Time Discretization Parameters Research Articles

A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen–Cahn equation

Abstract In this work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise $\dot{W}^\varepsilon $ tending to rough as $\varepsilon \rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sinica, 15, 407–438). We prove a posteriori error estimates for the $m$-dimensional problem, $m\leq 4$ for a general class of space-time finite element spaces. The a posteriori bound is growing only polynomially in $\varepsilon ^{-1}$ if the step length $h$ is bounded by a positive power of $\varepsilon $. This agrees with the restriction posed so far in the a priori error analysis of continuous finite element schemes for the $\varepsilon $-dependent deterministic Allen–Cahn or deterministic and stochastic Cahn–Hilliard equation. As an application, we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the a posteriori estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $\dot{W}^\varepsilon $ as defined in Weber (2010, On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. Henri Poincare Probab. Stat., 46, 965–975) through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one- and two-dimensional problems in various cases of interest, and compare with the deterministic ones.

Estimation of Infinitesimal Generators for Unknown Stochastic Hybrid Systems via Sampling: A Formal Approach

In this letter, we develop a data-driven framework with formal confidence bounds for the estimation of infinitesimal generators for continuous-time stochastic hybrid systems with unknown dynamics. The proposed approximation scheme employs both time discretization and sampling from the solution process, and estimates the infinitesimal generator of the solution process via a set of data collected from trajectories of systems. We assume some mild continuity assumptions on the dynamics of the system and quantify the closeness between the infinitesimal generator and its approximation while ensuring an a-priori guaranteed confidence bound. To provide a reasonable closeness precision, we discuss significant roles of both time discretization and number of data in our approximation scheme. In particular, for a fixed number of data, variance of the estimation converges to infinity when the time discretization goes to zero. The proposed approximation framework guides us how to jointly select a suitable data size and a time discretization parameter to cope with this counter-intuitive behavior. We demonstrate the effectiveness of our proposed results by applying them to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> jet engine compressor with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unknown</i> dynamics.

Asymptotic of the discrete volume-preserving fractional mean curvature flow via a nonlocal quantitative Alexandrov theorem

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball if the dimension N≤7 and the fractional esponent s≈1, or for any s∈(0,1) when N=2. As an intermediate result we establish a fractional quantitative Alexandrov type estimate for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero. Furthermore, analogous results for the classical perimeter are obtained in the limit s→1.

The Yoccoz–Birkeland livestock population model coupled with random price dynamics

We study a random version of the population-market model proposed by Arlot, Marmi, and Papini in Arlot et al. (2019). The latter model is based on the Yoccoz–Birkeland integral equation and describes a time evolution of livestock commodities prices which exhibits endogenous deterministic, stochastic behavior. We introduce a stochastic component inspired by the Black–Scholes market model into the price equation, and we prove the existence of a random attractor and a random invariant measure. We compute numerically the fractal dimension and the entropy of the random attractor, and show its convergence to the deterministic one as the volatility in the market equation tends to zero. We also investigate in detail the dependence of the attractor on the choice of the time-discretization parameter. We implement several statistical distances to quantify the similarity between the discretized systems’ attractors and the original ones. In particular, following a work by Cuturi (2013), we use the Sinkhorn distance. This distance is a discrete and penalized version of the Optimal Transport Distance between two measures, given a transport cost matrix.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl &amp; Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

Multilevel minimization for deep residual networks

We present a new multilevel minimization framework for the training of deep residual networks (ResNets), which has the potential to significantly reduce training time and effort. Our framework is based on the dynamical system’s viewpoint, which formulates a ResNet as the discretization of an initial value problem. The training process is then formulated as a time-dependent optimal control problem, which we discretize using different time-discretization parameters, eventually generating multilevel-hierarchy of auxiliary networks with different resolutions. The training of the original ResNet is then enhanced by training the auxiliary networks with reduced resolutions. By design, our framework is conveniently independent of the choice of the training strategy chosen on each level of the multilevel hierarchy. By means of numerical examples, we analyze the convergence behavior of the proposed method and demonstrate its robustness. For our examples we employ a multilevel gradient-based methods. Comparisons with standard single level methods show a speedup of more than factor three while achieving the same validation accuracy.

On the Choice of Interface Parameters in Robin–Robin Loosely Coupled Schemes for Fluid–Structure Interaction

We consider two loosely coupled schemes for the solution of the fluid–structure interaction problem in the presence of large added mass effect. In particular, we introduce the Robin–Robin and Robin–Neumann explicit schemes where suitable interface conditions of Robin type are used. For the estimate of interface Robin parameters which guarantee stability of the numerical solution, we propose a new strategy based on the optimization of the reduction factor of the corresponding strongly coupled (implicit) scheme, by means of the optimized Schwarz method. To check the suitability of our proposals, we show numerical results both in an ideal cylindrical domain and in a real human carotid. Our results showed the effectiveness of our proposal for the calibration of interface parameters, which leads to stable results and shows how the explicit solution tends to the implicit one for decreasing values of the time discretization parameter.

On the stability of a loosely-coupled scheme based on a Robin interface condition for fluid-structure interaction

We consider a loosely-coupled algorithm for fluid-structure interaction based on a Robin interface condition for the fluid problem (explicit Robin-Neumann scheme). We study the dependence of the stability of this method on the interface parameter in the Robin condition. In particular, for a model problem we find sufficient conditions for instability and stability of the method. In the latter case, we find a stability condition relating the time discretization parameter, the interface parameter, and the fluid and structure densities. Numerical experiments confirm the theoretical findings and highlight optimal choices of the interface parameter that guarantee accurate solutions.

Data-Driven Estimation of Infinitesimal Generators of Stochastic Systems

Abstract This paper is concerned with a data-driven approach for the estimation of infinitesimal generators of continuous-time stochastic systems with unknown dynamics. We first approximate the infinitesimal generator of the solution process via a set of data collected from trajectories of the unknown system. The approximation utilizes both time discretization and sampling from the solution process. Assuming proper continuity assumptions on dynamics of the system, we then quantify the closeness between the infinitesimal generator and its approximation while providing a priori guaranteed confidence bound. We demonstrate that both the time discretization and the number of data play significant roles in providing a reasonable closeness precision. Moreover, for a fixed size of data, variance of the estimation converges to infinity when the time discretization parameter goes to zero. The formulated error bound shows how to pick proper data size and time discretization jointly to prevent this counter-intuitive behavior. The proposed results are demonstrated on a case study.

On convergence of Chorin’s projection method to a Leray–Hopf weak solution

The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a $$C^5$$ -solution, while Temam demonstrated an abstract argument to obtain a Leray–Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the authors extend Chorin’s result with full details to obtain convergent finite difference approximation of a Leray–Hopf weak solution to the incompressible Navier–Stokes equations on an arbitrary bounded Lipschitz domain of $${{\mathbb {R}}}^3$$ with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong $$L^2$$ -convergence (up to a subsequence) under the scaling condition $$ h^{3-\alpha }\le \tau $$ (no upper bound is necessary), where $$h,\tau $$ are space, time discretization parameters, respectively, and $$\alpha \in (0,2]$$ is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.

Diffusion MRI simulation of realistic neurons with SpinDoctor and the Neuron Module

The diffusion MRI signal arising from neurons can be numerically simulated by solving the Bloch-Torrey partial differential equation. In this paper we present the Neuron Module that we implemented within the Matlab-based diffusion MRI simulation toolbox SpinDoctor. SpinDoctor uses finite element discretization and adaptive time integration to solve the Bloch-Torrey partial differential equation for general diffusion-encoding sequences, at multiple b-values and in multiple diffusion directions. In order to facilitate the diffusion MRI simulation of realistic neurons by the research community, we constructed finite element meshes for a group of 36 pyramidal neurons and a group of 29 spindle neurons whose morphological descriptions were found in the publicly available neuron repository NeuroMorpho.Org. These finite elements meshes range from having 15,163 nodes to 622,553 nodes. We also broke the neurons into the soma and dendrite branches and created finite elements meshes for these cell components. Through the Neuron Module, these neuron and cell components finite element meshes can be seamlessly coupled with the functionalities of SpinDoctor to provide the diffusion MRI signal attributable to spins inside neurons. We make these meshes and the source code of the Neuron Module available to the public as an open-source package.To illustrate some potential uses of the Neuron Module, we show numerical examples of the simulated diffusion MRI signals in multiple diffusion directions from whole neurons as well as from the soma and dendrite branches, and include a comparison of the high b-value behavior between dendrite branches and whole neurons. In addition, we demonstrate that the neuron meshes can be used to perform Monte-Carlo diffusion MRI simulations as well. We show that at equivalent accuracy, if only one gradient direction needs to be simulated, SpinDoctor is faster than a GPU implementation of Monte-Carlo, but if many gradient directions need to be simulated, there is a break-even point when the GPU implementation of Monte-Carlo becomes faster than SpinDoctor. Furthermore, we numerically compute the eigenfunctions and the eigenvalues of the Bloch-Torrey and the Laplace operators on the neuron geometries using a finite elements discretization, in order to give guidance in the choice of the space and time discretization parameters for both finite elements and Monte-Carlo approaches. Finally, we perform a statistical study on the set of 65 neurons to test some candidate biomakers that can potentially indicate the soma size. This preliminary study exemplifies the possible research that can be conducted using the Neuron Module.

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrodinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of $$L^2$$ compared to classical results in dimension d, which are limited to higher-order (sufficiently smooth) Sobolev spaces $$H^s$$ with $$s>d/2$$ . In particular, we are able to establish a global error estimate in $$L^2$$ for $$H^1$$ solutions which is roughly of order $$\tau ^{ {1\over 2} + { 5-d \over 12} }$$ in dimension $$d \le 3$$ ( $$\tau $$ denoting the time discretization parameter). This breaks the “natural order barrier” of $$\tau ^{1/2}$$ for $$H^1$$ solutions which holds for classical numerical schemes (even in combination with suitable filter functions).

Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier–Stokes–Voigt equations

We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional Navier–Stokes–Voigt equations with a quadratic objective functional and box control constraints. The space-time error estimates of order $$O(\sqrt{\tau }+h)$$ , where $$\tau $$ and h are respectively the time and space discretization parameters, are proved for the difference between the locally optimal controls and their discrete approximations.

Asymptotic preserving trigonometric integrators for the quantum Zakharov system

We present a new class of asymptotic preserving trigonometric integrators for the quantum Zakharov system. Their convergence holds in the strong quantum regime $$\vartheta = 1$$ as well as in the classical regime $$\vartheta \rightarrow 0$$ without imposing any step size restrictions. Moreover, the new schemes are asymptotic preserving and converge to the classical Zakharov system in the limit $$\vartheta \rightarrow 0$$ uniformly in the time discretization parameter. Numerical experiments underline the favorable error behavior of the new schemes with first- and second-order time convergence uniformly in $$\vartheta $$ , first-order asymptotic convergence in $$\vartheta $$ and long time structure preservation properties.

Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

AbstractWe investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

Formal and Efficient Synthesis for Continuous-Time Linear Stochastic Hybrid Processes

Stochastic processes are expressive mathematical tools for modeling real-world systems that are subject to uncertainty. It is hence crucial to be able to formally analyze the behavior of these processes, especially in safety-critical applications. Most of the existing formal methods are not designed for continuous-time processes, and those that are typically suffering from state explosion in practice. This article introduces a theoretical framework and a scalable computational method for formal analysis and control synthesis for switched diffusions, a class of stochastic models with linear dynamics that are continuous in both time and space domains; the focus is on safety with possible extensions to other properties. The proposed framework first constructs a finite abstraction in the form of an uncertain Markov process through discretization of both time and space domains. The errors caused by the discretization in each domain are formally characterized and cast into the abstraction model. Then, a strategy that maximizes the probability of the safety property and is robust against the errors is synthesized over the abstraction model. Finally, this robust strategy is mapped to a switching strategy for the stochastic processes that guarantees the safety property. The framework is demonstrated in three case studies, including one that illustrates the tradeoff of the error contribution by the time and space discretization parameters.

Numerical modeling of pulse propagation through a double-negative metamaterial with adjacent absorptive and gain Lorentz dispersions: A comparison

The state-of-the-art time-domain numerical techniques, the finite-deference time-domain (FDTD) method and the pseudospectral time-domain (PSTD) method, are applied to simulate the physical propagation process of a pulse through a double-negative metamaterial with adjacent absorptive and gain Lorentz dispersions. The algorithmic feasibility and modeling accuracy of the two numerical methods combined with the common techniques for Lorentz dispersion implementation are demonstrated from the obtained simulation results and compared with the analytical solution. It is shown that the FDTD method with material parameters averaged at the slab’s two interfaces works better than that without averaging and also better than the case of PSTD method with the same time and space discretization parameters. The proposed techniques are useful for the ultra-accurate modeling of pulse interaction with metamaterials.

Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls

The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ C h 2 , then L 2 error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.

A construction of a uniform continuous minimizing movement associated with a singular functional

A minimizing movement is constructed associated with a singular functional introduced by Alt and Caffarelli in order to study a free boundary problem. The main purpose of the present research is to construct a minimizing movement, which is uniformly continuous with respect to both time and space variables. The strategy is to regularize the singular term of time discretized functionals, and then to pass to the limit in the regularization parameter in the sense of € \Gamma -convergence keeping the time discretization parameter fixed.

Nonlinear damped partial differential equations and their uniform discretizations

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is then inferred from the previous results, by adapting the ideas to deal with linear systems.Our results cover, for instance, the Schrödinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, the nonlinear transport equation, as well as certain classes of equations with nonlocal terms.