosity solutions of the continuous model. To illustrate the performance of the approximation schemes, we support our theoretical analysis with numerical simulations. This article is part of the theme issue 'Non-smooth variational problems and applications'.

An asymptotic-preserving discretization scheme for gas transport in pipe networks

Abstract We consider the simulation of barotropic flow of gas in long pipes and pipe networks. Based on a Hamiltonian reformulation of the governing system, a fully discrete approximation scheme is proposed using mixed finite elements in space and an implicit Euler method in time. Assuming the existence of a smooth subsonic solution bounded away from vacuum, a full convergence analysis is presented based on relative energy estimates. Particular attention is paid to establishing error bounds that are uniform in the friction parameter. As a consequence, the method and results also cover the parabolic problem arising in the asymptotic large friction limit. The error estimates are derived in detail for a single pipe, but using appropriate coupling conditions and the particular structure of the problem and its discretization, the main results directly generalize to pipe networks. Numerical tests are presented for illustration.

Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms

In the past few years deep artificial neural networks (DNNs) have been successfully employed in a large number of computational problems including, e.g., language processing, image recognition, fraud detection, and computational advertisement. Recently, it has also been proposed in the scientific literature to reformulate high-dimensional partial differential equations (PDEs) as stochastic learning problems and to employ DNNs together with stochastic gradient descent methods to approximate the solutions of such high-dimensional PDEs. There are also a few mathematical convergence results in the scientific literature which show that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters employed to describe the DNN grows at most polynomially both in the PDE dimension d in {mathbb {N}} and the reciprocal of the prescribed approximation accuracy varepsilon > 0. One key argument in most of these results is, first, to employ a Monte Carlo approximation scheme which can approximate the solution of the PDE under consideration at a fixed space-time point without the curse of dimensionality and, thereafter, to prove then that DNNs are flexible enough to mimic the behaviour of the employed approximation scheme. Having this in mind, one could aim for a general abstract result which shows under suitable assumptions that if a certain function can be approximated by any kind of (Monte Carlo) approximation scheme without the curse of dimensionality, then the function can also be approximated with DNNs without the curse of dimensionality. It is a subject of this article to make a first step towards this direction. In particular, the main result of this paper, roughly speaking, shows that if a function can be approximated by means of some suitable discrete approximation scheme without the curse of dimensionality and if there exist DNNs which satisfy certain regularity properties and which approximate this discrete approximation scheme without the curse of dimensionality, then the function itself can also be approximated with DNNs without the curse of dimensionality. Moreover, for the number of real parameters used to describe such approximating DNNs we provide an explicit upper bound for the optimal exponent of the dimension d in {mathbb {N}} of the function under consideration as well as an explicit lower bound for the optimal exponent of the prescribed approximation accuracy varepsilon >0. As an application of this result we derive that solutions of suitable Kolmogorov PDEs can be approximated with DNNs without the curse of dimensionality.

Stability and convergence of Galerkin schemes for parabolic equations with application to Kolmogorov pricing equations in time-inhomogeneous Lévy models

Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time dependent coercive operators. We consider abstract Galerkin approximations in space combined with theta-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular the results apply to partial integro differential equations for option pricing in time-inhomogeneous Levy models and allows for a large variety of option types and models. The derivation builds on the strong foundation laid out by von Petersdorff and Schwab (2003) who provide the respective results for the time-homogeneous case. We discuss the assumptions in the context of option pricing.

Numerical analysis of a dynamic viscoplastic contact problem

In this paper, we study a dynamic contact problem with Clarke subdifferential boundary conditions. The material is assumed to be viscoplastic which has an implicit expression of the stress field in constitutive law. The weak form of the model is governed by an evolutionary hemivariational inequality coupled with an integral equation. We study a fully discrete approximation scheme of the problem and bound the errors. Under appropriate solution regularity assumptions, optimal-order error estimates can be derived. Finally, a numerical example is also included to support our theoretical analysis. Particularly, it gives numerical evidence on the theoretically predicted optimal convergence order.

Discrete Approximation Scheme in Distributionally Robust Optimization

Discrete Approximation Scheme in Distributionally Robust Optimization

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

We consider Galerkin approximations of eigenvalue problems for holomorphic Fredholm operator functions for which the operators do not have the structure “coercive+compact”. In this case the regularity (in the vocabulary of discrete approximation schemes) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. The technique is based on the knowledge of a suitable Test function operator. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework can be successfully applied to analyze e.g. complex scaling/perfectly matched layer methods, problems involving sign-changing coefficients due to meta-materials and also (boundary element) approximations of Maxwell-type equations. We demonstrate the application of our framework to the Maxwell eigenvalue problem for a conductive material.

Discrete approximation of dynamic phase-field fracture in visco-elastic materials

<p style='text-indent:20px;'>This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.</p>

On the observability inequalities of time discrete 2‐D integro‐differential systems in square domains

AbstractIn this article, we consider an observability inequality of time discrete approximation schemes for a class of integro‐differential equations in square domains. The equation is discretized in time by the back‐ward Euler method in combination with Lubich's convolution quadrature. We derive uniform observability inequality for time discretization schemes in which the high frequency components have been filtered. In this way, the well‐known exact observability inequality of the integro‐differential systems (see Loreti and Sforza, Evolution Equations and Control Theory, 2018, Vol. 7, No. 1, pp. 61–77, Thm. 1.2) can be reproduced as the limit, as the time step Δt → 0. The time discrete observability inequality is established by means of a time‐discrete version of the classical harmonic analysis approach. Numerical experiments are presented to show the theoretical analysis.

Existence, uniqueness, and numerical approximations for stochastic Burgers equations

In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.

Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations

Smoothed particle hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the mathematics of SPH, a nonlocal relaxation to the conventional linear steady-state Stokes system for incompressible viscous flows. The nonlocal continuum model is characterized by a smoothing length \(\delta \) which measures the range of nonlocal interactions. It serves as a bridge between the discrete approximation schemes that involve a nonlocal integral relaxation and the local continuum models. We show that for a class of carefully chosen nonlocal operators, the resulting nonlocal Stokes equation is well-posed and recovers the original Stokes equation in the local limit when \(\delta \) approaches zero. For some other commonly used smooth kernels, there are risks in getting ill-posed continuum models that could lead to computational difficulties in practice. This leads us to discuss the implications of our finding on the design of numerical methods.

A variational scheme for hyperbolic obstacle problems

We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions.

Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids

Abstract Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\varOmega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully discrete approximation scheme, using a spatial mixed finite element approximation on general shape-regular simplicial meshes combined with backward Euler time-stepping. We consider the case when the velocity field belongs to the space of solenoidal functions contained in $\textrm{L}^\infty (0,T;\textrm{L}^2(\varOmega )^d)\cap \textrm{L}^q(0,T;\textrm{W}^{1,q}_0(\varOmega )^d)$ with $q\in \left (2d/(d+2), \infty \right )$, which is the maximal range of $q$ with respect to existence of weak solutions. In order to facilitate passage to the limit with the discretization parameters for the sub-range $q \in \left (2d/(d+2), (3d+2)/(d+2) \right )$, we introduce a regularization of the momentum equation by means of a penalty term, and first show convergence of a subsequence of approximate solutions to a weak solution of the regularized problem; we then pass to the limit with the regularization parameter. This is achieved by the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness techniques. For $q \geq (3d+2)/(d+2)$ convergence of a subsequence of approximate solutions to a weak solution can be shown directly, without the regularization term.

On the Observability of Time Discrete Integro-differential Systems

In this paper we study the observability properties of time discrete approximation schemes for some integro-differential equations. The equation is discretized in time by the back-ward Euler method in combination with convolution quadrature. We prove uniform observability results for time discretization schemes in which the high frequency components have been filtered. In this way, the well-known exact observability estimates of the integro-differential systems can be reproduced as the limit, as the time step $$ {\varDelta }t\rightarrow 0 $$ . The discrete observability estimates are established by means of a time-discrete version of the classical harmonic analysis approach.

Dispersive and dissipative properties of the fully discrete bicompact schemes of the fourth order of spatial approximation for hyperbolic equations

The Fourier analysis of fully discrete bicompact fourth-order spatial approximation schemes for hyperbolic equations is presented. This analysis is carried out on the example of a model linear advection equation. The results of Fourier analysis are presented as graphs of the dependence of the dispersion and dissipative characteristics of the bicompact schemes on the dimensionless wave number and the Courant number. The dispersion and dissipative properties of bicompact schemes are compared with those of other widely used difference schemes for hyperbolic equations. It is shown that bicompact schemes have one of the best spectral resolutions among the difference schemes being compared.

Hermite spectral method for Long-Short wave equations

We are concerned with the initial boundary value problem of the Long-Short wave equations on the whole line. A fully discrete spectral approximation scheme is structured by means of Hermite functions in space and central difference in time. A priori estimates are established which are crucial to study the numerical stability and convergence of the fully discrete scheme. Then, unconditionally numerical stability is proved in a space of $H^1({\Bbb R})$ for the envelope of the short wave and in a space of $L^2({\Bbb R})$ for the amplitude of the long wave. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented and numerical results are illustrated to agree well with the convergence order of the discrete scheme.