Evaluating the Frechet derivative of the matrix exponential

1992 ◽  
Vol 63 (1) ◽  
pp. 213-226 ◽  
Author(s):  
Roy Mathias
Keyword(s):  
Fréchet Derivative ◽  
Matrix Exponential ◽  
Frechet Derivative ◽  
The Matrix

scholarly journals Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

2013 ◽  
Vol 35 (4) ◽  
pp. C394-C410 ◽  
Author(s):  
Awad H. Al-Mohy ◽  
Nicholas J. Higham ◽  
Samuel D. Relton
Keyword(s):  
Condition Number ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
Matrix Logarithm ◽  
The Matrix

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

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.

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

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.

A Splitting method for the calculation of the matrix exponential

Analysis ◽  
1994 ◽  
Vol 14 (2-3) ◽  
pp. 103-112 ◽  
Author(s):  
Eberhard U. Stichel
Keyword(s):  
Splitting Method ◽  
Matrix Exponential ◽  
The Matrix

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

The Matrix Exponential

SIAM Review ◽  
1996 ◽  
Vol 38 (3) ◽  
pp. 507-512 ◽  
Author(s):  
I. E. Leonard
Keyword(s):  
Matrix Exponential ◽  
The Matrix
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