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.

Reliability of the two-unit priority standby system revisited

2021 ◽  
pp. 1748006X2110518
Author(s):  
Serkan Eryilmaz ◽  
Maxim Finkelstein
Keyword(s):  
Laplace Transform ◽  
Reliability Assessment ◽  
Repair Time ◽  
Matrix Exponential ◽  
Exponential Distributions ◽  
Reliability Indices ◽  
Cold Standby ◽  
The Matrix ◽  
The Laplace Transform ◽  
Standby System

This paper deals with reliability assessment of the repairable two-unit cold standby system when the first, main unit has the better performance level than the second one. Therefore, after its repair, the main unit is always switched into operation. The new Laplace transform representation for the system’s lifetime is obtained for arbitrary operation and repair time distributions of the units. For some particular cases, the Laplace transform of the system is shown to be rational, which enables the use of the matrix-exponential distributions for obtaining relevant reliability indices. The discrete setup of the model is also considered through the corresponding matrix-geometric distributions, which are the discrete analogs of the matrix-exponential distributions.

Generalized coupled transient thermoelastic problem of multilayered spheres by the hybrid numerical method

2003 ◽  
Vol 217 (12) ◽  
pp. 1315-1323
Author(s):  
Z Y Lee ◽  
C L Chang
Keyword(s):  
Laplace Transform ◽  
Finite Difference Methods ◽  
Generalized Thermoelasticity ◽  
Inverse Laplace Transform ◽  
Hybrid Numerical Method ◽  
The Matrix ◽  
The Laplace Transform ◽  
Computational Procedures ◽  
Thermoelastic Problems ◽  
Matrix Similarity

This paper deals with axisymmetric quasi-static coupled thermoelastic problems for multilayered spheres. Laplace transforms and finite difference methods are used to analyse the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. Solutions are obtained for the temperature and thermal deformation distributions for the transient and steady state. It is demonstrated that the computational procedures established in this paper are capable of solving the generalized thermoelasticity problem of multilayered spheres.

scholarly journals Introducing the Dynamic Destination Satisfaction Method: An Analytical Tool to Track Tourism Destination Satisfaction Trends with Repeated Cross-sectional Data

2020 ◽  
pp. 004728752095820
Author(s):  
Andrea Guizzardi ◽  
Marcello M. Mariani
Keyword(s):  
Large Data ◽  
System Level ◽  
Set Covering ◽  
Tourism Destination ◽  
Cross Sectional ◽  
Data Set ◽  
Individual Level ◽  
Specific Attribute ◽  
The Matrix

This study introduces a new method, named Dynamic Destination Satisfaction Method (DDSME), to model tourists’ satisfaction with a destination (and its attributes), breaking it down into an individual-level component (linked to the specific individual tourists’ perceptions) and a system-level (time-related) component (common to all the tourists). Moreover, this work develops a matrix “entropy/trend accuracy” that destination managers can use to understand to what extent managing a specific attribute has increased tourists’ satisfaction with the destination over multiyear time spans. We test the innovative method on a large data set, covering the period 1997-2015 and including almost 0.8 million observations. By doing so, we analyze tourists’ satisfaction with tourism-related sectors and attributes of Italy as an inbound tourism destination and we use the matrix to map out destination attributes over time. The findings indicate that courtesy, art, and food are strategic attributes to enhance satisfaction in the long term.

Faster calculation of the full matrix for least-squares refinement

1999 ◽  
Vol 55 (4) ◽  
pp. 695-699 ◽  
Author(s):  
David H. Templeton
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Large Data ◽  
Bragg Angle ◽  
Large Crystal ◽  
Data Set ◽  
Full Matrix ◽  
Space Groups ◽  
The Matrix ◽  
Atomic Coordinates

Equations derived from a statistical model and valid for all space groups give estimates of the elements of the matrix for least-squares refinement of atomic coordinates and isotropic thermal parameters for a large crystal structure with many diffraction data. The elements are functions of the lengths and directions of Patterson vectors and the distribution of weights as a function of Bragg angle. For a large data set, this matrix, or a portion of it, can be calculated in a time that is approximately proportional to the number of elements calculated and independent of the number of reflections.

scholarly journals Identification of Excitation Parameters Based on TLS - ESPRIT

2018 ◽  
Vol 228 ◽  
pp. 01015
Author(s):  
Guoqiang Xie ◽  
li Zou ◽  
Kansheng Yu ◽  
Jin Zou ◽  
Zhicheng Wang ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Impulse Response ◽  
Least Squares Method ◽  
Matrix Pencil ◽  
System Response ◽  
A Algorithm ◽  
Excitation System ◽  
System Parameters ◽  
The Matrix ◽  
The Laplace Transform

In order to accurately detect excitation system parameters, this paper presents a algorithm based on the TLS-ESPRIT. The matrix pencil algorithm is used to extract the frequency and damping of each component of system response. So it’s necessary to apply the Laplace transform for s function of excitation system. After getting the Laplace transform of a function f(t), as mean as impulse response for excitation system. Then the magnitude and phase of each component of impulse response are estimated by least squares method, thus achieving the excitation system parameters. In the end, the simulation results show when SNR is between 35dB to 30dB, it still accurately identified the parameters.

Analytical Expressions for the Relaxation Moduli of Linear Viscoelastic Composites With Periodic Microstructure

1995 ◽  
Vol 62 (3) ◽  
pp. 786-793 ◽  
Author(s):  
R. Luciano ◽  
E. J. Barbero
Keyword(s):  
Laplace Transform ◽  
Viscoelastic Behavior ◽  
Relaxation Functions ◽  
Periodic Microstructure ◽  
The Matrix ◽  
The Laplace Transform ◽  
Linear Viscoelastic ◽  
The Time Domain ◽  
Viscoelastic Composites ◽  
Long Fibers

In this paper the viscoelastostatic problem of composite materials with periodic microstructure is studied. The matrix is assumed linear viscoelastic and the fibers elastic. The correspondence principle in viscoelasticity is applied and the problem in the Laplace domain is solved by using the Fourier series technique and assuming the Laplace transform of the homogenization eigenstrain piecewise constant in the space. Formulas for the Laplace transform of the relaxation functions of the composite are obtained in terms of the properties of the matrix and the fibers and in function of nine triple series which take into account the geometry of the inclusions. The inversion to the time domain of the relaxation and the creep functions of composites reinforced by long fibers is carried out analytically when the four-parameter model is used to represent the viscoelastic behavior of the matrix. Finally, comparisons with experimental results are presented.

scholarly journals Accuracy and Computational Efficiency of a Graphical Modeling Approach to Linkage Disequilibrium Estimation

2011 ◽  
Vol 10 (1) ◽  
Author(s):  
Haley J Abel ◽  
Alun Thomas
Keyword(s):  
Linkage Disequilibrium ◽  
Missing Data ◽  
Computational Efficiency ◽  
Graphical Model ◽  
Optimal Parameter ◽  
Model Fitting ◽  
Large Data ◽  
Large Data Sets ◽  
Data Sets ◽  
Data Set

We develop recent work on using graphical models for linkage disequilibrium to provide efficient programs for model fitting, phasing, and imputation of missing data in large data sets. Two important features contribute to the computational efficiency: the separation of the model fitting and phasing-imputation processes into different programs, and holding in memory only the data within a moving window of loci during model fitting. Optimal parameter values were chosen by cross-validation to maximize the probability of correctly imputing masked genotypes. The best accuracy obtained is slightly below than that from the Beagle program of Browning and Browning, and our fitting program is slower. However, for large data sets, it uses less storage. For a reference set of n individuals genotyped at m markers, the time and storage required for fitting a graphical model are approximately O(nm) and O(n+m), respectively. To impute the phases and missing data on n individuals using an already fitted graphical model requires O(nm) time and O(m) storage. While the times for fitting and imputation are both O(nm), the imputation process is considerably faster; thus, once a model is estimated from a reference data set, the marginal cost of phasing and imputing further samples is very low.

Quaternion domain Laplace transform of the pulse vector

2021 ◽  
Author(s):  
V.M. Sovetov
Keyword(s):  
Laplace Transform ◽  
Transfer Functions ◽  
Mimo Systems ◽  
Matrix Multiplication ◽  
Time Shift ◽  
Complex Frequency ◽  
Hypercomplex Numbers ◽  
Radio Engineering ◽  
The Matrix ◽  
The Laplace Transform

The Laplace transform (LT) is widely used in radio engineering for signal and circuit analysis. The PL greatly facilitates the solution of differential equations, the calculation of transfer functions, the finding of impulse responses, etc. Multiple-Input Multiple-Output (MIMO) systems are becoming more common today. With input influences on such systems, at the output signals are obtained, the elements of which are closely related to each other, and changes in some influencing elements of the input vector change the values of others. Such changes are usually associated with the preservation of the vector norm during transformation. Obviously, this completely changes the shape of the output response and, accordingly, its spectrum. To calculate such changes, it is possible to use the usual PL of real signals and the corresponding theorems. However, this approach requires a significant investment of time and computational resources. If you change the amplitude, shape, time shift of at least one pulse, you will have to repeat all the calculations again. Quaternion transformations, including the Laplace transform, have been studied in many works. However, these studies are often of a general theoretical nature or are used only to obtain the Fractional Quaternion Laplace Transform of 2D images. To calculate the LT of the impulse vector when using the MIMO scheme, it is proposed to use hypercomplex numbers, in the particular case, quaternions. Quaternion is a hypercomplex number with one scalar and three imaginary numbers i, j, k. To get rid of operations with imaginary numbers, the quaternion is represented as an orthogonal 4×4 matrix. The matrix, in turn, is decomposed into 4 basis matrices. Moreover, operations with matrices correspond to operations with imaginary units and the quaternion as a whole. It is shown that the quaternionic Laplace transform (QLT) of the vector is represented as a one-dimensional integral from 0 to ∞ of the vector. In this case, the matrix exponent in the power of the quaternion frequency matrix S = Eσ + 1/√3(I + J + K)ω is used as the transformation kernel, where E, I, J, K are basis matrices. The main properties of the QLT are considered. It is shown that in terms of the notation form, the properties of the QLT correspond to the properties of the LT of real functions, taking into account the non-commutativity of matrix multiplication. Therefore, to calculate the QLT, it is possible to use the well-known expressions for the LT of real pulses with the replacement of the complex frequency s by the matrix of quaternion frequencies S. Expressions for the QLT are obtained for the pulse vectors, which are often used to solve radio engineering problems. It is shown that for σ = 0 these expressions correspond to the quaternionic Fourier transform of the vector pulses. In general, vector pulses can have different delays, amplitudes and shapes. Expressions are obtained for finding the QLT of such vectors.

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