Inverse Iteration Research Articles

Neutron transport criticality calculations using a parallel monolithic multilevel Schwarz preconditioner together with a nonlinear diffusion acceleration method

The multigroup neutron transport criticality calculations using modern supercomputers have been widely employed in a nuclear reactor analysis for studying whether or not a system is self-sustaining. However, the design and development of efficient parallel algorithms for the transport criticality calculations is challenging especially when the number of processor cores is large and an unstructured mesh is adopted. In particular, both the compute time and the memory usage have to be carefully taken into consideration due to the dimensionality of the neutron transport equations. In this paper, we study a monolithic multilevel Schwarz preconditioner for the transport criticality calculations based on a nonlinear diffusion acceleration (NDA) method. In NDA, the multigroup nonlinear diffusion equations are computed using an inexact Jacobian-free Newton method with an initial guess generated from a few inverse power iterations. The computed scalar fluxes and eigenvalue are used to evaluate the fission and scattering terms of the transport equations, and then the nonlinear system of transport equations is simplified to a linear system of equations. The linear systems of equations arising from the discretizations of the nonlinear diffusion equations and the transport equations need to be efficiently solved. We propose a monolithic multilevel Schwarz method that is capable of efficiently handling the systems of linear equations for both the transport system and the diffusion system. However, in the multilevel method, algebraically constructing coarse spaces is expensive and often unscalable. We study a subspace-based coarsening algorithm to address such a challenge by exploring the matrix structures of the transport equations and the nonlinear diffusion equations. We numerically demonstrate that the monolithic multilevel preconditioner with the subspace-based coarsening algorithm is twice as fast as that equipped with an unmodified coarsening approach on thousands of processor cores for an unstructured mesh neutron transport problem with billions of unknowns.

Quantum inverse iteration algorithm for programmable quantum simulators

We propose a quantum inverse iteration algorithm, which can be used to estimate ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares the approximate ground state. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device, and can be used for analog quantum simulators. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models, which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.

A generalized Newton iteration for computing the solution of the inverse Henderson problem

ABSTRACT We develop a generalized Newton scheme called IHNC (inverse hypernetted-chain iteration) for the construction of effective pair potentials for systems of interacting point-like particles. The construction is realized in such a way that the distribution of the particles matches a given radial distribution function. The IHNC iteration uses the hypernetted-chain integral equation for an approximate evaluation of the inverse of the Jacobian of the forward operator. In contrast to the full Newton method realized in the Inverse Monte Carlo (IMC) scheme, the IHNC algorithm requires only a single molecular dynamics computation of the radial distribution function per iteration step and no further expensive cross-correlations. Numerical experiments are shown to demonstrate that the method is as efficient as the IMC scheme, and that it easily allows to incorporate thermodynamical constraints.

逆向迭代离轴非球面检测方法

逆向迭代离轴非球面检测方法

Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

The current paper proposes a parametric eigenvalue problem for the Helmholtz equation on the plane based on two well-known physical models of emission from two-dimensional (2-D) microcavity lasers. The first model has been developed for passive, either lossy or lossless, open cavities and is usually referred to as the Complex-Frequency Eigenvalue Problem. The second model, named the Lasing Eigenvalue Problem (LEP), has been tailored to characterize emission from open cavities, filled in with gain material, on the threshold of nonattenuating in time radiation. Our generalized model contains both of them as special cases. We reduce the original problem to a nonlinear eigenvalue problem for a set of boundary integral equations with weakly singular kernels and formulate it as a parametric eigenvalue problem for a holomorphic Fredholm operator-valued function, which is convenient for mathematical and numerical analysis. Using this formulation and fundamental results of the theory of holomorphic operator-valued functions, we study the properties of the spectrum. For numerical solution of the problem, we propose a Nyström method, prove its convergence, and derive error estimates in the eigenvalue approximation. Combining this discretization with the residual inverse iteration technique, we compute approximate solutions of LEP and compare them with the exact ones and with the results obtained using other numerical methods.

Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency

We propose a new normalized Sobolev gradient flow for the Gross--Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in $H^1$ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.

The revisited total least squares problems with linear equality constraint

The total least squares problem with linear equality constraint is proved to be approximated by an unconstrained total least squares problem with a large weight on the constraint. A criterion for choosing the weighting factor is given, and a QR-based inverse (QR-INV) iteration method is presented. Numerical results show that the QR-INV method is more efficient than the standard QR-SVD procedure and Schaffrin's inverse iteration method, especially for large and sparse matrices.

Quick and Highly Efficient Modal Analysis Method Based on the Reanalysis Technique for Large Complex Structure and Topology Optimization

Modal analysis is widely used to investigate the dynamic characteristics of large and complex structures. For finite element models, iterative solvers are needed to precisely calculate eigenpairs or frequency and vibration mode. However, in cases such as large-scale analysis or reanalysis studies, or optimization design of a huge structure, computational cost may become too time consuming. This paper focuses on the quick structural modal analysis based on the reanalysis technique for large complex structures. Based on the stiffness and mass matrix of the analytical structures, a high precision and efficiency eigensolution is generated by the proposed modal analysis method (the Pseudo Random Independent and Coupling Inverse Iteration (PRICII) method), which combines the pseudo random number initialization, ICII (Independent and Coupling Inverse Iteration) strategy with the double Rayleigh–Ritz analysis. By comparing with the Subspace iteration method, Lanczos method, etc. the large-scale numerical examples show that the actual computational savings of the proposed method are usually higher than 75% with sufficient precision. Also, its applications in topology optimization are greatly effective.

Elemental Image Generation Algorithm Using Reverse Iteration Along Optical Path

We propose a fast elemental image generation algorithm for integral imaging display based on depth information and a two-dimensional scene image. Existing elemental image generation algorithms establish many-to-one mappings from the scene to the elemental image, thereby including redundant calculations. The improved algorithm entails establishing a single mapping from elemental image to the scene. Each pixel in the elemental image matches a unique scene pixel by inverse iteration along the optical path. This establishes a high precision matching and eliminates hole caused by depth discontinuities. In this paper, we obtain a unique mapping solution by inverse iteration along the optical path using a one-pixel step. The time complexity of this algorithm is mainly driven by the pixel sum of the elemental image array. We generate the elemental image seven times faster than the original algorithm. The larger total quantity of the scene pixels further demonstrates the advantages of our algorithm. The experiments show that the update rate can reach more than 30 fps according to the VCD standard (352×240 resolution) using four parallel calculation threads with a four-core Intel® Core(TM) i7-7500U 2.70 GHz CPU despite various delay factors. The results verify the correctness of the theoretical analysis.

Regularity of implicit solution mapping to parametric generalized equation

This paper concerns the study of both local and global metric regularity/Lipschitz-like properties concerning the behavior of the implicit solution mapping associated to parametric generalized equation in metric space. We extend some implicit multifunction results to the addition of two multifunctions both depending on parameters. Through the approach of inverse mapping iteration, several results are established regarding the relations between the (partial) metric regularity/Lipschitz-like moduli of multifunctions used as the defining form of the generalized equation and the corresponding implicit solution mapping, the proof of which is completely self-contained. Finally, a local Lyusternik–Graves Theorem is obtained as an application.

Shifted inverse iteration based multigrid methods for the quad-curl eigenvalue problem

This paper is devoted to multgrid methods for solving the quad-curl eigenvalue problem. The multgrid methods based on the Rayleigh quotient iteration and the inverse iteration with fixed shift are proposed. We prove the error estimates for both methods. Numerical experiments confirm our theoretical analysis and validate the efficiency of both methods.

A Strategy for Uncertainty Evaluation of Airport Operation Safety Based on Interval Inverse Power Method

In order to maintain a safe and suitable state of civil airports, it is necessary to conduct an airport operation safety evaluation. Aiming at uncertainty of airport operation safety evaluation and the deficiencies of analytic hierarchy process, we propose an improved analytic hierarchy process for solving this problem. By estimating the spectral radius of the interval judgment matrix and using it as shift, we propose an interval inverse power method with shift, and apply this algorithm to airport operation safety evaluation. The numerical examples show that this method can comprehensively consider the certainty and uncertainty of airport safety evaluation and compute the combined weight and relative importance of the evaluating indicators. Our method is more flexible and effective than traditional method, and is very suitable for uncertainty evaluation of airport operation safety.

NtkFoam: An OpenFOAM based neutron transport kinetics solver for nuclear reactor simulation

Due to the complexity of detailed nuclear reactor numerical simulation, especially the complicated geometry and multi-physics coupling properties of the advanced reactor, a nuclear reactor kinetics solver for nuclear reactor engineering and design needs to be developed. Based on the open source C++ software OpenFOAM, this work establishes a neutron transport kinetics solver, namely ntkFoam, for the nuclear reactor kinetics simulation, from the governing equations to the detailed implementations. The coupling between multi-group neutron transport equations and the delayed neutron precursor balance equations are considered with using the finite volume method. By introducing the inverse power method to the OpenFOAM, the k-eigenvalue problems are calculated, and by coupling the neutron transport and delayed neutron precursor calculations using the Euler implicit scheme, the transient kinetics problems are simulated. Numerical results show that the proposed ntkFoam can simulate the multi-group neutron transport kinetics problem accurately and flexibility, and both the regular and irregular mesh configurations can be adopted. This work can provide some new perspectives and foundations to nuclear reactor coupling calculations.

Analytical model for the winding process-induced residual stresses of the multilayered filament wound cylindrical composite parts

The residual stresses due to winding tensions have significant influence on the product quality of the filament wound composite parts. In this paper, an analytical model is proposed to calculate the winding process-induced residual stresses for the multilayered filament wound composite parts with metal linear based on the three-dimensional thick-walled cylinder theory. In the modeling of winding process, the contribution of the current outmost winding tension to the residual stresses of the previously wound composite layers is first calculated and then the total residual stresses is obtained based on the elastic superposition principle. Moreover, the winding tensions are modified for equal residual stress in fibers based on the inverse iteration principle. Compared with the results in other work and obtained by finite element method, it shows that the proposed analytical model is of high accuracy and can be used to calculate the residual stresses due to winding tensions for the multilayered filament wound composite parts. The expected winding tensions for the equal residual fiber stresses can be quickly obtained after several iterations. Moreover, the effects of thickness of metal linear on the winding tensions distributions are further investigated based on the proposed model and the iterative algorithm.

Convergence and Preconditioning of Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

Abstract This paper focuses on the inner iteration that arises in inexact inverse subspace iteration for computing a small deflating subspace of a large matrix pencil. First, it is shown that the method achieves linear rate of convergence if the inner iteration is performed with increasing accuracy. Then, as inner iteration, block-GMRES is used with preconditioners generalizing the one by Robbé, Sadkane and Spence [Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 2009, 1, 92–113]. It is shown that the preconditioners help to maintain the number of iterations needed by block-GMRES to approximately a small constant. The efficiency of the preconditioners is illustrated by numerical examples.

Coupling calculation and analysis of three‐dimensional temperature and fluid field for high‐power high‐speed permanent magnet machine

In order to accurately estimate the temperature rise for high-power high-speed permanent magnet machines (HSPMMs), a novel temperature calculation method considering the non-linear variation of material properties with temperature is proposed based on multi-physics co-simulation analysis. According to the theory of computational fluid dynamics and heat transfer, the computation model of fluid–solid–heat coupling heat transfer is established, and the coupled field is calculated using finite volume method with fundamental assumptions and corresponding boundary conditions. With the influences from temperature gradient and water flow rate considered, the heat transfer coefficients of water pipe surfaces are obtained by the application of the inverse iteration method. Thus, HSPMM temperature and fluid field can be simulated numerically by the finite volume methods, while the spatial temperature distributions for the machine main components are analysed in this study. The 1.12 MW, 18,000 rpm HSPMM is prototyped with experiments conducted on it, while the test data are then compared with the calculated results, which validate the correctness of the solution method of the coupled field.

Cooperative multi-agent search using Bayesian approach with connectivity maintenance

PurposeThis paper aims to present a distributed Bayesian approach with connectivity maintenance to manage a multi-agent network search for a target on a two-dimensional plane.Design/methodology/approachThe Bayesian framework is used to compute the local probability density functions (PDFs) of the target and obtain the global PDF with the consensus algorithm. An inverse power iteration algorithm is introduced to estimate the algebraic connectivity λ2 of the network. Based on the estimated λ2, the authors design a potential field for the connectivity maintenance. Then, based on the detection probability function, the authors design a potential field for the search target. The authors combine the two potential fields and design a distributed gradient-based control for the agents.FindingsThe inverse power iteration algorithm can distributed estimate the algebraic connectivity by the agents. The agents can efficient search the target with connectivity maintenance with the designed distributed gradient-based search algorithm.Originality/valuePrevious study has paid little attention to the multi-agent search problem with connectivity maintenance. Our algorithm guarantees that the strongly connected graph of the multi-agent communication topology is always established while performing the distributed target search problem.

Power iteration and inverse power iteration for eigenvalue complementarity problem

SummaryIn this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.

Lightweight Topology Optimization with Buckling and Frequency Constraints Using the Independent Continuous Mapping Method

This research focuses on the lightweight topology optimization method for structures under the premise of meeting the requirements of stability and vibration characteristics. A new topology optimization model with the constraints of natural frequencies and critical buckling loads and the objective of minimizing the structural volume is established and solved based on the independent continuous mapping method. The eigenvalue equations and composite exponential filter functions are applied to convert the optimization formulation into a continuous, solvable mathematical programming model. In the process of topology optimization, suitable initial values of the filter functions are chosen to avoid local modes, and the dynamic frequency gap constraints are added in the optimal model to prevent mode switches. Furthermore, for the optimal structures with grey elements obtained by the continuous optimization model, the bisection–inverse iteration is applied to search the optimal discrete structures. Finally, a detailed scheme is given for the buckling and frequency topology optimization problem. Numerical examples illustrate that the modelling method of minimizing the economic index with given performance requirements is practical and feasible for multi-performance topology optimization problems.

Asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems

We consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients. With these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stochastic function space, and the approximation space for the underlying spatial function space. Sparse polynomial approximation is employed to obtain the first one, while classical finite elements are employed to obtain the latter. An error analysis is presented for the asymptotic convergence of the spectral inverse iteration to the smallest eigenvalue and the associated eigenvector of the problem. A series of detailed numerical experiments supports the conclusions of this analysis.