Sharp Error Estimates Research Articles

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Non-radiating sources in elastodynamics and their applications in the exterior cloaking

Non-radiating sources in elastodynamics and their applications in the exterior cloaking

Sharp Error Estimate of Variable Time-Step Imex Bdf2 Scheme for Parabolic Integro-Differential Equations with Initial Singularity Arising in Finance

Sharp Error Estimate of Variable Time-Step Imex Bdf2 Scheme for Parabolic Integro-Differential Equations with Initial Singularity Arising in Finance

Robust mixed finite element methods for a quad-curl singular perturbation problem

Robust mixed finite element methods for a quad-curl singular perturbation problem

Towards optimal sensor placement for inverse problems in spaces of measures

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

An Adaptive PML Finite Volume Algorithm for the Scattering by Periodic Gratings

In this paper, we studied the finite volume formulations for solving the diffraction grating problem that is truncated by the perfectly matched layer (PML) technique. Based on a reliable the a-posteriori error estimate, an adaptive PML finite volume method is discussed for the numerical approximation of the diffraction grating problem. The PML parameters are obtained numerically by sharp a-posteriori error estimates of the PML finite volume method such as the thickness of the layer and the medium property. It is worth mentioning in the a-posteriori error estimates that we derive the error representation formula and use a [Formula: see text]-orthogonality property of the residual similar to the Galerkin orthogonality used in the finite element method. Furthermore, the lower bound is established to demonstrate the efficiency of the a-posteriori error estimates. Numerical experiments are given to illustrate the accuracy and robustness of our adaptive algorithm.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

An IMEX-based approach for the pricing of equity warrants under fractional Brownian motion models

In this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models. doi: 10.1017/S1446181123000159

Darcy’s Law for Porous Media with Multiple Microstructures

In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar’s method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.

Sharp error estimates for spatial-temporal finite difference approximations to fractional sub-diffusion equation without regularity assumption on the exact solution

Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $$u\in C^{4}({\bar{\varOmega }})$$ is needed to preserve $${\mathcal {O}}(h^{2})$$ convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where u is the exact solution and h is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach $${\mathcal {O}}(h^{\min (\sigma +\frac{1}{2}-\epsilon ,2)})$$ in both $$l^{2}$$ -norm and $$l^{\infty }$$ -norm in one-dimensional domain when the initial value and source term are both in $${\dot{H}}^{\sigma }(\varOmega )$$ but without any regularity assumption on the exact solution, where $$\sigma \ge 0$$ and $$\epsilon >0$$ being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to $${\mathcal {O}}(h^{2})$$ in $$l^{2}$$ -norm and $${\mathcal {O}}(h^{\min (\sigma +\frac{3}{2}-\epsilon ,2)})$$ in $$l^{\infty }$$ -norm. It is worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann–Liouville fractional derivative, and $${\mathcal {O}}(\tau ^{2})$$ convergence is obtained for all $$\alpha \in (0,1)$$ . Finally, some numerical experiments verify the effectiveness of the built theory.

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.

A Compact Scheme Combining the Fast Time Stepping Method for Solving 2D Fractional Subdiffusion Equations

In this paper, in order to improve the calculation accuracy and efficiency of α-order Caputo fractional derivative (0 < α ≤ 1), we developed a compact scheme combining the fast time stepping method for solving 2D fractional nonlinear subdiffusion equations. In the temporal direction, a time stepping method was applied. It can reach second-order accuracy. In the spatial direction, we utilized the compact difference scheme, which can reach fourth-order accuracy. Some properties of coefficients are given, which are essential for the theoretical analysis. Meanwhile, we rigorously proved the unconditional stability of the proposed scheme and gave the sharp error estimate. To overcome the intensive computation caused by the fractional operators, we combined a fast algorithm, which can reduce the computational complexity from O(N2) to O(Nlog(N)), where N represents the number of time steps. Considering that the solution of the subdiffusion equation is weakly regular in most cases, we added correction terms to ensure that the solution can achieve the optimal convergence accuracy.

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.

AN IMEX-BASED APPROACH FOR THE PRICING OF EQUITY WARRANTS UNDER FRACTIONAL BROWNIAN MOTION MODELS

AbstractIn this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models.

Fast compact finite difference schemes on graded meshes for fourth-order multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions

In this paper, fast compact finite difference schemes are derived for fourth-order multi-term fractional sub-diffusion equations with initial singularity under the first Dirichlet boundary conditions. In contrast to the direct scheme, the fast algorithm adopted to approximate the Caputo derivative reduces the computation costs effectively. Sharp error estimate of the proposed scheme for linear models is rigorously presented by the energy method. Furthermore, to handle more intricate nonlinear models, a crucial Grönwall inequality is then deduced to analyse the stability and convergence of the linearized scheme. It is worth pointing out that the Grönwall inequality is also helpful in numerical analysis of multi-step schemes for other problems, such as, integro-differential equations with multiple fractional derivatives. Ultimately, numerical examples are provided to verify the efficiency of the established difference schemes.

A Gaussian Method for the Square Root of Accretive Operators

Abstract We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss–Legendre rule applied to a special integral formulation of the fractional power. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite-dimensional case is also considered. In this setting, the convergence is shown to be of exponential type. The method is also tested for the computation of a generic fractional power.

Two energy stable variable-step L1 schemes for the time-fractional MBE model without slope selection

A variable-step L1 scheme is proposed for the time-fractional molecular beam epitaxy model without slope selection. By taking advantage of the convex splitting of nonlinear bulk, a sharp L 2 norm error estimate is established under a convergence-solvability-stability (CSS) consistent time-step constraint, that is, the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter ϵ → 0 + . By using the discrete gradient structure of L1 formula, we build up a discrete variational energy dissipation law, which is asymptotically compatible with the classical energy dissipation law, as the fractional order α → 1 − . The stability and convergence of the stabilized convex splitting scheme are also investigated. The resulting scheme is shown to unconditionally preserve the variational energy dissipation law by adding a ( 2 − α ) -order stabilization term like τ n 1 − α ( ϕ n − ϕ n − 1 ) . Numerical experiments are presented to support our theoretical results.

Modal approximation for time-domain elastic scattering from metamaterial quasiparticles

This paper aims at quantitatively understanding the elastic wave scattering due to negative metamaterial structures under wide-band signals in the time domain. Specifically, we establish the modal expansion for the time-dependent field scattered by metamaterial quasiparticles in elastodynamics. By Fourier transform, we first analyze the modal expansion in the time-harmonic regime. With the presence of quasiparticles, we validate such an expansion in the static regime via quantitatively analyzing the spectral properties of the Neumann-Poincaré operator associated with the elastostatic system. We then approximate the incident field with a finite number of modes and apply perturbation theory to obtain such an expansion in the perturbative regime. In addition, we give polariton resonances as simple poles for the elastostatic system. Finally, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates. Dans cet article, nous nous concernons sur les comportements quantitatives de la diffusion d'ondes élatiques issu des structures métamatérielles négatives, ainsi sous les signales de bandes larges dans le domain temporel. Plus précisément, nous avons établi un développement modal pour le champ de diffusion élastody- namique, qui dépend du temps et issu des quasi-particules métamatérielles. Par la transformé de Fourier, nous analysons en première le développement modal dans le régime harmonique en temps. Avec la présence des quasi-particules, nous vérifions un tel développement dans le régime statique via les analyses quantitatives des propriétés spectrales de l'opérateur de Neumann-Poincaré associé au systèm élastostatique. En suite, nous avons donné une approximation du champ incident avec d'un nombre fini de modes et avoir obtenu un tel développement dans le régime pertubatif en appliquant la théorie de perturbation. En plus, nous avons donné les résonnances de polarisation comme des pôles simples pour le system élastique avec une fréquance non nulle. Finalement, nous avons montré que la partie basse-fréquancielle du champ diffusé dans le domain temporel peut être approché en utilisant le développement modal de résonnance avec des éstimations d'erreurs précises.

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

Local Convergence of the FEM for the Integral Fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.