Representation of the local error for higher-order exponential splitting schemes involving two or three sub-operators

2015 ◽  
Author(s):  
Winfried Auzinger ◽  
Othmar Koch ◽  
Mechthild Thalhammer
Keyword(s):  
Higher Order ◽  
Local Error ◽  
Splitting Schemes ◽  
Exponential Splitting

scholarly journals Computing Exponential for Iterative Splitting Methods: Algorithms and Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Jürgen Geiser
Keyword(s):  
Splitting Method ◽  
Real Life ◽  
Higher Order ◽  
Splitting Methods ◽  
Matrix Exponential ◽  
Approximation Schemes ◽  
Theoretical Point ◽  
Splitting Scheme ◽  
Splitting Schemes ◽  
Practical Applications

Iterative splitting methods have a huge amount to compute matrix exponential. Here, the acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of view, iterative splitting methods are at least alternating Picards fix-point iteration schemes. For practical applications, it is important to compute very fast matrix exponentials. In this paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme. First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential. In this notation, we consider fast approximation schemes to the integral formulations, also known as -functions. Second, the error analysis is explained and applied to the integral formulations. The novelty is to compute cheaply the decoupled exp-matrices and apply only cheap matrix-vector multiplications for the higher-order terms. In general, we discuss an elegant way of embedding recently survey on methods for computing matrix exponential with respect to iterative splitting schemes. We present numerical benchmark examples, that compared standard splitting schemes with the higher-order iterative schemes. A real-life application in contaminant transport as a two phase model is discussed and the fast computations of the operator splitting method is explained.

Higher order weak linearizations of stochastically driven nonlinear oscillators

2007 ◽  
Vol 463 (2083) ◽  
pp. 1827-1856 ◽  
Author(s):  
Nilanjan Saha ◽  
D Roy
Keyword(s):  
Vector Fields ◽  
Probability Distributions ◽  
Strong Solutions ◽  
Nonlinear Oscillators ◽  
Higher Order ◽  
Computational Effort ◽  
Local Error ◽  
True Solution ◽  
Multiplicative Noises ◽  
Backward Euler

We present derivative-free weak and strong solutions of stochastically driven nonlinear oscillators of engineering interest using higher order forms of the locally transversal linearization (LTL) method. Unlike strong solutions, weak stochastic solutions attempt to predict only mathematical expectations of functions of the true solution and are obtainable with much less computational effort. The linearized equations corresponding to the higher order implicit LTL schemes are arrived at using backward Euler–Maruyama and Newmark expansions in order to conditionally replace nonlinear drift and multiplicative diffusion vector fields. We also briefly describe alterations through which explicit forms of such higher order linearizations are obtained. In the weak forms, the Gaussian stochastic integrals, appearing in the linearized solutions, are replaced by random variables with simpler and discrete probability distributions. The resulting local approximations to the true solution are of significantly higher formal order of accuracy, as reflected through local error estimates. Numerical illustrations are presently provided for the Duffing and Van der Pol oscillators driven by additive and multiplicative noises, which are indicative of the numerical accuracy, computational speed and algorithmic simplicity.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

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