Iterative Equation Solver Research Articles

An iterative equation solver with low sensitivity on the initial value

The objective of this publication is to reduce the sensitivity of iterative equation solvers on the initial value. To this end, at the hand of Newton’s method, we exemplify how to reformulate the initial problem by means of a set of generalized moment generating functions. The approach allows to choose that very function, which is best approximated by a linear function and thus allows to set up an efficient iteration procedure. As a result of this, the number of iterations required to meet a given precision goal is significantly reduced in comparison to Newton’s method especially for large deviations between the initial value and the actual root. At the hand of seven academic examples and three applications we demonstrate that the computing time of the discussed approach reveals a far lower susceptibility on the initial value when compared to results from Newton’s method. This insensitivity offers the prospect to implement iterative equation solvers for applications with strict real-time requirements such as power system simulation or on-demand control algorithms on embedded systems with low computing power. We are confident that the devised methodology may be generalized to other well-established iteration algorithms.

Fast Iterative Integral Equation Solver for Acoustic Scattering by Inhomogeneous Objects Using the Butterfly Approximation.

The acoustic scattering by highly inhomogeneous objects is analyzed by a method-of-moment solver for the volume integral equation. To enable the treatment of acoustically large scatterers of various topologies, the iterative numerical solution of the resulting system is accelerated via a kernel independent algebraic compression scheme: blocks of the hierarchically partitioned moment stiffness matrix are expressed in butterfly form that, for volume problems, scales favorably compared to the popular low-rank approximation. A detailed description of the algorithm, as implemented in this work, is provided. Validations of the numerical formulation, parameter tuning, and performance study of the fast method for acoustically large objects are presented, in various settings and for a range of examples, representative of biomedical and oceanographic applications.

Very high thermoelectric power factor near magic angle in twisted bilayer graphene

Recent research on twisted bilayer graphene (TBG) uncovered that its twist-angle-dependent electronic structure leads to a host of unique properties, such as superconductivity, correlated insulating states, and magnetism. The flat bands that emerge at low twist angles in TBG result in sharp features in the electronic density of states, affecting transport. Here we show that they lead to superior and tuneable thermoelectric (TE) performance. Combining an iterative Boltzmann transport equation solver and electronic structure from an exact continuum model, we calculate TE transport properties of TBG at different twist angles, carrier densities, and temperatures. Our simulations show the room-temperature TE power factor (PF) in TBG reaches 40 mW m−1 K−2, significantly higher than single-layer graphene and among the highest reported to date. The peak PF is observed near the magic angle, at a twist angle of ≈1.3∘, and near complete band filling. We elucidate that its dependence is driven by two opposing forces: the band gap between the flat and remote bands suppresses bipolar transport and improves the Seebeck coefficient but the gap decreases with twist angle; on the other hand, the Fermi velocity, which impacts conductivity, is smallest at the magic angle and rises with twist angle. We observed a further increase in the PF of TBG with decreasing temperature. The strong TE performance, along with the ability to fine-tune transport using twist angle, makes TBG an interesting candidate for future research and applications in energy conversion and thermal sensing and management.

Solving Combined Field Integral Equation With Deep Neural Network for 2-D Conducting Object

Solving the combined field integral equation (CFIE) for the large-scale scattering problem is computationally expensive. In this letter, we investigate the feasibility of applying deep learning to solve the CFIE for 2-D perfect electrically conducting objects. Inspired by the conjugate gradient method, an iterative deep neural network is designed to learn the manner of solving the surface current density from the CFIE, with the input being the coefficient matrix of the equation. This process involves physics through surface integration and need less iterations than the conventional iterative equation solver. In numerical tests, we evaluate the network's performance by comparing the predicted surface current density and bistatic scattering cross section with the solutions rigorously computed. This method provides an insight into applying machine learning techniques together with electromagnetic (EM) physics to fast EM computation with the same level of accuracy as traditional method.

An iterative three-dimensional parabolic equation solver for propagation above irregular boundaries.

This paper describes the development of an iterative three-dimensional parabolic equation solver that takes into account the effects of irregular boundaries and refraction from a layered atmosphere. A terrain-following coordinate transformation, based on the well-known Beilis-Tappert mapping, is applied to the narrow-angle parabolic equation in an inhomogeneous media. The main advantage of this approach, which has been used in two dimensions in the past, is the simplification of the impedance boundary condition at the earth's surface. The transformed initial-boundary value problem is discretized using the Crank-Nicholson marching scheme in the propagating direction and second-order finite-differences in the transversal plane. The proposed method relies on an efficient iterative fixed-point solver, which involves the inversion of tridiagonal matrices only. The accuracy of the method is evaluated through a comparison with boundary element simulations in a homogeneous atmosphere above a Gaussian hill. Results show that transversal scattering occurs in the shadow zone of the obstacle where the two-dimensional parabolic equation underestimates the pressure amplitude. The model is particularly suited for the simulation of infrasound in a three-dimensional environment with realistic topographies.

On the convergence of spectral deferred correction methods

In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE) solver is typically described wherein a low-order approximation (such as forward or backward Euler) is lifted to higher order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right hand side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as difference between the current and previous iterate always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying "solver" is inconsistent the underlying ODE. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers to which we present some examples.

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.

Implementations of parallel software for crack analyses based on the improved XFEM

The extended finite element method (XFEM) has been a state-of-the-art tool for crack analyses. However, the method has been long hampered in engineering applications by two major issues. One is the global stiffness matrix in static analyses being highly ill-conditioned. The other one is the inconsistent energy transfer between the neighboring time steps in dynamic analyses. The first issue leads to bad convergence in large-scale engineering applications where iterative equation solvers are mainly used. The second issue causes poor accuracy and difficulties in implementations of dynamic analyses. In order to overcome these two issues, the improved XFEM is proposed, based on an extra-dof-free PU approximation. Thereby, parallel software for crack analyses, PANDA_Fracture, is developed based on the proposed method with an HPC framework for numerical applications—JAUMIN. As numerical examples indicate, the software PANDA_Fracture has several distinguished features. Firstly, it introduces highly accurate crack tip enrichment and supports arbitrary crack propagation. Secondly, it supports both implicit and explicit dynamic analyses. Thirdly, it supports multiple crack analyses with high efficiency. Fourthly, it scales up to one thousand CPU cores for high-performance 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.

A new simple multidomain fast multipole boundary element method

A simple multidomain fast multipole boundary element method (BEM) for solving potential problems is presented in this paper, which can be applied to solve a true multidomain problem or a large-scale single domain problem using the domain decomposition technique. In this multidomain BEM, the coefficient matrix is formed simply by assembling the coefficient matrices of each subdomain and the interface conditions between subdomains without eliminating any unknown variables on the interfaces. Compared with other conventional multidomain BEM approaches, this new approach is more efficient with the fast multipole method, regardless how the subdomains are connected. Instead of solving the linear system of equations directly, the entire coefficient matrix is partitioned and decomposed using Schur complement in this new approach. Numerical results show that the new multidomain fast multipole BEM uses fewer iterations in most cases with the iterative equation solver and less CPU time than the traditional fast multipole BEM in solving large-scale BEM models. A large-scale fuel cell model with more than 6 million elements was solved successfully on a cluster within 3 h using the new multidomain fast multipole BEM.

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.

A New Equation Solver for Modeling Turbulent Flow in Coupled Matrix-Conduit Flow Models.

Karst aquifers represent dual flow systems consisting of a highly conductive conduit system embedded in a less permeable rock matrix. Hybrid models iteratively coupling both flow systems generally consume much time, especially because of the nonlinearity of turbulent conduit flow. To reduce calculation times compared to those of existing approaches, a new iterative equation solver for the conduit system is developed based on an approximated Newton-Raphson expression and a Gauß-Seidel or successive over-relaxation scheme with a single iteration step at the innermost level. It is implemented and tested in the research code CAVE but should be easily adaptable to similar models such as the Conduit Flow Process for MODFLOW-2005. It substantially reduces the computational effort as demonstrated by steady-state benchmark scenarios as well as by transient karst genesis simulations. Water balance errors are found to be acceptable in most of the test cases. However, the performance and accuracy may deteriorate under unfavorable conditions such as sudden, strong changes of the flow field at some stages of the karst genesis simulations.

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.

Application of a Coupled Eulerian–Lagrangian approach on pile installation problems under partially drained conditions

A numerical technique for modeling the soil as a two-phase medium by means of a user material subroutine is presented. The approach is applied to a classical FE analysis in a Lagrangian formulation using an explicit time integration rule and to a Coupled Eulerian–Lagrangian approach. The application of these two approaches to the problem of pile jacking into fully saturated soil under partially drained conditions is investigated and compared to the solution obtained by use of an iterative equation solver. The influence of the installation process as well as the influence of the permeability on the surrounding soil is investigated.

Impact of data distribution on the parallel performance of iterative linear solvers with emphasis on CFD of incompressible flows

A parallel data structure that gives optimized memory layout for problems involving iterative solution of sparse linear systems is developed, and its efficient implementation is presented. The proposed method assigns a processor to a problem subdomain, and sorts data based on the shared entries with the adjacent subdomains. Matrix---vector-product communication overhead is reduced and parallel scalability is improved by overlapping inter-processor communications and local computations. The proposed method simplifies the implementation of parallel iterative linear equation solver algorithms and reduces the computational cost of vector inner products and matrix---vector products. Numerical results demonstrate very good performance of the proposed technique.

An easy method to accelerate an iterative algebraic equation solver

This article proposes to add a single function calls to an iterative algebraic equation solver with an order n convergence rate, and to raise (for n>1) the order of convergence to 2n−1.

A new preconditioning technique for implicitly coupled multidomain simulations with applications to hemodynamics

In cardiovascular blood flow simulations a large portion of computational resources is dedicated to solve the linear system of equations. Boundary conditions in these applications are critical for obtaining accurate and physiologically realistic solutions, and pose numerical challenges due to the coupling between flow and pressure. Using an implicit time integration setting can lead to an ill-conditioned tangent matrix that causes deterioration in performance of traditional iterative linear equation solvers (LS). In this paper we present a novel and efficient preconditioner (PC) for this class of problems that exploits the strong coupling between the flow and pressure. We implement this PC in a LS algorithm designed for solving systems of equations governing incompressible flows. Excellent efficiency and stability properties of the proposed method are illustrated on a set of clinically relevant hemodynamics simulations.

Hybrid conjugate gradient-Occam algorithms for inversion of multifrequency and multitransmitter EM data

We describe novel hybrid algorithms for inversion of electromagnetic geophysical data, combining the computational and storage efficiency of a conjugate gradient approach with an Occam scheme for regularization and step-length control. The basic algorithm is based on the observation that iterative solution of the symmetric (Gauss-Newton) normal equations with conjugate gradients effectively generates a sequence of sensitivities for different linear combinations of the data, allowing construction of the Jacobian for a projection of the original full data space. The Occam scheme can then be applied to this projected problem, with the tradeoff parameter chosen by assessing fit to the full data set. For EM geophysical problems with multiple transmitters (either multiple frequencies or source geometries) an extension of the basic hybrid algorithm is possible. In this case multiple forward and adjoint solutions (one each for each transmitter) are required for each step in the iterative normal equation solver, and each corresponds to the sensitivity for a separate linear combination of data. From the perspective of the hybrid approach, with conjugate gradients generating an approximation to the full Jacobian, it is advantageous to save all of the component sensitivities, and use these to solve the projected problem in a larger subspace. We illustrate the algorithms on a simple problem, 2-D magnetotelluric inversion, using synthetic data. Both the basic and modified hybrid schemes produce essentially the same result as an Occam inversion based on a full calculation of the Jacobian, and the modified scheme requires significantly fewer steps (relative to the basic hybrid scheme) to converge to an adequate solution to the normal equations. The algorithms are expected to be useful primarily for 3-D inverse problems for which the computational burden is heavily dominated by solution to the forward and adjoint problems.

A simulation methodology for heat and cold distribution in thermo-hydronic networks

This paper presents a simulation methodology to analyze hydronic heat distribution systems in a fast and user friendly way. As suggested in its name, the “Base Circuit Methodology” (BCM) is based on the observation that thermo-hydronic networks can be built up as a modular composition of elementary “Base Circuits” (BCs). Once the hydronic and thermodynamic behavior of such basic components is described in a set of dedicated equations, complex thermal distribution networks can easily be modeled by connecting the basic sub models. In addition to control performance simulations (accuracy, stability, speed) the BCM puts extra effort into energy efficiency analysis. In fact, every BC is a local sub unit in which heat flows are gathered, divided or changed in terms of temperature and/or flow. Therefore the BCM model setup yields the opportunity to analyze the net heat transport and its adaptations while crossing the network. Doing so, system designers get the efficiency variables more structured, leading to straightforward abilities to optimize heat and cold distribution. Practical examples prove the benefits of the methodology. Moreover, a test installation was built in which flows, pressures, and temperatures are confronted with the simulation results. The simulations are processed by means of the iterative equation solver EES (Engineering Equation Solver; ©F-chart) which has been experienced as a very compliant software package. As a result the methodology is delivered as a validated and open source library.