Matrix Exponential Research Articles

Improved Fourier Modal Method Analyzing 2-D Ultrathin Periodic Structures Without Solving Eigenvalues

Intensive computation for solving eigenvalues of large eigenmatrix restricted the development of Fourier modal method (FMM) in analyzing periodic structures, especially 2-D ultrathin periodic structures. To alleviate the problem of intensive computation in the traditional FMM algorithm when analyzing 2-D ultrathin periodic structures, an efficient FMM algorithm without solving eigenvalues is proposed in this letter. Based on the first-order Taylor Expansion, the exponential term matrix in the traditional algorithm can be replaced by two simple diagonal matrices, which avoids the large dimensional matrix inversion process, and the newly formed execution equation is only related to the eigenmatrix. Two numerical examples show that the results of the traditional FMM algorithm, improved FMM algorithm and CST simulation, are highly consistent. Compared with the traditional algorithm, the improved algorithm can greatly alleviate the computing-intensive problems, and the improved algorithm can significantly reduce the CPU time by more than 90% and reduce computation memory by about 70%.

Appearing ZGV point in the first flexural branch of Lamb waves in multilayered plates

Appearing the zero group velocity (ZGV) point belonging to the first quasi flexural branch (A1) of Lamb waves propagating in the multilayered traction-free plates containing homogeneous isotropic layers is observed and analyzed; in contrast to the previously studied homogeneous isotropic plates admitting ZGV points that may belong to the higher flexural modes (A2, A3, …) and higher symmetric modes. The detected ZGV point in a multilayered traction-free plate leads to the transverse quasi resonance occurred at the lower frequency than the previously studied resonance associated with the first symmetric mode (S1) and especially higher symmetric and flexural modes. The analysis is done by the combined method comprising Cauchy sextic formalism and the exponential fundamental matrix method.

Stability analysis of delayed Markovian jump systems with delay switching and state signals and applications

AbstractThis article addresses the asymptotical mean square stability problem of continuous‐time delayed Markovian jump systems (MJSs). Different from the existing delayed MJS models, time‐varying delays are contained in not only system state but also switching signal and could lead to great difficulties. An auxiliary system approach is developed to handle the difficulties, in which the traditionally delayed state and a new delayed exponential matrix are both involved. Based on the established model, sufficient stability conditions with solvable forms are given by using a Lyapunov function approach and some novel techniques. More extensions about an MJS under sampled switching and state signals are further considered, and so does the stabilization problem. A numerical example is offered to verify the effectiveness and superiority of the methods proposed in this study.

The Study of A System of Non-Homogeneous Equations Using Matrix Exponential Method and Analysis With Scilab Software 6.1.1

The study for a system of non-homogeneous equations (coupled differential equations) with initial conditions using Matrix Exponential Method has been done through Scilab software as well as manually. We observe that both the procedures for the equations have been satisfied with the initial conditions. Eigen values and corresponding eigen vectors have been determined to get the complete solutions; and the solutions for x and y also have been found manually. The graphs are plotted with Scilab software 6.1.1. A linear relation between x & y is observed by the new outcomes when they plotted. The study of variation of x, y with t shows the curves increase linearly for limited values of t also the study between t, x+y & t, x-y shows that the curves of both are the mirror image about x-axis. There are much more applications of matrix exponential method in science, engineering and computational fields.

Fast algorithm for quantum polar decomposition and applications

The polar decomposition of a matrix is a key element in the quantum linear algebra toolbox. We show that the problem of quantum polar decomposition, recently studied in Lloyd et al. [arXiv:2006.00841], has a simple and concise implementation via the quantum singular value transform (QSVT). We focus on the applications to pretty-good measurements, a close-to-optimal measurement to distinguish quantum states, and the quantum Procrustes problem, the task of learning an optimal unitary mapping between given ``input'' and ``output'' quantum states. By transforming the state-preparation unitaries into a block-encoding, a prerequisite for QSVT, we develop algorithms for these problems whose gate complexity exhibits a polynomial advantage in the size and condition number of the input compared to alternative approaches for the same problem settings [Lloyd et al., arXiv:2006.00841; Gily\'en et al., arXiv:2006.16924]. For these applications of the polar decomposition, we also obtain an exponential speedup in precision compared to Lloyd et al. [arXiv:2006.00841], as the block-encodings remove the need for the costly density matrix exponentiation step. We contribute a rigorous analysis of the approach of Lloyd et al. [arXiv:2006.00841].

3D large-scale transient electromagnetic modeling using a Shift-and-Invert Krylov subspace method

In this paper, the combination of the Shift-and-Invert Krylov (SAI-Krylov) subspace method and the preconditioned conjugate gradient method (PCG) solves the problem of 3D large-scale time-domain transient electromagnetic (TEM) forward modeling. Firstly, the Galerkin method is used to obtain the spatial discretization governing equation of the electric field from the governing equation of the electric field. The electric field can be expressed in the form of a matrix exponential function using the model reduction method, according to the homogeneous ODE. The TEM response of small and medium-scale models can be solved quickly by using the SAI-Krylov technique, selecting the optimal shift γopt, and combining with the direct method solver. The orthogonal basis of the SAI-Krylov subspace is difficult to build by the direct method due to the high coefficient matrix order of the large-scale TEM forward modeling problem, but it can be solved successfully using the PCG method. Since the optimal shift γopt in the SAI-Krylov method applies to the direct method solver, the computation efficiency is quite low if directly applied to the PCG method. In conclusion, it is necessary to re-select the optimal shift γopt when solving the orthogonal basis of SAI-Krylov subspace using the PCG method. In this paper, the optimal shift γopt is selected based on the relationship between the condition number of the coefficient matrix and shift γ to reduce the conditional number of the coefficient matrix and minimize the number of iterations in the calculation process. Furthermore, the tolerance ε and relaxation factors ω in the PCG method are chosen based on the 5% error threshold. By reselecting the shift γ, the calculation time is reduced by 2/3. Finally, the validity and applicability of the proposed method are verified by the homogeneous half-space model with different resistivity, the 3D vertical contact zone model, and the large-scale complex sulfide ore body model.

Model order reduction of linear and bilinear systems via low-rank Gramian approximation

In this paper, we propose a series of model order reduction algorithms based on low-rank Gramian approximation for linear and bilinear systems. The main idea of the approach for linear systems is to use approximate low-rank factors of the controllability and observability Gramians to generate approximate balanced system for the large-scale system. Then, the reduced-order models are obtained by truncating the states corresponding to the smaller approximate Hankel singular values. The low-rank factors are constructed directly from the Laguerre functions expansion coefficient vectors of the matrix exponential functions by solving a recurrence formula instead of Lyapunov equations. In addition, the reduction procedure is modified with the idea of dominant subspace projection method to produce a stable reduced model under certain conditions. Furthermore, our algorithms are extended to bilinear systems successfully, with a series of corresponding algorithms for bilinear systems derived. Finally, numerical experiments are provided to demonstrate the effectiveness of the proposed methods.

Technical Note—On Matrix Exponential Differentiation with Application to Weighted Sum Distributions

On Modeling the Probability Distribution of Stochastic SumsIn the “Technical Note—On Matrix Exponential Differentiation with Application to Weighted Sum Distributions,” Das, Tsai, Kyriakou, and Fusai propose an efficient methodology for approximating the unknown probability distribution of a weighted stochastic sum or time integral. Resulting from earlier contributions based on continuous-time Markov chain approximations of one-dimensional Markov processes is the Laplace transform of the unknown distribution available in exponential matrix form. In this paper, the authors develop a bona fide Pearson curve-fitting approach to this distribution based on the moments, which they recover from the derivatives of the Laplace transform. Motivated by the computational hurdles toward this, they derive computationally efficient closed-form expressions for the derivatives of the matrix exponential. They then apply to pricing average-based options.

A study on the solution of the spatial kinetics equations in the neutron diffusion theory

In this work, recent studies on the solution of the spatial kinetics equations in the neutron diffusion theory are presented. A nodal technique for the treatment of spatial variables is applied, resulting in the definition of average quantities with respect to neutron fluxes, delayed neutron precursor concentrations, and current densities. The current densities are expressed in terms of the average neutron fluxes to adjust the number of additional unknowns that originate in the nodal scheme. Thus, two systems of coupled differential equations, dependent on the temporal variable and involving the neutron fluxes and delayed neutron precursor concentrations must be solved. A proposal for an iterative scheme with independent solutions of these two systems is developed. Analytical solutions are derived for the delayed neutron precursor concentrations equations. For neutron fluxes equations, numerical and analytical solutions are discussed. The latter use Padé approximations and the Schur–Parlett algorithm to evaluate matrix exponentials. Numerical results obtained through the proposed techniques are compared with results available in the literature, showing satisfactory agreement and contributing to establishing benchmark results. The proposed approach of solving the two systems independently lead to a reduction of the stiffness of the system, a relevant issue for the global approach, and required lower computational costs.

Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector

The action of the matrix exponential on a vector eAtv, A∈Cn×n, v∈Cn, appears in problems that arise in mathematics, physics, and engineering, such as the solution of systems of linear ordinary differential equations with constant coefficients. Nowadays, several state-of-the-art approximations are available for estimating this type of action. In this work, two Taylor algorithms are proposed for computing eAv, which make use of the scaling and recovering technique based on a backward or forward error analysis. A battery of highly heterogeneous test matrices has been used in the different experiments performed to compare the numerical and computational properties of these algorithms, implemented in the MATLAB language. In general, both of them improve on those already existing in the literature, in terms of accuracy and response time. Moreover, a high-performance computing version that is able to take advantage of the computational power of a GPU platform has been developed, making it possible to tackle high dimension problems at an execution time significantly reduced.

Extreme Value Analysis for a Markov Additive Process Driven by a Nonirreducible Background Chain

A common assumption in the vast literature on the extremes of spectrally one-sided Markov additive processes (MAPs) is that the continuous-time Markov chain that serves as the background process is irreducible. In the present paper, we consider, motivated by, for example, applications in credit risk, the case in which the irreducibility condition has been lifted, thus allowing the presence of one or more transient classes. More specifically, we consider the distribution of the maximum when the MAP under study has only positive jumps (the spectrally positive case) or negative jumps (the spectrally negative case). The methodology used relies on two crucial previous results: (i) the Wiener–Hopf decomposition for Lévy processes and, in particular, its explicit form in spectrally one-sided cases and (ii) a result on the number of singularities of the matrix exponent of a spectrally one-sided MAP. In both the spectrally positive and negative cases, we derive a system of linear equations of which the solution characterizes the distribution of the maximum of the process. As a by-product of our results, we develop a procedure for calculating the maximum of a spectrally one-sided Lévy process over a phase-type distributed time interval.

Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion

The extension from the planar case to three dimensions of the clothoid curve (Euler spiral) is herein presented, that is, a curve parametrised by arc length, whose curvature and torsion are linear (affine) functions of the arc length. The problem is modelled as a linear time variant system and its stability is studied with Lyapunov techniques. Closed form solutions in terms of the standard Fresnel integrals are for the first time presented and they are valid for a wide family of clothoids; for the remaining cases, numerical methods of order four, based on Lie algebra, Magnus Expansions and Commutator-Free Expansions are provided. These geometric integrators have been optimised to be easily implemented with few lines of code and require only matrix-vector products (avoiding explicit matrix exponentials at each iteration). The achieved computational efficiency has a dramatic impact for several real-time applications, especially for designing trajectories for flying vehicles, as previous approaches were fully numeric and could not take advantage of the present closed form solutions, which render the computation of the 3D clothoid as computationally expensive as its planar companion.

Faster quantum and classical SDP approximations for quadratic binary optimization

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.

A μ-mode integrator for solving evolution equations in Kronecker form

In this paper, we propose a μ-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how μ-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schrödinger equations, and we show that the μ-mode integrator can significantly outperform numerical methods well-established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem.

3D hygro-elastic shell model for the analysis of composite and sandwich structures

The present work proposes the study of the hygrometric loading effects in the static analysis of multilayered composite and sandwich plates and shells. The employed model is based on a general exact 3D shell theory valid for one-layered and multilayered sandwich and composite plates, cylinders and cylindrical/spherical shell panels. The employed 3D equilibrium equations are developed in an orthogonal mixed curvilinear coordinate system for spherical shells in order to obtain the other simpler geometries as particular cases. A layer-wise approach is employed and equilibrium equations are solved in the thickness direction via the exponential matrix method. The partial derivatives in the in-plane directions are exactly calculated by assuming simply-supported edges and harmonic forms for the variables. The presented 3D shell model is extended to hygro-elastic cases by considering moisture content amplitudes imposed at the external surfaces of the structures in steady-state conditions. The moisture content profile through the thickness of the structures can be included in the 3D shell model using three different methodologies: it can be calculated by solving the steady-state 3D or 1D version of the Fick diffusion law, or it can be “a priori” assumed as linear through the thickness. Therefore, the developed system includes three non-homogeneous second order differential equilibrium equations that are solved after the introduction of opportune mathematical layers to consider the curvature effects through the thickness. The reduction to a first order differential equation system is obtained by redoubling the number of variables. The exponential matrix method is employed to accurately define both general and particular solutions. The implemented 3D hygro-elastic shell model is firstly validated and then it is employed with confidence for new benchmarks where the effects of thickness ratio, geometry, lamination scheme, material, thickness layer and imposed moisture content values at the external surfaces are investigated for composite and sandwich plates and shells. Showed results demonstrate the importance of an accurate developing of the elastic part of the 3D shell model combined with an appropriate evaluation of the moisture content profile through the thickness. Assumed linear moisture content profiles are correct only for thin one-layered isotropic structures, the use of the 1D Fick diffusion law allows only the correct definition of the material layer effect, the use of its 3D version allows the correct definition of both material and thickness layer effects.

Demonstration of Density Matrix Exponentiation Using a Superconducting Quantum Processor

Density matrix exponentiation may offer a natively quantum approach to programming quantum computers. A new experiment presents the first demonstration of the protocol in a superconducting quantum processor.

3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

3-D Large-Scale TEM Modeling Using Restarting Polynomial Krylov Method

The transient electromagnetic (TEM) method is widely used in near-surface geophysical prospecting. The high-precision forward and inversion of large-scale complex models is a challenging problem. It is difficult to solve the large-scale problems using the direct solvers due to the large memory requirements. In this article, we present a restarting polynomial Krylov method for modeling 3-D large-scale TEM responses. The mimetic finite volume method is carried out for spatial discretization. The step-off TEM response then can be expressed as a matrix exponential function. The restarting polynomial Krylov method is used to solve the matrix exponential function. For a given restart subspace dimension, the residual is used to obtain the forward response at any time that meets the given accuracy. This method does not need to solve large-scale linear equations. Furthermore, the memory usage is mainly determined by the number of spatial discrete grids and the restart subspace order. The numerical experiments of large-scale model demonstrated that the method is accurate and uses limited memory.

On the Construction of the Kernel Matrix by Primitive BCH Codes for Polar Codes

The polar codes defined by the kernel matrix are a class of codes with low coding-decoding complexity and can achieve the Shannon limit. In this paper, a novel method to construct the 2n-dimensional kernel matrix is proposed, that is based on primitive BCH codes that make use of the interception, the direct sum and adding a row and a column. For ensuring polarization of the kernel matrix, a solution is also put forward when the partial distances of the constructed kernel matrix exceed their upper bound. And the lower bound of exponent of the 2n-dimensional kernel matrix is obtained. The lower bound of exponent of our constructed kernel matrix is tighter than Gilbert-Varshamov (G-V) type, and the scaling exponent is better in the case of 16-dimensional.

Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times

<p style='text-indent:20px;'>We study the <inline-formula><tex-math id="M2">\begin{document}$ MAP/M/s+G $\end{document}</tex-math></inline-formula> queueing model that arises in various multi-server engineering problems including telephone call centers, under the assumption of MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest via the steady-state solution of a certain Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid, for which case discretization of the patience time distribution is required giving rise to a computational complexity depending linearly on the number of discretization levels. Additionally, a novel method is proposed to accurately obtain the first passage time distributions for the virtual and actual waiting times again using CFFQs while approximating the deterministic time horizons by Erlang distributions or more efficient Concentrated Matrix Exponential (CME) distributions. Numerical results are presented to validate the effectiveness of the proposed numerical method.</p>