Jacobi-Davidson Algorithm Research Articles

Structure of the impurity band in heavily doped nonmagnetic semiconductors

We study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band.

Flutter Prediction Using Reduced-Order Modeling with Error Estimation

This paper presents a methodology for dynamic aeroelastic analysis of aircraft based on model order reduction with error estimation. A projection-based model order reduction approach is used to create an aerodynamic reduced-order model (ROM), which is coupled to a structural model to create an aeroelastic ROM. The governing aerodynamic equations are the linearized semidiscrete Euler equations. Flutter analysis is conducted by analyzing the eigenvalues of the aeroelastic ROM. A dual-weighted residual-based error estimator is presented which approximates the error in the eigenvalues obtained from the reduced eigenproblem relative to the eigenvalues from the high-dimensional aeroelastic model. The error estimator thus allows for the construction of aeroelastic ROMs with select eigenvalues that satisfy a user-prescribed accuracy. The aerodynamic ROM is constructed using approximate high-dimensional aeroelastic eigenvectors computed using the two-sided Jacobi–Davidson algorithm. Dynamic aeroelastic analyses are presented for the NACA 64A010 Isogai case, the AGARD 445.6 wing model, and a NACA 0012 benchmark case. The error estimator is shown to have good agreement with the exact error. For the test cases presented in this work, the cost of computing the flutter point at a given Mach number is equivalent to the cost of approximately 5 to 12 steady nonlinear flow evaluations of the high-dimensional Euler equations.

A Riccati-type algorithm for solving generalized Hermitian eigenvalue problems

The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.

An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo.

Although the linear method is one of the most robust algorithms for optimizing nonlinearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact that at each optimization step, a generalized eigenvalue problem is solved in which the Hamiltonian and overlap matrices are stored in memory. Here, we demonstrate that by applying the Jacobi-Davidson algorithm, one can solve the generalized eigenvalue problem iteratively without having to build and store the matrices in question. The resulting direct linear method greatly lowers the cost and improves the scaling of the algorithm with respect to the number of parameters. To further improve the efficiency of optimization for wavefunctions with a large number of parameters, we use the first order method AMSGrad far from the minimum as it is very inexpensive and only switch to the direct linear method near the end of the optimization where methods such as AMSGrad have long convergence tails. We apply this improved optimizer to wavefunctions with real and orbital space Jastrow factors applied to a symmetry-projected generalized Hartree-Fock reference. Systems addressed include atomic systems such as beryllium and neon, molecular systems such as the carbon dimer and iron(ii) porphyrin, and model systems such as the Hubbard model and hydrogen chains.

Multi‐level preconditioning and eigenvector approximation for axisymmetric Maxwell eigenproblems

AbstractWe compute resonant modes in axisymmetric radio frequency cavities by solving a few generalized eigenvalue problems using the symmetric Jacobi–Davidson algorithm. In particular, we focus on the parallelization and the use of the multi‐level preconditioner for the correction equation. Additionally, we compute eigenvector approximations using the solutions of the problem on a coarser mesh and present the results on the problem of computing a few lowest modes of a real‐world cavity.

Entanglement and co-tunneling of two equivalent protons in hydrogen bond pairs.

A theoretical study is reported of a system of two identical symmetric hydrogen bonds, weakly coupled such that the two mobile protons can move either separately (stepwise) or together (concerted). It is modeled by two equivalent quartic potentials interacting through dipolar and quadrupolar coupling terms. The tunneling Hamiltonian has two imaginary modes (reaction coordinates) and a potential with a single maximum that may turn into a saddle-point of second order and two sets of (inequivalent) minima. Diagonalization is achieved via a modified Jacobi-Davidson algorithm. From this Hamiltonian the mechanism of proton transfer is derived. To find out whether the two protons move stepwise or concerted, a new tool is introduced, based on the distribution of the probability flux in the dividing plane of the transfer mode. While stepwise transfer dominates for very weak coupling, it is found that concerted transfer (co-tunneling) always occurs, even when the coupling vanishes since the symmetry of the Hamiltonian imposes permanent entanglement on the motions of the two protons. We quantify this entanglement and show that, for a wide range of parameters of interest, the lowest pair of states of the Hamiltonian represents a perfect example of highly entangled quantum states in continuous variables. The method is applied to the molecule porphycene for which the observed tunneling splitting is calculated in satisfactory agreement with experiment, and the mechanism of double-proton tunneling is found to be predominantly concerted. We show that, under normal conditions, when they are in the ground state, the two porphycene protons are highly entangled, which may have interesting applications. The treatment also identifies the conditions under which such a system can be handled by conventional one-instanton techniques.

A parallel two-level polynomial Jacobi–Davidson algorithm for large sparse PDE eigenvalue problems

Many scientific and engineering applications require accurate, fast, robust, and scalable numerical solution of large sparse algebraic polynomial eigenvalue problems (PEVP’s) that arise from some appropriate discretization of partial differential equations. The polynomial Jacobi-Davidson (PJD) algorithm has been numerically shown as a promising approach for the PEVP’s to finding the interior spectrum. The PJD algorithm is a subspace method, which extracts the candidate eigenpair from a search space and the space updated by embedding the solution of the correction equation at the JD iteration. In this research, we develop and study the two-level PJD algorithm for PEVP’s with emphasis on the application of the dissipative acoustic cubic eigenvalue problem. The proposed two-level PJD algorithm consists of two important ingredients: A good initial basis for the search space is constructed on the fine-level by using the interpolation of the coarse solution of the same eigenvalue problem in order to enhance the robustness of the algorithm. Also, an efficient and scalable two-level preconditioner based on the Schwarz framework is used for the correction equation. Some numerical examples obtained on a parallel cluster of computers are given in order to demonstrate the robustness and scalability of our PJD algorithm.

Parallel two-level domain decomposition based Jacobi–Davidson algorithms for pyramidal quantum dot simulation

We consider a quintic polynomial eigenvalue problem arising from the finite volume discretization of a quantum dot simulation problem. The problem is solved by the Jacobi–Davidson (JD) algorithm. Our focus is on how to achieve the quadratic convergence of JD in a way that is not only efficient but also scalable when the number of processor cores is large. For this purpose, we develop a projected two-level Schwarz preconditioned JD algorithm that exploits multilevel domain decomposition techniques. The pyramidal quantum dot calculation is carefully studied to illustrate the efficiency of the proposed method. Numerical experiments confirm that the proposed method has a good scalability for problems with hundreds of millions of unknowns on a parallel computer with more than 10,000 processor cores.

Parallel Jacobi–Davidson with block FSAI preconditioning and controlled inner iterations

SummaryThe Jacobi–Davidson (JD) algorithm is considered one of the most efficient eigensolvers currently available for non‐Hermitian problems. It can be viewed as a coupled inner‐outer iteration, where the inner one expands the search subspace and the outer one reduces the eigenpair residual. One of the difficulties in the JD efficient use stems from the definition of the most appropriate inner tolerance, so as to avoid useless extra work and keep the number of outer iterations under control. To this aim, the use of an efficient preconditioner for the inner iterative solver is of paramount importance. The present paper describes a fresh implementation of the JD algorithm with controlled inner iterations and block factorized sparse approximate inverse preconditioning for non‐Hermitian eigenproblems in a parallel computational environment. The algorithm performance is investigated by comparison with a freely available software package such as SLEPc. The results show that combining the inner tolerance control with an efficient preconditioning technique can allow for a significant improvement of the JD performance, preserving a good scalability. Copyright © 2016 John Wiley & Sons, Ltd.

Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

SummaryIn this work, we consider the numerical solution of large nonlinear eigenvalue problems that arise in thermoacoustic simulations involved in the stability analysis of large combustion devices. We briefly introduce the physical modeling that leads to a nonlinear eigenvalue problem that is solved using a nonlinear fixed point iteration scheme. Each step of this nonlinear method requires the solution of a complex non‐Hermitian linear eigenvalue problem. We review a set of state‐of‐the‐art eigensolvers and discuss strategies to recycle spectral information from one nonlinear step to the next. More precisely, we consider the Jacobi–Davidson algorithm, the Implicitly Restarted Arnoldi method, the Krylov–Schur solver and its block‐variant, and the subspace iteration method with Chebyshev acceleration. On a small test example, we study the relevance of the different approaches and illustrate on a large industrial test case the performance of the parallel solvers best suited to recycle spectral information for large‐scale thermoacoustic stability analysis. Copyright © 2015 John Wiley & Sons, Ltd.

Tunneling splitting in double-proton transfer: direct diagonalization results for porphycene.

Zero-point and excited level splittings due to double-proton tunneling are calculated for porphycene and the results are compared with experiment. The calculation makes use of a multidimensional imaginary-mode Hamiltonian, diagonalized directly by an effective reduction of its dimensionality. Porphycene has a complex potential energy surface with nine stationary configurations that allow a variety of tunneling paths, many of which include classically accessible regions. A symmetry-based approach is used to show that the zero-point level, although located above the cis minimum, corresponds to concerted tunneling along a direct trans - trans path; a corresponding cis - cis path is predicted at higher energy. This supports the conclusion of a previous paper [Z. Smedarchina, W. Siebrand, and A. Fernández-Ramos, J. Chem. Phys. 127, 174513 (2007)] based on the instanton approach to a model Hamiltonian of correlated double-proton transfer. A multidimensional tunneling Hamiltonian is then generated, based on a double-minimum potential along the coordinate of concerted proton motion, which is newly evaluated at the RI-CC2/cc-pVTZ level of theory. To make it suitable for diagonalization, its dimensionality is reduced by treating fast weakly coupled modes in the adiabatic approximation. This results in a coordinate-dependent mass of tunneling, which is included in a unique Hermitian form into the kinetic energy operator. The reduced Hamiltonian contains three symmetric and one antisymmetric mode coupled to the tunneling mode and is diagonalized by a modified Jacobi-Davidson algorithm implemented in the Jadamilu software for sparse matrices. The results are in satisfactory agreement with the observed splitting of the zero-point level and several vibrational fundamentals after a partial reassignment, imposed by recently derived selection rules. They also agree well with instanton calculations based on the same Hamiltonian.

3-dimensional eigenmodal analysis of plasmonic nanostructures

We introduce a 3-dimensional electromagnetic eigenmodal algorithm for the theoretical analysis of resonating nano-optical structures. The method, a variant of the Jacobi-Davidson algorithm, solves the electric field vector wave, or curl-curl, equation for the electromagnetic eigenmodes of resonant optical structures with a finite element method. In particular, the method includes transparent boundary conditions that enable the analysis of resonating structures in unbounded space. We demonstrate the performance of the method. First, we calculate the modes of several dielectric resonator antennas and compare them to theoretically determined results. Second, we calculate the modes of a nano-cuboid and compare them to theoretically determined results. Third, we numerically analyze spherical nanoparticles and compare the result to the theoretical Mie solution. Fourth, we analyze optical dipole antenna configurations in order to assess the method's capability for solving technologically relevant problems.

Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

SUMMARYThe choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large‐size sparse eigenproblems. Although incomplete factorizations with partial fill‐in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large‐size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi–Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid‐dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd.

A parallel polynomial Jacobi–Davidson approach for dissipative acoustic eigenvalue problems

We consider a rational algebraic large sparse eigenvalue problem arising in the discretization of the finite element method for the dissipative acoustic model in the pressure formulation. The presence of nonlinearity due to the frequency-dependent impedance poses a challenge in developing an efficient numerical algorithm for solving such eigenvalue problems. In this article, we reformulate the rational eigenvalue problem as a cubic eigenvalue problem and then solve the resulting cubic eigenvalue problem by a parallel restricted additive Schwarz preconditioned Jacobi–Davidson algorithm (ASPJD). To validate the ASPJD-based eigensolver, we numerically demonstrate the optimal convergence rate of our discretization scheme and show that ASPJD converges successfully to all target eigenvalues. The extraneous root introduced by the problem reformulation does not cause any observed side effect that produces an undesirable oscillatory convergence behavior. By performing intensive numerical experiments, we identify an efficient correction-equation solver, an effective algorithmic parameter setting, and an optimal mesh partitioning. Furthermore, the numerical results suggest that the ASPJD-based eigensolver with an optimal mesh partitioning results in superlinear scalability on a distributed and parallel computing cluster scaling up to 192 processors.

A parallel additive Schwarz preconditioned Jacobi–Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

We develop a parallel Jacobi–Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi–Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc’s efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrödinger’s equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi–Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.

3-D eigenmode calculation of metallic nano-structures

Abstract. In the calculation of eigenfrequencies of 3-D metallic nanostructures occurs the challenge that the material parameters depend on the desired eigenfrequency. We propose a formulation where this leads to a polynomial eigenvalue problem which can be tackled by different solving strategies. A comparison between a Newton-type method and a Jacobi-Davidson algorithm is given.

PHOENIX: MHD spectral code for rotating laboratory and gravitating astrophysical plasmas

The new PHOENIX code is discussed together with a sample of many new results that are obtained concerning magnetohydrodynamic (MHD) spectra of axisymmetric plasmas where flow and gravity are consistently taken into account. PHOENIX, developed from the CASTOR code [W. Kerner, J.P. Goedbloed, G.T.A. Huysmans, S. Poedts, E. Schwarz, J. Comput. Phys. 142 (1998) 271], incorporates purely toroidal, or both toroidal and poloidal flow and external gravitational fields to compute the entire ideal or resistive MHD spectrum for general tokamak or accretion disk configurations. These equilibria are computed by means of FINESSE [A.J.C. Beliën, M.A. Botchev, J.P. Goedbloed, B. van der Holst, R. Keppens, J. Comp. Physics 182 (2002) 91], which discriminates between the different elliptic flow regimes that may occur. PHOENIX makes use of a finite element method in combination with a spectral method for the discretization. This leads to a large generalized eigenvalue problem, which is solved by means of Jacobi–Davidson algorithm [G.L.G. Sleijpen, H.A. van der Vorst, SIAM J. Matrix Anal. Appl. 17 (1996) 401]. PHOENIX is compared with CASTOR, PEST-1 and ERATO for an internal mode of Soloviev equilibria. Furthermore, the resistive internal kink mode has been computed to demonstrate that the code can accurately handle small values for the resistivity. A new reference test case for a Soloviev-like equilibrium with toroidal flow shows that, on a particular unstable mode, the flow has a quantifiable stabilizing effect regardless of the direction of the flow. PHOENIX reproduces the Toroidal Flow induced Alfvén Eigenmode (TFAE, [B. van der Holst, A.J.C. Beliën, J.P. Goedbloed, Phys. Rev. Lett. 84 (2000) 2865]) where finite resistivity in combination with equilibrium flow effects causes resonant damping. Localized ideal gap modes are presented for tokamak plasmas with toroidal and poloidal flow. Finally, we demonstrate the ability to spectrally diagnose magnetized accretion disk equilibria where gravity acts together with either purely toroidal flow or both toroidal and poloidal flow. These cases show that the MHD continua can be unstable or overstable due to the presence of a gravitational field together with equilibrium flow-driven dynamics [J.P. Goedbloed, A.J.C. Beliën, B. van der Holst, R. Keppens, Phys. Plasmas 11 (2004) 28].

On solving complex-symmetric eigenvalue problems arising in the design of axisymmetric VCSEL devices

We investigate the Jacobi-Davidson algorithm for solving the time-harmonic Maxwell equation in axisymmetric vertical cavity surface emitting lasers (VCSELs). We compare various strategies in the extraction and extension phase of the algorithm and discuss their properties with complex-symmetric eigenvalue problems obtained from PML encoated resonator cavities.

A full-configuration interaction “nuclear orbital” method to study doped HeN3 clusters (N⩽4)

An efficient full configuration interaction (FCI) treatment, based on the Jacobi-Davidson algorithm, is developed in order to study small doped (3)He(N) clusters. The state of each He atom in a given cluster is described by a set of wave-functions which by extention of the quantum-chemistry notation are caller here "nuclear orbitals". The FCI treatment is applied to the calculation of binding energies and helium natural orbitals of (3)He(N)...Br(2)(X) complexes. In agreement with our previous calculations using a Hartree-Fock approach [Phys. Rev. Lett. 93, 053401 (2004)], in which the He-He interaction is modified at small distances to account for short-range correlation effects, the lowest-energy states of each multiplet are found to be very close in energy. The natural orbital analysis, in turn, indicates the adequacy of the "nuclear orbital" approach in these systems.

A finite integration method for conformal, structured-grid, electromagnetic simulation

We describe a numerical scheme for solving Maxwell’s equations in the frequency domain on a conformal, structured, non-orthogonal, multi-block mesh. By considering Maxwell’s equations in a volume parameterized by dimensionless curvilinear coordinates, we obtain a set of tensor equations that are a continuum analogue of common circuit equations, and that separate the metrical and metric-free parts of Maxwell’s equations and the material constitutive relations. We discretize these equations using a new formulation that treats the electric field and magnetic induction using simple basis-function representations to obtain a discrete form of Faraday’s law of induction, but that uses finite integral representations for the displacement current and magnetic field to obtain a discrete form of Ampere’s law, as in the finite integration technique [T. Weiland, A discretization method for the solution of Maxwell’s equations for six-component fields, Electron. Commun. (AE U) 31 (1977) 116; T. Weiland, Time domain electromagnetic field computation with finite difference methods, Int. J. Numer. Model: Electron. Netw. Dev. Field 9 (1996) 295–319]. We thereby derive new projection operators for the discrete tensor material equations and obtain a compact numerical scheme for the discrete differential operators. This scheme is shown to exhibit significantly reduced numerical dispersion when compared to the standard linear finite element method. We take advantage of the mesh structure on a block-by-block basis to implement these numerical operators efficiently, and achieve computational speed with modest memory requirements when compared to explicit sparse matrix storage. Using the Jacobi–Davidson [G.L.G. Sleijpen, H.A. van der Vorst, A Jacobi–Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (2) (1996) 401–425; S.J. Cooke, B. Levush, Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the Jacobi–Davidson algorithm, J. Comput. Phys. 157 (1) (2000) 350–370] and quasi-minimal residual [R.W. Freund, N.M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315–339] iterative matrix solution algorithms, we solve the resulting discrete matrix eigenvalue equations and demonstrate the convergence characteristics of the algorithm. We validate the model for three-dimensional electromagnetic problems, both cavity eigenvalue solutions and a waveguide scattering matrix calculation.