Defect-based local error estimators for high-order splitting methods involving three linear operators

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting methods of lower order, e.g. second-order. Our aim is to develop higher-order ADI methods, which are performed by Richardson extrapolation, Crank-Nicolson methods and higher-order LOD methods, based on locally higher-order methods. We discuss the new theoretical results of the stability and consistency of the ADI methods. The main idea is to apply a higher- order time discretization and combine it with the ADI methods. We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives. The higher-order methods are unconditionally stable. Some numerical experiments verify our results. AMS subject classifications: 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50, 80A20, 80M25

We consider initial value problems (IVPs) where we are interested in a quantity of interest (QoI) that is the integral in time of a functional of the solution. For these, we analyze goal oriented time adaptive methods that use only local error estimates. A local error estimate and timestep controller for step-wise contributions to the QoI are derived. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical tests verify these results. We compare performance with classical local error based time-adaptivity and a posteriori based adaptivity using the dual-weighted residual (DWR) method. For dissipative problems, local error based methods show better performance than DWR and the goal oriented method shows good results in most examples, with significant speedups in some cases.

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise.

In this paper, we are concerned with the derivation of a local error representation for exponential operator splitting methods when applied to evolutionary problems that involve critical parameters. Employing an abstract formulation of differential equations on function spaces, our framework includes Schrodinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients. We illustrate the general mechanism on the basis of the first-order Lie splitting and the second-order Strang splitting method. Further, we specify the local error representation for a fourth-order splitting scheme by Yoshida. From the given error estimate it is concluded that higher-order exponential operator splitting methods are favourable for the time-integration of linear Schrodinger equations in the semi-classical regime with critical parameter 0<e≪1, provided that the time stepsize h is sufficiently smaller than $\sqrt[p]{\varepsilon}$ , where p denotes the order of the splitting method.

We consider adaptive time discretization methods for ordinary differential equations where one aims to control the error in a quantity of interest of the form , with j : ℝd → ℝ. In this setting we propose a new timestep controller based on local error estimates of the quantity of interest. The new method converges when the tolerance goes to zero.We experimentally compare the new scheme with the classic norm‐based time‐adaptivity based on local error estimates as well as the dual‐weighted residual (DWR) method. The results show significantly lower efficiency for the DWR method. The local error based schemes are similarly efficient, with the new scheme showing significant improvement in some cases. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Automatic time step control algorithms for structural dynamics

Collections of linear or nonlinear operator equations Au = f are considered which may represent (i) differential or integral equations or (ii) finite-dimensional approximations. Input sets of coefficients a or data f are admitted. The envelope of the set of solutions is to be constructed where this boundary refers (i) to the ränge of values of the solutions or (ii) to a finite-dimensional space. The construction employs either topological boundary mapping or truncated Taylor expansions. Estimates of the local procedural errors are due to suitable a priori sets and interval mathematics. The relation between local and global error estimates is due to boundary mapping or an auxiliary inverse-monotone operator B. The operator B is constructed for the case of arbitrary linear ordinary differential equations with boundary or initial conditions, provided the admitted A satisfy a mild condition.KeywordsBoundary MappingOperator EquationRemainder TermDiscrete AnalogyOuter ApproximationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the efficiency of algorithms simulating a system evolving with Hamiltonian $H=\sum_{j=1}^m H_j$. We consider high order splitting methods that play a key role in quantum Hamiltonian simulation. We obtain upper bounds on the number of exponentials required to approximate $e^{-iHt}$ with error $\e$. Moreover, we derive the order of the splitting method that optimizes the cost of the resulting algorithm. We show significant speedups relative to previously known results.

&lt;abstract&gt;&lt;p&gt;We systematically test families of explicit adaptive step size controllers for solving the diffusion or heat equation. After discretizing the space variables as in the conventional method of lines, we are left with a system of ordinary differential equations (ODEs). Different methods for estimating the local error and techniques for changing the step size when solving a system of ODEs were suggested previously by researchers. In this paper, those local error estimators and techniques are used to generate different types of adaptive step size controllers. Those controllers are applied to a system of ODEs resulting from discretizing diffusion equations. The performances of the controllers were compared in the cases of three different experiments. The first and the second system are heat conduction in homogeneous and inhomogeneous media, while the third one contains a moving heat source that can correspond to a welding process.&lt;/p&gt;&lt;/abstract&gt;

This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are local error estimates in the sense of the local error in the numerical solution of ordinary differential equations. They represent a different approach to the general problem of error estimation, an approach which involves only a single discretized equation instead of a converging family of such equations.

In this study, we give an a posteriori error analysis on the weighted essentially nonoscillatory schemes for the nonlinear scalar conservation laws. This analysis is based on the new concept of numerical smoothness, with some new error analysis mechanisms developed for the finite difference and finite volume discretizations. The local error estimate is of optimal order in space and time. The global error estimate grows linearly in time, because of the direct application of the L1 ‐contraction between entropy solutions in the error propagation analysis. As a beginning, we only deal with smooth solutions in this article. Within the same error propagation framework, when we deal with piecewise smooth solutions later, we only need to work on estimating the local error where smoothness is lost. The smoothness indicators not only serve the purpose of local error estimation, but also serve as a monitor on both the possible numerical instability and the expected solution shapening. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

A local error estimation and adaptive meshing method developed by some of the authors for finite element analysis of 2D electrostatic and magnetostatic problems is now extended to 3D problems. The problems are formulated in terms of scalar potentials and discretized on a tetrahedral mesh using linear shape functions. Local error is estimated by approximately solving an independent differential problem in each tetrahedral element. The elements selected for refinement are subdivided by adding nodes inside the element, or along an edge, depending on their geometrical quality factor.

Geophysical Inverse Theory and Regularization Problems

Due to the accuracy limitation of the navigation system, deviations between the real trajectory and the measured one appear inevitably in airborne synthetic aperture radar (SAR), which degrades the image quality dramatically. To improve the focusing performance, these trajectory deviations should be well estimated and compensated. In this paper, a data-based autofocus approach is proposed to correct the residual 3-D trajectory deviations. This new approach mainly contains two processing stages. The first stage is the local phase error estimation procedure involving small images autofocusing. A gradient function considering smoothness regularization is developed to efficiently achieve the sharpness-maximizing local phase error functions. In the second stage, the local phase error functions are combined to retrieve the residual 3-D trajectory deviations by a proposed weighted total least square method. This approach has been applied on highly squinted and large-swath airborne SAR raw data, respectively. Both real data experiments generate well-focused SAR images by the estimated trajectory parameters, and thus, validate the effectiveness of the proposed autofocus approach.

An a posteriori error estimation strategy for the boundary element solution of two-dimensional heat conduction problem is presented. The error estimate formulation is based on the boundary element equations and employs a least squares fit of the solutions obtained from these equations. A local error estimate is evaluated for the unknown quantity in each element. The global error estimates are found from local (element) quantities. The range and accuracy of the proposed strategy is very good as demonstrated by problems containing smooth and non-smooth solutions.