Efficient Preconditioning Techniques Research Articles

Accelerating self-consistent field iterations in Kohn-Sham density functional theory using a low-rank approximation of the dielectric matrix

We present an efficient preconditioning technique for accelerating the fixed point iteration in real-space Kohn-Sham density functional theory (DFT) calculations. The preconditioner uses a low rank approximation of the dielectric matrix (LRDM) based on G\^ateaux derivatives of the residual of fixed point iteration along appropriately chosen direction functions. We develop a computationally efficient method to evaluate these G\^ateaux derivatives in conjunction with the Chebyshev filtered subspace iteration procedure, an approach widely used in large-scale Kohn-Sham DFT calculations. Further, we propose a variant of LRDM preconditioner based on adaptive accumulation of low-rank approximations from previous SCF iterations, and also extend the LRDM preconditioner to spin-polarized Kohn-Sham DFT calculations. We demonstrate the robustness and efficiency of the LRDM preconditioner against other widely used preconditioners on a range of benchmark systems with sizes ranging from $\sim$ 100-1100 atoms ($\sim$ 500--20,000 electrons). The benchmark systems include various combinations of metal-insulating-semiconducting heterogeneous material systems, nanoparticles with localized $d$ orbitals near the Fermi energy, nanofilm with metal dopants, and magnetic systems. In all benchmark systems, the LRDM preconditioner converges robustly within 20--30 iterations. In contrast, other widely used preconditioners show slow convergence in many cases, as well as divergence of the fixed point iteration in some cases. Finally, we demonstrate the computational efficiency afforded by the LRDM method, with up to 3.4$\times$ reduction in computational cost for the total ground-state calculation compared to other preconditioners.

An efficient high-order numerical solver for diffusion equations with strong anisotropy

In this paper, we present an interior penalty discontinuous Galerkin finite element scheme for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. We demonstrate the accuracy produced by the high-order scheme and develop an efficient preconditioning technique to solve the corresponding linear system, which is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the accuracy and efficiency of the proposed algorithm.

Multidirectional sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems

This paper explores a family of generalized sweeping preconditioners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity.

Efficient Steady-State Solver for the Hierarchical Equations of Motion Approach: Formulation and Application to Charge Transport through Nanosystems.

An iterative approach is introduced, which allows the efficient solution of the hierarchical equations of motion (HEOM) for the steady-state of open quantum systems. The approach combines the method of matrix equations with an efficient preconditioning technique to reduce the numerical effort of solving the HEOM. Illustrative applications to simulate nonequilibrium charge transport in single-molecule junctions demonstrate the performance of the method.

Efficient preconditioning techniques for velocity tracking of Stokes control problem

We are concerned with robust iterative solution methods for solving the Stokes optimal control problems. Two efficient preconditioners are proposed for the discretized saddle point linear systems arising from the velocity tracking of the Stokes control problem. The proposed preconditioners are similar in structure to and can be viewed as modifications of the preconditioner in Axelsson et al. (2017) [2], which are economic to implement in an inner-outer framework within Krylov acceleration. They can lead to similar tight and problem independent eigenvalue distribution results for the preconditioned matrices, which yield rates of convergence independent of both the regularization parameter and refinement level. Moreover, we also give inexact variants of the proposed preconditioners, which avoid the inner-outer implementations utilizing preconditioned GMRES methods as inner loops. Numerical experiments indicate that the proposed preconditioners demonstrate robust performance and comparable to some existing preconditioners when used to accelerate the Krylov subspace methods.

Extended Lagrangian Born-Oppenheimer molecular dynamics using a Krylov subspace approximation.

It is shown how the electronic equations of motion in extended Lagrangian Born-Oppenheimer molecular dynamics simulations [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008); J. Chem. Phys. 147, 054103 (2017)] can be integrated using low-rank approximations of the inverse Jacobian kernel. This kernel determines the metric tensor in the harmonic oscillator extension of the Lagrangian that drives the evolution of the electronic degrees of freedom. The proposed kernel approximation is derived from a pseudoinverse of a low-rank estimate of the Jacobian, which is expressed in terms of a generalized set of directional derivatives with directions that are given from a Krylov subspace approximation. The approach allows a tunable and adaptive approximation that can take advantage of efficient preconditioning techniques. The proposed kernel approximation for the integration of the electronic equations of motion makes it possible to apply extended Lagrangian first-principles molecular dynamics simulations to a broader range of problems, including reactive chemical systems with numerically sensitive and unsteady charge solutions. This can be achieved without requiring exact full calculations of the inverse Jacobian kernel in each time step or relying on iterative non-linear self-consistent field optimization of the electronic ground state prior to the force evaluations as in regular direct Born-Oppenheimer molecular dynamics. The low-rank approximation of the Jacobian is directly related to Broyden's class of quasi-Newton algorithms and Jacobian-free Newton-Krylov methods and provides a complementary formulation for the solution of nonlinear systems of equations.

An efficient preconditioning method for state box-constrained optimal control problems

Abstract An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrix blocks is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau–Yosida penalization method. It is shown that there occur very few nonlinear iteration steps and also few iterations to solve the arising linearized equations on the fine mesh. This holds for a wide range of the penalization and discretization parameters. The arising nonlinearity can be handled with a hybrid nonlinear-linear procedure that raises the computational efficiency of the overall solution method.

BFGS‐like updates of constraint preconditioners for sequences of KKT linear systems in quadratic programming

SummaryWe focus on efficient preconditioning techniques for sequences of Karush‐Kuhn‐Tucker (KKT) linear systems arising from the interior point (IP) solution of large convex quadratic programming problems. Constraint preconditioners (CPs), although very effective in accelerating Krylov methods in the solution of KKT systems, have a very high computational cost in some instances, because their factorization may be the most time‐consuming task at each IP iteration. We overcome this problem by computing the CP from scratch only at selected IP iterations and by updating the last computed CP at the remaining iterations, via suitable low‐rank modifications based on a BFGS‐like formula. This work extends the limited‐memory preconditioners (LMPs) for symmetric positive definite matrices proposed by Gratton, Sartenaer and Tshimanga in 2011, by exploiting specific features of KKT systems and CPs. We prove that the updated preconditioners still belong to the class of exact CPs, thus allowing the use of the conjugate gradient method. Furthermore, they have the property of increasing the number of unit eigenvalues of the preconditioned matrix as compared with the generally used CPs. Numerical experiments are reported, which show the effectiveness of our updating technique when the cost for the factorization of the CP is high.

A saddle point least squares approach for primal mixed formulations of second order PDEs

We present a Saddle Point Least Squares (SPLS) method for discretizing second order elliptic problems written as primal mixed variational formulations. A stability LBB condition and a data compatibility condition at the continuous level are automatically satisfied. The proposed discretization method follows a general SPLS approach and has the advantage that a discrete inf–sup condition is automatically satisfied for standard choices of the test and trial spaces. For the proposed iterative processes a nodal basis for the trial space is not required. Efficient preconditioning techniques that involve inversion only on the test space can be considered. Stability and approximation properties for two choices of discrete spaces are investigated. Applications of the new approach include discretization of second order problems with highly oscillatory coefficient, interface problems, and higher order approximation of the flux for elliptic problems with smooth coefficients.

Development and verification of MOC code based on Krylov subspace and efficient preconditioning techniques

Derived from the conventional characteristic sweeping operations, the method of characteristics (MOC) with matrix form has more favorable performance compared with its traditional implementation potentially. However, the advantage depends heavily upon the efficiency of the linear algebraic solver. In current study, a simple and efficient preconditioning technique is implemented in the restart version of Generalized Minimal RESidual algorithm (GMRES) to accelerate the resulted linear system. A complete code including geometry processing and algebraic solver has been developed and verified with the reference benchmark problem. Numerical results demonstrate the code can model the reference problem accurately, and the proposed preconditioning techniques based on the typical iterative method can decrease dramatically the number of iterations without introducing additional computation and storage expense.

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.

Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

Legendre---Gauss---Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive the main properties of the associated dyadic grids needed for preconditioning the systems arising from $$hp$$hp- or even spectral (conforming or Discontinuous Galerkin type) discretizations for second order elliptic problems in a way that is fully robust with respect to varying polynomial degrees. To establish these properties requires revisiting some refined properties of LGL grids and their relatives.

Preconditioning the Incompressible Navier-Stokes Equations with Variable Viscosity

The aim of the project is to construct, analyse and implement fast and reliable numerical solution methods to simulate multi-phase flow, modeled by a coupled system consisting of the time-dependent Cahn-Hilliard and incompressible Navier-Stokes equations with variable viscosity and variable density. This thesis mainly discusses the efficient solution methods for the latter equations aiming at constructing preconditioners, which are numerically and computationally efficient, and robust with respect to various problem, discretization and method parameters.In this work we start by considering the stationary Navier-Stokes problem with constant viscosity. The system matrix arising from the finite element discretization of the linearized Navier-Stokes problem is nonsymmetric of saddle point form, and solving systems with it is the inner kernel of the simulations of numerous physical processes, modeled by the Navier-Stokes equations. Aiming at reducing the simulation time, in this thesis we consider iterative solution methods with efficient preconditioners. When discretized with the finite element method, both the Cahn-Hilliard equations and the stationary Navier-Stokes equations with constant viscosity give raise to linear algebraic systems with nonsymmetric matrices of two-by-two block form. In Paper I we study both problems and apply a common general framework to construct a preconditioner, based on the matrix structure. As a part of the general framework, we use the so-called element-by-element Schur complement approximation. The implementation of this approximation is rather cheap. However, the numerical experiments, provided in the paper, show that the preconditioner is not fully robust with respect to the problem and discretization parameters, in this case the viscosity and the mesh size. On the other hand, for not very convection-dominated flows, i.e., when the viscosity is not very small, this approximation does not depend on the mesh size and works efficiently. Considering the stationary Navier-Stokes equations with constant viscosity, aiming at finding a preconditioner which is fully robust to the problem and discretization parameters, in Paper II we turn to the so-called augmented Lagrangian (AL) approach, where the linear system is transformed into an equivalent one and then the transformed system is iteratively solved with the AL type preconditioner. The analysis in Paper II focuses on two issues, (1) the influence of a scalar method parameter (a stabilization constant in the AL method) on the convergence rate of the preconditioned method and (2) the choice of a matrix parameter for the AL method, which involves an approximation of the inverse of the finite element mass matrix. In Paper III we consider the stationary Navier-Stokes problem with variable viscosity. We show that the known efficient preconditioning techniques in particular, those for the AL method, derived for constant viscosity, can be straightforwardly applicable also in this case.One often used technique to solve the incompressible Navier-Stokes problem with variable density is via operator splitting, i.e., decoupling of the solutions for density, velocity and pressure. The operator splitting technique introduces an additional error, namely the splitting error, which should be also considered, together with discretization errors in space and time. Insuring the accuracy of the splitting scheme usually induces additional constrains on the size of the time-step. Aiming at fast numerical simulations and using large time-steps may require to use higher order time-discretization methods. The latter issue and its impact on the preconditioned iterative solution methods for the arising linear systems are envisioned as possible directions for future research.When modeling multi-phase flows, the Navier-Stokes equations should be considered in their full complexity, namely, the time-dependence, variable viscosity and variable density formulation. Up to the knowledge of the author, there are not many studies considering all aspects simultaneously. Issues on this topic, in particular on the construction of efficient preconditioners of the arising matrices need to be further studied.

Data-sparse LU-decomposition preconditioning combined with multilevel fast multipole method for electromagnetic scattering problems

Owing to the robust of the multilevel fast multipole method (MLFMM), large dense linear systems, arising from discretised electric field integral equations of electromagnetic scattering problems, are generally solved by a Krylov iterative solver. An efficient preconditioning technique based on hierarchical LU-decomposition (ℋ-LU) is proposed to accelerate the convergence rate of the Krylov iterations. The ℋ-LU preconditioner is constructed from the available sparse near-field matrix and provides a data-sparse way to approximate its LU-factors, which can be computed and stored in ℋ-matrix arithmetic with almost linear complexity. The accuracy of the approximation is adjustable that will guarantee a bounded number of iterations. Some means are introduced to further lower the computational cost of the ℋ-LU. Numerical examples are provided to illustrate that the resulting ℋ-LU preconditioner is effective and flexible with MLFMM to reduce both the iteration number and the overall simulation time significantly.

Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations

AbstractWe develop an efficient preconditioning techniques for the solution of large linearized stationary and non‐stationary incompressible Navier–Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non‐stationary case. The time discretization procedure uses the Gear scheme and the second‐order Taylor–Hood element P2−P1 is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r∇(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P1−P2 hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd.

Efficient preconditioning for image reconstruction with radial basis functions

Radial basis functions are a popular basis for interpolating scattered data during the image reconstruction process in graphic analysis. In this context, the solution of a linear system of equations represents the most time-consuming operation. In this paper an efficient preconditioning technique is proposed to solve these linear systems of equations. This algorithm consists of an iterative method which enforces at each iteration a projection of the residual onto a suitable subspace called coarse space. This constraint is ensured by solving an auxiliary problem at each iteration without regularisation. As increasing the number of the coarse space basis functions increases the computational cost of the algorithm, correct selection of coarse space basis is addressed in the paper. Numerical results illustrate the convergence properties of the proposed method with wavelet-like basis functions and regular distributed radial basis functions for image reconstruction.

Efficient 3-D extraction of interconnect capacitance considering floating metal fills with boundary element method

Inserting dummy (area fill) metals is necessary to reduce the pattern-dependent variation of dielectric thickness in the chemical-mechanical polishing (CMP) process. Such floating dummy metals affect interconnect capacitance and, therefore, signal delay and crosstalk significantly. To take the floating dummies into account, an efficient method for three-dimensional (3-D) capacitance extraction based on boundary element method is proposed. By introducing a floating condition into the direct boundary integral equation (BIE) and adopting an efficient preconditioning technique, and the quasi-multiple medium (QMM) acceleration, the method achieves very high computational speed. For some typical structures of area fill, the presented algorithm has shown over 1000/spl times/ speedup over the industry-standard Raphael while preserving high accuracy. Compared with the recently proposed PASCAL in the work of Park et al. (2000), the proposed method also has about ten times speedup. Since the dummies are not regarded as normal electrodes in capacitance extraction, the proposed method is much more efficient than the conventional method, especially in cases with a large number of floating dummies.

Preconditioning for a Class of Spectral Differentiation Matrices

We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton's method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator

An efficient preconditioning technique using Krylov subspace methods for 3D characteristics solvers

Abstract The Generalized Minimal RESidual ( GMRES ) method, using a Krylov subspace projection, is adapted and implemented to accelerate a 3D iterative transport solver based on the characteristics method. Another acceleration technique called the self-collision rebalancing technique ( SCR ) can also be used to accelerate the solution or as a left preconditioner for GMRES . The GMRES method is usually used to solve a linear algebraic system ( A x = b ) . It uses K ( r ( o ) , A ) as projection subspace and A K ( r ( o ) , A ) for the orthogonalization of the residual. This paper compares the performance of these two combined methods on various problems. To implement the GMRES iterative method, the characteristics equations are derived in linear algebra formalism by using the equivalence between the method of characteristics and the method of collision probability to end up with a linear algebraic system involving fluxes and currents. Numerical results show good performance of the GMRES technique especially for the cases presenting large material heterogeneity with a scattering ratio close to 1. Similarly, the SCR preconditioning slightly increases the GMRES efficiency.

An efficient preconditioning technique for numerical simulation of hydrodynamic model semiconductor devices

AbstractWe investigate the application of preconditioned generalized minimal residual (GMRES) algorithm to the equations of hydrodynamic model of semiconductor devices. An introduction to such a model is presented. We use finite‐element method P1‐isoP2 element to discretize the equations. A preconditioning technique is proposed. The CPU times are presented for n+‐n‐n+ diodes and 0.25 μm gate length Si MESFETs by using the preconditioned GMRES algorithm and the GMRES algorithm. The numerical results show that the preconditioning technique accelerates effectively the velocity of convergence. Copyright © 2003 John Wiley & Sons, Ltd.