On the algorithm by Al-Mohy and Higham for computing the action of the matrix exponential: A posteriori roundoff error estimation

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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Optimal parameter selection in Weeks’ method for numerical Laplace transform inversion based on machine learning

Journal of Algorithms & Computational Technology ◽

10.1177/1748302621999621 ◽

2021 ◽

Vol 15 ◽

pp. 174830262199962

Author(s):

Patrick O Kano ◽

Moysey Brio ◽

Jacob Bailey

Keyword(s):

Laplace Transform ◽

Optimal Parameter ◽

Large Data ◽

Matrix Exponential ◽

Resolvent Matrix ◽

Data Set ◽

Network Training ◽

The Matrix ◽

The Laplace Transform ◽

Space Function

The Weeks method for the numerical inversion of the Laplace transform utilizes a Möbius transformation which is parameterized by two real quantities, σ and b. Proper selection of these parameters depends highly on the Laplace space function F( s) and is generally a nontrivial task. In this paper, a convolutional neural network is trained to determine optimal values for these parameters for the specific case of the matrix exponential. The matrix exponential eA is estimated by numerically inverting the corresponding resolvent matrix [Formula: see text] via the Weeks method at [Formula: see text] pairs provided by the network. For illustration, classes of square real matrices of size three to six are studied. For these small matrices, the Cayley-Hamilton theorem and rational approximations can be utilized to obtain values to compare with the results from the network derived estimates. The network learned by minimizing the error of the matrix exponentials from the Weeks method over a large data set spanning [Formula: see text] pairs. Network training using the Jacobi identity as a metric was found to yield a self-contained approach that does not require a truth matrix exponential for comparison.

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Effective Convergence Checks for Verifying Finite Element Stresses at Three-Dimensional Stress Concentrations

Journal of Verification Validation and Uncertainty Quantification ◽

10.1115/1.4042515 ◽

2018 ◽

Vol 3 (3) ◽

Author(s):

J. R. Beisheim ◽

G. B. Sinclair ◽

P. J. Roache

Keyword(s):

Finite Element ◽

Error Estimation ◽

A Posteriori Error Estimates ◽

Three Dimensional ◽

Posteriori Error ◽

Test Problems ◽

A Posteriori ◽

Stress Concentrations ◽

Element Analysis ◽

A Posteriori Error

Current computational capabilities facilitate the application of finite element analysis (FEA) to three-dimensional geometries to determine peak stresses. The three-dimensional stress concentrations so quantified are useful in practice provided the discretization error attending their determination with finite elements has been sufficiently controlled. Here, we provide some convergence checks and companion a posteriori error estimates that can be used to verify such three-dimensional FEA, and thus enable engineers to control discretization errors. These checks are designed to promote conservative error estimation. They are applied to twelve three-dimensional test problems that have exact solutions for their peak stresses. Error levels in the FEA of these peak stresses are classified in accordance with: 1–5%, satisfactory; 1/5–1%, good; and <1/5%, excellent. The present convergence checks result in 111 error assessments for the test problems. For these 111, errors are assessed as being at the same level as true exact errors on 99 occasions, one level worse for the other 12. Hence, stress error estimation that is largely reasonably accurate (89%), and otherwise modestly conservative (11%).

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