The Problem of Aperture Delay in Digital Measurement Systems and its Analytic Solution by the Matrix Exponential Method

2017 ◽  
Vol 60 (9) ◽  
pp. 874-880
Author(s):  
A. G. Nagiev ◽  
V. V. Sadikhov ◽  
G. A. Nagiev
Keyword(s):  
Analytic Solution ◽  
Matrix Exponential ◽  
Digital Measurement ◽  
Measurement Systems ◽  
The Matrix ◽  
Exponential Method

Generalized Polar Decompositions in Matrix Exponential

2011 ◽  
Vol 130-134 ◽  
pp. 3023-3026
Author(s):  
Yi Min Tian ◽  
Ao Zhang
Keyword(s):  
Differential Equation ◽  
Lie Group ◽  
Analytic Solution ◽  
Matrix Exponential ◽  
Group Method ◽  
Lie Group Method ◽  
The Matrix ◽  
Numeric Solution ◽  
Basic Ideas ◽  
Solution Manifold

Matrix exponential computstion is a difficulty thing when the order of the matrix get big and big after discretion. When we use Lie group method to get numeric solution of a differential equation, we often face this problem.Li group method is a kind of prosperous method, its basic ideas is to keep the numeric solution in a manifold which is less than the Euclid space while bigger than the analytic solution manifold, so we can get more exact numeric solution than other method. So we discussed the generalized polar decompositions method for matrix exponential.

scholarly journals MATRIX EXPONENTIAL METHODS FOR PARALLEL COMPUTING OF ISOTOPIC DEPLETION AND SPECIES TRANSPORT FOR MOLTEN SALT REACTOR ANALYSIS

2021 ◽  
Vol 247 ◽  
pp. 06047
Author(s):  
Zack Taylor ◽  
Benjamin Collins ◽  
Ivan Maldonado
Keyword(s):  
Differential Equations ◽  
Molten Salt ◽  
Fission Products ◽  
Matrix Exponential ◽  
Fissile Material ◽  
Species Transport ◽  
The Matrix ◽  
Exponential Method ◽  
Reactor Loop ◽  
Exponential Methods

Matrix exponential methods have long been utilized for isotopic depletion in nuclear fuel calculations. In this paper we discuss the development of such methods in addition to species transport for liquid fueled molten salt reactors (MSRs). Conventional nuclear reactors work with fixed fuel assemblies in which fission products and fissile material do not transport throughout the core. Liquid fueled molten salt reactors work in a much different way, allowing for material to transport throughout the primary reactor loop. Because of this, fission product transport must be taken into account. The set of partial differential equations that apply are discretized into systems of first order ordinary differential equations (ODEs). The exact solution to the set of ODEs is herein being estimated using the matrix exponential method known as the Chebychev Rational Approximation Method (CRAM).

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
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