Iterative Linear Equation Research Articles

Decentralized ϵ‐Nash strategy for linear quadratic mean field games using a successive approximation approach

AbstractThis paper presents a successive approximation method for decentralized strategy design in the large‐scale linear quadratic (LQ) Gaussian game. The strategy consists of transforming the original mean field game (MFG) problem into solving decoupled ordinary differential equations (ODEs) whose numerical solutions are obtained by a new two‐loop iteration algorithm. It should be noted that we employ the augmented model technique and the LQ framework to derive these low‐dimensional solvable ODEs, which is the cornerstone of constructing the decentralized ‐Nash strategy. In addition, the quadratic ODEs contained therein are approximately solved for by a sequence of iterative linear ordinary differential equations (LODEs) with guaranteed convergence. A numerical example is given to show the effectiveness of the proposed algorithm.

Fast Desynchronization Algorithms for Decentralized Medium Access Control Based on Iterative Linear Equation Solvers

We tackle the problem of having multiple transmitters cooperating to be desynchronized using a distributed algorithm. Although this problem can also be found in surveillance, it has the most impact in achieving a fair access to a wireless shared communication medium at the medium access control layer in the context of wireless sensor networks. In this article, we first theoretically investigate the convergence rate of various optimization algorithms, giving closed-form expressions for the parameters achieving the best worst-case convergence rate. We then show that a recently proposed time-varying parameters Nesterov algorithm applied to this problem has worse performance assuming one can determine the number of sensors in the network. In order to remove such an assumption, the problem is seen as the solution of a linear equation corresponding to the first optimality condition. Both theoretically and in simulation, we show that using the Gauss–Seidel method improves the speed of convergence, although its performance degrades for large network sizes. In simulations, it is shown the behavior for various number of wireless devices, emphasizing how the algorithms actually perform in comparison with their worst-case theoretical rates for different network sizes.

Representation of the solutions of the linear iterative equation by means of series

Representation of the solutions of the linear iterative equation by means of series

Unifying relations between iterative linear equation solvers and explicit Euler approximations for associated parabolic regularized equations

Iterative methods to solve linear equation systems are widely used in computational physics, engineering and many areas of applied mathematics. In recent works, performance improvements have been achieved based on modifications of several classes of iterative algorithms by various research communities driven by different perspectives and applications. This note presents a brief analysis of conventional and unifying perspectives by highlighting relations between several well-known iterative methods to solve linear equation systems and explicit Euler approximations of the associated parabolic regularized equations. Special cases of equivalence and general relations between different iterative methods such as Jacobi iterations, Richardson iterations, Steepest Descent and Quasi-Newton methods are shown and discussed. The results and discussion extend the conventional perspectives on these iterative methods and give way to intuitive physical interpretations and analogies. The accessibly presented relations give complementary educational insights and aim to inspire transdisciplinary developments of new iterative methods, solvers and preconditioners.

Indirect Visual Simultaneous Localization and Mapping Based on Linear Models

This paper proposes an indirect visual simultaneous localization and mapping (V-SLAM) for large-scale outdoor environments based on linear models. With the scale-invariant feature transform (SIFT) algorithm, a design combining key-frame insertion and map management is proposed to avoid redundant computation and unreliable landmarks. Then, an iterative linear equation is employed to update landmarks. With the supplement of reliable map data, absolute camera poses can be obtained using a simple linear equation based on a linearity index. In addition, a linear model helps the system to detect candidates of looped frames and discard invalid loops by means of an outlier weight function. If a loop is detected, an improved trajectory bending algorithm is introduced to optimize the estimations of poses and landmarks. To evaluate the effectiveness of the proposed approach, extensive experiments are conducted using various sequences from a well-known public dataset. Experimental results show that the proposed V-SLAM outperforms other existing methods in terms of various metrics.

New numerical process solving nonlinear infinite-dimensional equations

Solving a nonlinear equation in a functional space requires two processes: discretization and linearization. In recent paper Grammant et al. (J Integral Equ Appl 26:413–436, 2014), the authors study the difference between applying them in one and in the other order. Linearizing first the nonlinear problem and discretizing the linear problem will be called option (B). Discretizing first the nonlinear problem and linearizing the discrete nonlinear problem will be called option (C). In this paper, we propose a new numerical process equivalent to the option (B): we linearize first the original nonlinear problem with an alternative linearization scheme than that used in the option (B), then we discretize the resulting iterative linear equations using a projection method to implement the corresponding finite-dimensional problem. The aims of this new process are intended to get weaker the theoretical assumptions and to give a powerful numerical performance. We give sufficient conditions to deal with the convergence results. Finally, as a numerical application, we solve a system of Fredholm equations of the second kind. The accuracy and efficiency of this process are illustrated in some numerical examples.

Estimation of Calculation Time of Iterative Linear Equation Solvers Using Deep Learning － Proposal of Learning Model and its Feasibility Evaluation －

Estimation of Calculation Time of Iterative Linear Equation Solvers Using Deep Learning － Proposal of Learning Model and its Feasibility Evaluation －

Homogenization of the elasticity problem with periodically located cracks

In this paper we study the nonlinear problem of elasticity with periodically located cracks. On the edges of these cracks non-penetration conditions are given, which leads to a variational inequality. The period of distribution of cracks, as well as their sizes, depends on the small parameter.In this work the corresponding varational problem and its homogenized problems are presented. We derive the necessary estimates for the theorem about convergence of the solutions of the variational inequality to the solution of the homogenised problem. For the first corrector of the classical asymptotic expansion we construct a penalty equation and a linear iterative equation in integral form. The convergence theorems for corresponding problems are formulated.

Вариационная задача для упругого тела с малыми периодически расположенными трещинами

Рассматривается нелинейная задача о равновесии упругого тела с периодически расположенными трещинами. При этом на краях трещин ставятся односторонние ограничения, что приводит к вариационному неравенству. Период распределения трещин, а также их размеры зависят от малого параметра. Поведение решения задачи с периодически расположенными трещинами определяется двумя первыми членами u0(x), u1(x, y) асимптотического разложения. В статье изучается решение вариационного неравенства на ячейке периодичности (локальная задача). Для первого корректора u1(x, y) строится уравнение со штрафом и линейное итерационное уравнение в интегральной форме. Доказано, что последовательность решений задачи со штрафом при стремлении малого параметра регуляризации к нулю сходится к решению задачи на ячейке. Показано, что приближенное решение итерационного уравнения сходится сильно к решению уравнения со штрафом.

Simulations of Core Collapse Supernova Explosion on PEZY-SC Processors and GPUs

The core collapse supernovae are one of key phenomena to understand the history of the Universe and the origin of heavy elements. To understand their explosion mechanism, large scale numerical simulations are essential that require to solve a multi-physics system described by coupled equations of hydrodynamics and neutrino-radiation transfer in multidimensions. Since the neutrino transfer is governed by the Boltzmann equation in six-dimensional space, necessary computational resource rapidly increases as the number of grids in simulations grows. So far numerical studies have been performed mostly on massively parallel computers and only a few studies have been made using accelerator architectures, such as GPUs, despite their large potential. The PEZY-SC processors are novel many-core architecture that have tremendous potential for scientific high-performance computing. While they share typical features with GPUs, one needs to port and optimize an application considering the difference in their multi-level structure of cores and caches to fully make use of their computational performance. In this work, we apply the PEZY-SC processors to the numerical computations of neutrino radiation hydrodynamics under spherically symmetry as a prototype of the multi-dimensional supernova simulations. Adopting the implicit scheme for discretized time evolution, the bottlenecks of the simulations are as follows: (a) An iterative linear equation solver for the coefficient matrix in the evolution equation, (b) Computation of collision term in the Boltzmann equation, and (c) Inversion of a block diagonal matrix used for preconditioning. The most time-consuming part is typically (a), while (b) and (c) are non-negligible. The steps (a) and (c) concern a block tridiagonal matrix composed of O(1000) block matrices each having O(500) ranks. We offload these hot spots to PEZY-SC processors employing PEZY-CL (a variant of OpenCL) in our simulation code based on our previous studies to exploit the GPUs using NVIDIA CUDA framework. We measure the performance on Suiren Blue (wth PEZY-SC) and Suiren2 (PEZY-SC2) systems at KEK and compared with the result on a system with NVIDIA P100 processors. The achieved performance is sufficient for simulations with improved resolutions than the previous model size, which are not sufficient for recent observational progress. We also discuss the similarity and difference of these architectures and how to ease the porting effort as a prospect for application to the multi-dimension simulations.

Democratization of HPC cloud services with automated parallel solvers and application containers

SummaryIn this paper, we investigate several design choices for HPC services at different layers of the cloud computing architecture to simplify and broaden its use cases. We start with the platform‐as‐a‐service (PaaS) layer and compare direct and iterative parallel linear equation solvers. We observe that several matrix properties that can be identified before starting long‐running solvers can help HPC services automatically select the amount of computing resources per job, such that the job latency is minimized and the overall job throughput is maximized. As a proof of concept, we use classical problems in structural mechanics and mesh these problems with increasing granularities leading to various matrix sizes, ie, largest having 1 billion non‐zero elements. In addition to matrix size, we take into account matrix condition numbers, preconditioning effects, and solver types and execute these finite element analysis (FEA) over an IBM HPC cluster. Next, we focus on the infrastructure‐as‐a‐service (IaaS) layer and explore HPC application performance, load isolation, and deployment issues using application containers (Docker) while also comparing them to physical and virtual machines (VM) over a public cloud.

Lipschitzian solutions to linear iterative equations revisited

We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions varphi of equations of the form φ(x)=∫Ωg(ω)φ(f(x,ω))μ(dω)+F(x),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\varphi (x)=\\int _{\\Omega }g(\\omega )\\varphi \\big (f(x,\\omega )\\big )\\mu (d\\omega )+F(x), \\end{aligned}$$\\end{document}where mu is a measure on a sigma -algebra of subsets of Omega and int _{Omega }g(omega )mu (domega )!=!1.

Lipschitzian solutions to linear iterative equations

Lipschitzian solutions to linear iterative equations

A novel lattice Boltzmann model for the solid–liquid phase change with the convection heat transfer in the porous media

A novel lattice Boltzmann (LB) model with double distribution functions is proposed to simulate the solid–liquid phase change problems with the convection heat transfer in the porous media at the representative elementary volume scale. A generalized LB model is adopted to simulate the velocity field of the liquid region in the presence of the porous media. Based on the framework of a LB model for nonlinear convection–diffusion equation, an LB evolution equation without a phase change source term, as well as a newly modified equilibrium distribution function including the effective total enthalpy and thermal capacity ratio, are constructed to solve the effective total enthalpy energy conservation equation for the porous media. The macroscopic effective total enthalpy energy conservation equation can be exactly recovered from the present LB model by the Chapman–Enskog procedure. As compared with the most previous models, the present model can provide a higher computational efficiency, because the iterative step or linear equations solving procedure for the latent heat source term can be avoided. Furthermore, the temperature and heat flux continuity conditions at the interface between phases of different thermophysical properties are inherently satisfied for the present model. In such a case, the differences in both thermal conductivity and specific heat between phases, as well as the variation of porosity can be tackled by independently adjusting the relaxation time and the thermal capacity ratio included in the equilibrium distribution function, respectively. Additionally, the numerical diffusion across the phase interface for temperature, induced by solid–liquid phase change, can be dramatically reduced in the present model through the MRT collision scheme with a special configuration for the relaxation matrix. Numerical tests for several benchmark problems are carried out to validate the wide practicability and the accuracy of the present model. Very good agreements with analytical solutions or previous experimental and numerical results can be achieved, demonstrating the present model can be served as a promising numerical technique for simulating the transient solid–liquid phase change problems in the porous media.

A time-integration method for stable simulation of extremely deformable hyperelastic objects

This paper presents a time integration method for realtime simulation of extremely deformable objects subject to geometrically nonlinear hyperelasticity. In the presented method, the equation of motion of the system is discretized by the backward Euler method, and linearly approximated through the first-order Taylor expansion. The approximate linear equation is solved with the quasi-minimal residual method (QMR), which is an iterative linear equation solver for non-symmetric or indefinite matrices. The solution is then corrected considering the nonlinear term that is omitted at the Taylor expansion. The method does not demand the constitutive law to guarantee the positive definiteness of the stiffness matrix. Experimental results show that the presented method realizes stable behavior of the simulated model under such deformation that the tetrahedral elements are almost flattened. It is also shown that QMR outperforms the biconjugate gradient stabilized method (BiCGStab) in this application.

Four-Component Damped Density Functional Response Theory Study of UV/Vis Absorption Spectra and Phosphorescence Parameters of Group 12 Metal-Substituted Porphyrins.

The influences of group 12 (Zn, Cd, Hg) metal-substitution on the valence spectra and phosphorescence parameters of porphyrins (P) have been investigated in a relativistic setting. In order to obtain valence spectra, this study reports the first application of the damped linear response function, or complex polarization propagator, in the four-component density functional theory framework [as formulated in Villaume et al. J. Chem. Phys. 2010 , 133 , 064105 ]. It is shown that the steep increase in the density of states as due to the inclusion of spin-orbit coupling yields only minor changes in overall computational costs involved with the solution of the set of linear response equations. Comparing single-frequency to multifrequency spectral calculations, it is noted that the number of iterations in the iterative linear equation solver per frequency grid-point decreases monotonously from 30 to 0.74 as the number of frequency points goes from one to 19. The main heavy-atom effect on the UV/vis-absorption spectra is indirect and attributed to the change of point group symmetry due to metal-substitution, and it is noted that substitutions using heavier atoms yield small red-shifts of the intense Soret-band. Concerning phosphorescence parameters, the adoption of a four-component relativistic setting enables the calculation of such properties at a linear order of response theory, and any higher-order response functions do not need to be considered-a real, conventional, form of linear response theory has been used for the calculation of these parameters. For the substituted porphyrins, electronic coupling between the lowest triplet states is strong and results in theoretical estimates of lifetimes that are sensitive to the wave function and electron density parametrization. With this in mind, we report our best estimates of the phosphorescence lifetimes to be 460, 13.8, 11.2, and 0.00155 s for H2P, ZnP, CdP, and HgP, respectively, with the corresponding transition energies being equal to 1.46, 1.50, 1.38, and 0.89 eV.

On Lipschitzian solutions to an inhomogeneous linear iterative equation

Based on iteration of random-valued functions we study the problems of existence, uniqueness and continuous dependence of Lipschitzian solutions \({\varphi}\) of the equation $$\varphi(x)=F(x)-\int_{\Omega} \varphi(f(x, \omega))P(d\omega),$$ where P is a probability measure on a \({\sigma}\) -algebra of subsets of \({\Omega}\) .

Deterministic Random Walk: A New Preconditioner for Power Grid Analysis

Iterative linear equation solvers rely on high-quality preconditioners to achieve fast convergence. For sparse symmetric systems arising from large power grid analysis problems, however, preconditioners generated by traditional incomplete Cholesky factorization are usually of low quality, resulting in slow convergence. On the other hand, preconditioners generated by random walks are quite effective to reduce the number of iterations, though requiring considerable amount of time to compute in a stochastic manner. We propose in this paper a new preconditioning technique for power grid analysis, named deterministic random walk that combines the advantages of the above two approaches. Our proposed algorithm computes the preconditioners in a deterministic manner to reduce computation time, while achieving similar quality as stochastic random walk preconditioning by modifying fill-ins to compensate dropped entries. We have proved that for such compensation scheme, our algorithm will always succeed, which otherwise cannot be guaranteed by traditional incomplete factorizations. We demonstrate that by incorporating our proposed preconditioner, a conjugate gradient solver is able to outperform a state-of-the-art algebraic multigrid preconditioned solver for dc analysis, and is very efficient for transient simulation on public IBM power grid benchmarks.

Symmetry Lie algebras and properties of linear ordinary differential equations with maximal dimension

Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the first-order source equation are obtained. Based on certain properties of iterative equations, finding the solutions is reduced to finding solutions of the second-order source equation. We have therefore found classes of solutions to the source equations by letting the parameters of the source equation be functions of a specific type such as monomials, functions of exponential and logarithmic type.

Scalability of OpenFOAM for bio-medical flow simulations

We study a bio-medical fluid flow simulation using the incompressible, laminar OpenFOAM flow solver icoFoam using iterative linear equation solver and direct solvers (kernel class) such as SuperLU_DIST 3.3 and SuperLU_MCDT (Many-Core Distributed) for the large penta-diagonal and hepta-diagonal matrices coming from the simulation of blood flow in arteries with a structured mesh domain. A realistic simulation for the flow of blood in the heart or vessels in the whole body is a complex problem and may take a very long time, thousands of hours, for the main tasks such as pre-processing (meshing), decomposition and solving the large linear systems. Our aim is to test the potential scaling capability of the fluid solver for multi-petascale systems. We started from the relatively small instances for the whole simulation and solved large linear systems. We measured the wall clock time of single time steps of the simulation. This version gives important clues for a larger version of the problem. Later, we increase the problem size and the number of time steps to obtain a better picture gradually, in our general strategy. We test the performance of the solver icoFoam at TGCC Curie (a Tier-0 system) at CEA, France (see [31]). We achieved scaled speed-up for the largest matrices of 64 million $$\times 64$$×64 million in our dataset to run up to 16,384 cores. In other words, we find that the scalability improves as the problem size increases for this application. As the matrix size quadrupled, the speed-up improves at least 50 % near speed-up saturation point. This shows that there is no structural problem in the software up to this scale. This is an important and encouraging result for the problem. Moreover, we imbedded other direct solvers (kernel class) such as SuperLU_DIST 3.3 and SuperLU_MCDT in addition to the solvers provided by OpenFOAM. Since future exascale systems are expected to have heterogeneous and many-core distributed nodes, we believe that our SuperLU_MCDT software is a good candidate for future systems. SuperLU_MCDT worked up to 16,384 cores for the large penta-diagonal matrices for 2D problems and hepta-diagonal matrices for 3D problems, coming from the incompressible blood flow simulation, without any problem.