Large Eigenvalue Problems Research Articles

A Krylov eigenvalue solver based on filtered time domain solutions

This paper introduces a method for computing eigenvalues and eigenvectors of a generalized Hermitian, matrix eigenvalue problem. The work is focused on large scale eigenvalue problems, where the application of a direct inverse is out of reach. Instead, an explicit time-domain integrator for the corresponding wave problem is combined with a proper filtering and a Krylov iteration in order to solve for eigenvalues within a given region of interest. We report results of small scale model problems to confirm the reliability of the method, as well as the computation of acoustic resonances in a three dimensional model of a hunting horn to demonstrate the efficiency.

A rational‐Chebyshev projection method for nonlinear eigenvalue problems

AbstractThis article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function under consideration is approximated by a matrix polynomial in . The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

A computationally efficient [formula omitted]-spectral form for partial dispersion analyses within the wave finite element framework

This paper addresses the computation of frequency-dependent dispersion curves (i.e., k(ω)) and wave modes within the framework of the Wave Finite Element Method (WFEM) and in the context of high-dimensional periodic unit cell models. Numerous applications, ranging from phononics to vibroacoustics, now rely on dispersion analyses or wave expansion over a subset of eigensolutions – complex wavenumbers and Bloch waves – resulting from the resolution of an eigenvalue problem with a T-palindromic quadratic structure (T-PQEP). To exploit the structure of finite element models, various structure-preserving linearizations such as the Zhong-Williams and the (S+S−1)-transform have already been developed to achieve partial wave resolution of large T-PQEP, primarily targeting the dominating (least decaying) waves. In this paper we derive an alternative linearization of the T-PQEP for the k(ω) problem, which leads to enhanced targeting of the eigenvalues around the unit circle and reduces the inaccuracies induced by root multiplicity. A specific form of the problem is then proposed as an optimal compromise between ease of implementation, numerical stability, convergence and accuracy enhancement. The performance of our proposed linearization is compared against existing ones across various iterative eigensolvers, since the generalized eigenvalue problems involve complex non-hermitian matrices, which are not extensively included in eigensolvers. Results indicate that the proposed linearization should be favored for the WFEM, as it provides numerical enhancements in dispersion and wave vectors computation for large eigenvalue problems, as well as for further wave expansion applications.

GLT sequences and automatic computation of the symbol

Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions.

A robust, open-source implementation of the locally optimal block preconditioned conjugate gradient for large eigenvalue problems in quantum chemistry

We present two open-source implementations of the locally optimal block preconditioned conjugate gradient (lobpcg) algorithm to find a few eigenvalues and eigenvectors of large, possibly sparse matrices. We then test lobpcg for various quantum chemistry problems, encompassing medium to large, dense to sparse, well-behaved to ill-conditioned ones, where the standard method typically used is Davidson’s diagonalization. Numerical tests show that while Davidson’s method remains the best choice for most applications in quantum chemistry, LOBPCG represents a competitive alternative, especially when memory is an issue, and can even outperform Davidson for ill-conditioned, non-diagonally dominant problems.

The infinite Lanczos method for symmetric nonlinear eigenvalue problems

A new iterative method for solving large scale symmetric nonlinear eigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos method to this specific linearization, resulting in a short-term recurrence. We show how, under specific assumption on the starting vector, this method can be carried out in finite arithmetic and how the exploitation of the problem structure leads to improvements in terms of computation time. The eigenpair approximations are extracted with the nonlinear Rayleigh-Ritz procedure combined with a specific choice of the projection space. We illustrate how this extraction technique resolves the instability issues that may occur due to the loss of orthogonality in many standard Lanczos-type methods.

GCGE: a package for solving large scale eigenvalue problems by parallel block damping inverse power method

GCGE: a package for solving large scale eigenvalue problems by parallel block damping inverse power method

A Theory of Quantum Subspace Diagonalization

Quantum subspace diagonalization methods are an exciting new class of algorithms for solving large scale eigenvalue problems using quantum computers. Unfortunately, these methods require the solution of an ill-conditioned generalized eigenvalue problem, with a matrix pencil corrupted by a non-negligible amount of noise that is far above the machine precision. Despite pessimistic predictions from classical perturbation theories, these methods can perform reliably well if the generalized eigenvalue problem is solved using a standard truncation strategy. We provide a theoretical analysis of this surprising phenomenon, proving that under certain natural conditions, a quantum subspace diagonalization algorithm can accurately compute the smallest eigenvalue of a large Hermitian matrix. We give numerical experiments demonstrating the effectiveness of the theory and providing practical guidance for the choice of truncation level.

Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gierer–Meinhardt system in one-dimensional domain

In this paper, we study the existence and stability of multiple spikes pattern to the fractional Gierer–Meinhardt model with periodic boundary conditions and the fractional power [Formula: see text]. Specifically, we rigorously establish the existence of symmetric multiple spikes and asymmetric two-spikes solutions by the classical Lyapunov–Schmidt reduction method. We also investigate the stability of the constructed solution by studying its associated large and small eigenvalue problems, where we need to consider two nonlocal eigenvalue problems in their fractional versions. In the study of the large eigenvalue problem, the quantity [Formula: see text] is the critical threshold which determines the stability of [Formula: see text]-peaked solutions. For the symmetric two-spikes pattern we obtain the asymptotic expansion for the critical threshold [Formula: see text] up to the second order. Moreover, we provide some elementary properties of the Green’s function, including the first and second derivatives, they are linked to the location of the spikes and the stability. Among these properties on the Green’s function, we find out that the polygamma function [Formula: see text] plays a crucial role.

Accelerating an iterative eigensolver for nuclear structure configuration interaction calculations on GPUs using OpenACC

To accelerate the solution of large eigenvalue problems arising from many-body calculations in nuclear physics on distributed-memory parallel systems equipped with general-purpose Graphic Processing Units (GPUs), we modified a previously developed hybrid MPI/OpenMP implementation of an eigensolver written in Fortran 90 by using an OpenACC directives based programming model. Such an approach requires making minimal changes to the original code and enables a smooth migration of large-scale nuclear structure simulations from a distributed-memory many-core CPU system to a distributed GPU system. However, in order to make the OpenACC based eigensolver run efficiently on GPUs, we need to take into account the architectural differences between a many-core CPU and a GPU device. Consequently, the optimal way to insert OpenACC directives may be different from the original way of inserting OpenMP directives. We point out these differences in the implementation of sparse matrix–vector multiplications with multiple vectors, which constitutes the main cost of the eigensolver, as well as other differences in the preconditioning step and dense linear algebra operations. We compare the performance of the OpenACC based implementation executed on multiple GPUs with the performance on distributed-memory many-core CPUs, and demonstrate significant speedup achieved on GPUs compared to the on-node performance of a many-core CPU. We also show that the overall performance improvement of the eigensolver on multiple GPUs is more modest due to the communication overhead among different MPI ranks.

Accelerating molecular property calculations with semiempirical preconditioning.

Computing ab initio molecular linear response properties, e.g., electronic excitation energies and transition dipole moments, requires the solution of large eigenvalue problems or large systems of equations. These large eigenvalue problems or large systems of equations are commonly solved iteratively using Krylov space algorithms, such as the Davidson algorithm for eigenvalue problems. A critical ingredient in Krylov space algorithms is the preconditioner, which is used to generate optimal update vectors in each iteration. We propose to use semiempirical approximations as preconditioners to accelerate the calculation of ab initio properties. The crucial advantage to improving the preconditioner is that the converged result is unchanged, so there is no trade-off between accuracy and speedup. We demonstrate our approach by accelerating the calculation of electronic excitation energies and electric polarizabilities from linear response time-dependent density functional theory using the simplified time-dependent density functional theory semiempirical model. For excitation energies, the semiempirical preconditioner reduces the number of iterations on average by 37% and up to 70%. The semiempirical preconditioner reduces the number of iterations for computing the polarizability by 15% on average and up to 33%. Moreover, we show that the preconditioner can be further improved by tuning the empirical parameters that define the semiempirical model, leading to an additional reduction in the number of iterations by about 20%. Our approach bridges the gap between semiempirical models and ab initio methods and charts a path toward combining the speed of semiempirical models with the accuracy of ab initio methods.

Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems

<p style='text-indent:20px;'>An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient.</p>

Rational Spectral Filters with Optimal Convergence Rate

In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of so-called rational filters, we introduce a systematic method to design these filters by minimizing the worst-case convergence ratio and eliminate the parametric dependence on weight functions. Further, we provide an efficient way to deal with the box-constraints which play a central role for the use of iterative linear solvers in contour-based eigensolvers. Indeed, these parameter-free filters consistently minimize the number of iterations and the number of FLOPs to reach convergence in the eigensolver. As a byproduct, our rational filters allow for a simple solution to load balancing when the solution of an interior eigenproblem is approached by the slicing of the sought after spectral interval.

Homotopy perturbation technique for improving solutions of large quadratic eigenvalue problems: Application to friction-induced vibration

This paper puts forward a projection technique for accurately calculating solutions of large Quadratic Eigenvalue Problem. The aim here is to stabilize the complex eigensolutions whilst reducing residual errors, especially when considering significant damping contribution or asymmetric stiffness matrices. Hence, more confident results can be obtained in the frequency band of interest. To achieve this, high order modes, calculated using the homotopy perturbation technique, are introduced in the projection step of the classical method. This numerical proposal is a generalization of the classical projection, based only on normal modes of the associated undamped problem. To evaluate the efficiency of the suggested method, a finite element application dedicated to a friction-induced vibration problem is investigated.

Fast Generation of Stability Charts for Time-Delay Systems Using Continuation of Characteristic Roots

Abstract Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method can significantly reduce the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.

Arbitrary-level coarse mesh finite difference acceleration with hierarchical coarsening in space and energy

In this paper, we present an arbitrary-level coarse mesh finite difference (CMFD) scheme to accelerate discrete ordinates (SN) transport eigenvalue problems. It is a nested iterative scheme where a transport power iteration is accelerated with nested levels of CMFD eigenvalue problems whose space/energy grids can be hierarchically coarsened. The methodology provides a general framework independent of the space/angle discretization used to define the problem of interest. It is also readily extensible to solving diffusion eigenvalue problems. We present 2D and 3D numerical results on a representative quarter-core model from the VERA benchmark suite with different combinations of energy and spatial CMFD grids. We show that the methodology significantly reduces the diffusion and overall computational times required to solve large SN transport eigenvalue problems with CMFD on many parallel processes.

A parallel generalized conjugate gradient method for large scale eigenvalue problems

Based on damping blocked inverse power method, a type of generalized parallel conjugate gradient method is proposed for large scale eigenvalue problems. Techniques for orthogonalization and computing Rayleigh-Ritz problems are introduced to improve the stability, efficiency and scalability. Furthermore, a computing package is built based on the proposed method here. Some numerical tests are provided to validate the stability, efficiency and scalability of the method in this paper. The corresponding computing package can be downloaded from the web site: https://github.com/pase2017/GCGE-1.0 .

BDF: A relativistic electronic structure program package.

The BDF (Beijing Density Functional) program package is in the first place a platform for theoretical and methodological developments, standing out particularly in relativistic quantum chemical methods for chemistry and physics of atoms, molecules, and periodic solids containing heavy elements. These include the whole spectrum of relativistic Hamiltonians and their combinations with density functional theory for the electronic structure of ground states as well as time-dependent and static density functional linear response theories for electronically excited states and electric/magnetic properties. However, not to be confused by its name, BDF nowadays comprises also of standard and novel wave function-based correlation methods for the ground and excited states of strongly correlated systems of electrons [e.g., multireference configuration interaction, static-dynamic-static configuration interaction, static-dynamic-static second-order perturbation theory, n-electron valence second-order perturbation theory, iterative configuration interaction (iCI), iCI with selection plus PT2, and equation-of-motion coupled-cluster]. Additional features of BDF include a maximum occupation method for finding excited states of Hartree-Fock/Kohn-Sham (HF/KS) equations, a very efficient localization of HF/KS and complete active space self-consistent field orbitals, and a unique solver for exterior and interior roots of large matrix eigenvalue problems.

Electronic structure of semiconductor nanostructures: A modified localization landscape theory

In this paper we present a modified localization landscape theory to calculate localized/confined electron and hole states and the corresponding energy eigenvalues without solving a (large) eigenvalue problem. We motivate and demonstrate the benefit of solving $\hat{H}^2u=1$ in the modified localization landscape theory in comparison to $\hat{H}u=1$, solved in the localization landscape theory. We detail the advantages by fully analytic considerations before targeting the numerical calculation of electron and hole states and energies in III-N heterostructures. We further discuss how the solution of $\hat{H}^2u=1$ is used to extract an effective potential $W$ that is comparable to the effective potential obtained from $\hat{H}u=1$, ensuring that it can for instance be used to introduce quantum corrections to drift-diffusion transport calculations. Overall, we show that the proposed modified localization landscape theory keeps all the benefits of the recently introduced localization landscape theory but further improves factors such as convergence of the calculated energies and the robustness of the method against the chosen integration region for $u$ to obtain the corresponding energies. We find that this becomes especially important for here studied $c$-plane InGaN/GaN quantum wells with higher In contents. All these features make the proposed approach very attractive for calculation of localized states in highly disordered systems, where partitioning the systems into different subregions can be difficult.

On global convergence of subspace projection methods for Hermitian eigenvalue problems

We provide a concise overview of some iterative approaches, by which large and sparse eigenvalue problems are successfully dealt with. Our main goal in this paper is to prove the global convergence of a few state-of-the art iteration methods, such as the Jacobi–Davidson, the rational Krylov sequence and the Lanczos methods. We also derive some conditions to ensure global convergence of the corresponding inexact variants of the above-mentioned methods. These global convergence analyses result either in a depth understanding or in an efficient improvement to the original methods.