Incomplete Cholesky Preconditioner Research Articles

An Incomplete Cholesky Preconditioner Based on Orthogonal Approximations

An Incomplete Cholesky Preconditioner Based on Orthogonal Approximations

Fast finite element electrostatic analysis with domain decomposition method

Accurate simulation of electrostatic field in electronic devices is very important to improve their performance. When the traditional finite element method is used to analyze the electrostatic field of large and complex structures, the calculation speed is slow and the computing resources are greatly consumed. Therefore, in this paper, we investigate electrostatic analysis with a non-overlapping domain decomposition method based on the scalar finite element method under a novel transmission condition. In order to improve the efficiency of matrix solving and reduce the memory consumption, the block Jacobi preconditioner is introduced. In addition, the multifrontal block incomplete Cholesky preconditioner is utilized to further improve the computational performance based on the block Jacobi preconditioner. Several numerical examples are simulated and compared with the analytical solutions and the commercial software Maxwell 3D. The results demonstrate the accuracy and efficiency of the proposed domain decomposition method.

Linear programming with a feasible direction interior point technique for smooth optimization

We propose an adaptation of the Feasible Direction Interior Points Algorithm (FDIPA) of J. Herskovits, for solving large-scale linear programs. At each step, the solution of two linear systems with the same coefficient matrix is determined. This step involves a significant computational effort. Reducing the solution time of linear systems is, therefore, a way to improve the performance of the method. The linear systems to be solved are associated with definite positive symmetric matrices. Therefore, we use Split Preconditioned Conjugate Gradient (SPCG) method to solve them, together with an Incomplete Cholesky preconditioner using Matlab’s ICHOL function. We also propose to use the first iteration of the conjugate gradient, and to presolve before applying the algorithm, in order to reduce the computational cost. Following, we then provide mathematica proof that show that the iterations approach Karush–Kuhn–Tucker points of the problem under reasonable assumptions. Finally, numerical evidence show that the method not only works in theory but is also competitive with more advanced methods.

Adaptive rigidification of elastic solids

We present a method for reducing the computational cost of elastic solid simulation by treating connected sets of non-deforming elements as rigid bodies. Non-deforming elements are identified as those where the strain rate squared Frobenius norm falls below a threshold for several frames. Rigidification uses a breadth first search to identify connected components while avoiding connections that would form hinges between rigid components. Rigid elements become elastic again when their approximate strain velocity rises above a threshold, which is fast to compute using a single iteration of conjugate gradient with a fixed Laplacian-based incomplete Cholesky preconditioner. With rigidification, the system size to solve at each time step can be greatly reduced, and if all elastic element become rigid, it reduces to solving the rigid body system. We demonstrate our results on a variety of 2D and 3D examples, and show that our method is likewise especially beneficial in contact rich examples.

Acceleration and Parallelization of a Linear Equation Solver for Crack Growth Simulation Based on the Phase Field Model

We aim to accelerate the linear equation solver for crack growth simulation based on the phase field model. As a first step, we analyze the properties of the coefficient matrices and prove that they are symmetric positive definite. This justifies the use of the conjugate gradient method with the efficient incomplete Cholesky preconditioner. We then parallelize this preconditioner using so-called block multi-color ordering and evaluate its performance on multicore processors. The experimental results show that our solver scales well and achieves an acceleration of several times over the original solver based on the diagonally scaled CG method.

A high efficiency three‐dimensional finite‐element electromagnetic radiation simulation tool

In this article, a high efficiency three-dimensional electromagnetic radiation simulation tool based on finite element method (FEM) is presented. This simulation tool has the functions of human-computer interaction, constructing model, generating mesh, setting simulation parameters and making FEM analysis. In this simulation tool, the FEM analysis is the most critical part, so we mainly focus on the FEM electromagnetic radiation analysis technique in this article. In order to improve the efficiency of FEM analysis of electromagnetic radiation problems, we introduce the high order hierarchical vector basis function and the p-type multiplicative Schwarz (p-MUS) preconditioner to accelerate the iterative solution. What is more, we further improve the direct factorization process of p-MUS preconditioner by multifrontal block incomplete Cholesky preconditioner. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed tool.

An analysis of diagonal and incomplete Cholesky preconditioners for singularly perturbed problems on layer-adapted meshes

We investigate the solution of linear systems of equations that arise when singularly perturbed reaction–diffusion partial differential equations are solved using a standard finite difference method on layer adapted grids. It is known that there are difficulties in solving such systems by direct methods when the perturbation parameter, $$\varepsilon $$ , is small (MacLachlan and Madden in SIAM J Sci Comput 35(5):A2225–A2254, 2013). Therefore, iterative methods are natural choices. However, we show that, in two dimensions, the condition number of the coefficient matrix grows unboundedly when $$\varepsilon $$ tends to zero, and so unpreconditioned iterative schemes, such as the conjugate gradient algorithm, perform poorly with respect to $$\varepsilon $$ . We provide a careful analysis of diagonal and incomplete Cholesky preconditioning methods, and show that the condition number of the preconditioned linear system is independent of the perturbation parameter. We demonstrate numerically the surprising fact that these schemes are more efficient when $$\varepsilon $$ is small, than when $$\varepsilon $$ is $$\mathcal {O}(1)$$ . Furthermore, our analysis shows that when the singularly perturbed problem features no corner layers, an incomplete Cholesky preconditioner performs extremely well when $$\varepsilon \ll 1$$ . We provide numerical evidence that our findings extend to three-dimensional problems.

Accurate and fast three-dimensional free vibration analysis of large complex structures using the finite element method

Free vibration analysis of large complex structures with improved Lanczos iteration method is investigated through the finite element method. An important step in the traditional Lanczos iteration method for large generalized eigenvalue problems arising from free vibration analysis is that large-scale sparse linear system must be repeatedly solved within each step of the Lanczos iteration, which generally accounts for 70% to 90% of the calculation time of the eigenvalue problems. In the improved Lanczos iteration method given here, three-level preconditioner is proposed and utilized to improve the computational performance of Lanczos iteration. In this way, the multifrontal block incomplete Cholesky (MFBIC) preconditioner and p-type multigrid preconditioner based on higher-order hierarchical basis functions are proposed and utilized to improve the computational performance and decrease the memory requirements. To further reduce the memory consumption, the block Jacobi preconditioner is introduced in the Lanczos iteration, and a new three-level preconditioner that combines MFBIC, p-type multigrid and block Jacobi is proposed. The accuracy and performance of the improved method has been compared with exact solution, basic method and ANSYS. It is shown that this free vibration analysis method can be extremely useful to design structure dynamic characteristics of large complex structures.

“Traveling-wave tube mechanics simulator suite” a CAD/CAE integrated rapid redesign system of vibration analysis for traveling-wave tubes

Traveling-wave tubes (TWTs) are popularly used as the microwave power amplifier in civil and military applications and have a high mechanical sensitivity. It is necessary to understand their vibration characteristics under all operating conditions. To address this issue, a CAD/CAE integrated rapid redesign system of vibration analysis with unified data architecture (UDA) has been developed and presented in this paper. The UDA is presented to make CAD and CAE be an organic entity. This system is composed of the following four layers: (i) CAD/CAE integrated design environment, (ii) parametric modeling and fast modeling platform with custom-built tools, (iii) accurate and fast finite element free vibration analysis solver based on rapid solving techniques of equation, and (iv) fast finite element random vibration analysis solver. The CAD/CAE integrated system concept is typically applied in loop design-analysis-redesign procedures. In this CAD/CAE integrated design system, the designers can freely and rapidly complete component design and performance analysis simultaneously instead of using two different software or two interface environment. To improve the accuracy and efficiency of large-scale finite element vibration analysis, especially for free vibration analysis, an improved Lanczos iteration method with a new three-level preconditioner is developed and pseudo excitation method is introduced to perform random vibration analysis. The new three-level preconditioner is composed of a multifrontal block incomplete cholesky preconditioner, a p-type multigrid preconditioner, and a block Jacobi preconditioner. The developed system has been applied to the design of TWTs and been found to shorten the design cycle significantly compared with ANSYS.

Improvement of Block IC Preconditioner Using Fill-In Technique for Linear Systems Derived From Finite-Element Method Including Thin Elements

The block incomplete-Cholesky (IC) preconditioner is often used for parallel preconditioner in finite-element method (FEM) supported by distributed computing. When many thin elements are included in an analyzed mesh such as magnetic shielding, the convergence characteristic of preconditioned conjugate gradient method would be deteriorated. At that time, IC preconditioner with fill-in makes the convergence characteristic of linear solver much better. This paper demonstrates the validity of block IC with fill-in on linear systems arising in FEM with many flat elements. First, the ability to improve the convergence property using the fill-in strategy is verified by the simulation of magnetic shielding. Next, the parallelization performance of block IC with fill-in is illustrated by using finite-element mesh of laminated core model and induction heating cooker.

Updating preconditioners for modified least squares problems

In this paper, we analyze how to update incomplete Cholesky preconditioners to solve least squares problems using iterative methods when the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Our proposed method computes a low-rank update of the preconditioner using a bordering method which is inexpensive compared with the cost of computing a new preconditioner. Moreover, the numerical experiments presented show that this strategy gives, in many cases, a better preconditioner than other choices, including the computation of a new preconditioner from scratch or reusing an existing one.

On Efficiency Analysis of the OpenFOAM-Based Parallel Solver for Simulation of Heat Transfer in and Around the Electrical Power Cables

In this work, we study the efficiency of developed OpenFOAM-based parallel solver for the simulation of heat transfer in and around the electrical power cables. First benchmark problem considers three cables directly buried in the soil. We study and compare the efficiency of conjugate gradient solver with diagonal incomplete Cholesky (DIC) preconditioner, generalized geometric-algebraic multigrid GAMG solver from OpenFOAM and conjugate gradient solver with GAMG multigrid solver used as preconditioner. The convergence and parallel scalability of the solvers are presented and analyzed on quadrilateral and acute triangle meshes. Second benchmark problem considers a more complicated case, when cables are placed into plastic pipes, which are buried in the soil. Then a coupled multi-physics problem is solved, which describes the heat transfer in cables, air and soil. Non-standard parallelization approach is presented for multi-physics solver. We show the robustness of selected parallel preconditioners. Parallel numerical tests are performed on the cluster of multicore computers.

Applications of MMPBSA to Membrane Proteins I: Efficient Numerical Solutions of Periodic Poisson-Boltzmann Equation.

Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membranes into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multigrid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations.

A parallel finite-element time-domain method for transient electromagnetic simulation

The finite-difference time-domain (FDTD) method has been widely used to perform transient electromagnetic simulation. Compared to the traditional FDTD method, the finite-element time-domain (FETD) method with unstructured meshes and an adaptive time-stepping scheme has the potential to dramatically cut down the number of unknowns and the number of time steps. However, with a need to solve a large-scale unstructured sparse matrix at every time step, the FETD method is generally difficult to scale with the available parallel computing resources. To make efficient use of the multicore resources in current computing platforms, we designed a parallel FETD method that used multiple threads to accelerate the key steps for solving a large sparse matrix, including the construction of the preconditioner and matrix-vector multiplication. Using a customized incomplete Cholesky preconditioner, we achieved faster convergence than many existing preconditioners and a scalable parallel performance over multiple cores. To further reduce the total calculation time, we implemented an overlapping scheme and a preconditioner-reusing scheme. The overlapping scheme executed the solver computation and the preconditioner computation concurrently using different parts of the parallel resources. The preconditioner reusing scheme reduced the need to recompute preconditioners when increasing the time-step size, thus reducing the frequency of the less scalable preconditioner computations. It brought performance benefits by reducing the frequency of less scalable preconditioner computation. Computed parallel FETD results showed the same level of accuracy as FDTD results. In terms of parallel performance, using eight cores, the method cut execution time by a factor of 4 (i.e., 50 parallel efficiency). For a realistic salt model, we simulated TEM responses from 0.01 s to 30 s in 5 min, using hundreds of CPU cores.

An adaptive preconditioned conjugate gradient approach for extremely large optimisation problems arising in computational geomechanics

The feasibility of preconditioned conjugate gradient (PCG) for solving extremely large sparse linear systems arising in three-dimensional finite element limit analysis of geotechnical stability problems is studied. With problem size increasing beyond few millions of variables, the use of direct solvers becomes prohibitive (both in storage and computational times) even on modern computers. This motivates the study on possible employment of iterative methods to solve systems of equations resulting from interior point method-based limit analysis formulations. In this paper, an adaptive preconditioning technique based on incomplete Cholesky (IC) preconditioner, as well as a novel implementation scheme for preconditioned conjugate gradient method, is proposed to effectively deal with highly ill-conditioned systems, such as those which occur at last iterations of IPM methods. Numerical tests on some ultra-large scale sample geotechnical problems, along with comparison with other potential solution methods, verify the robustness and efficiency of the proposed approach.

A Cholesky preconditioned conjugate gradient algorithm on GPU for the 3D parabolic equation

For solving the large sparse linear systems to which numerical solution of the three-dimensional (3D) parabolic equation leads, an efficient parallel preconditioned conjugate gradient algorithm with the modified incomplete Cholesky (MIC) preconditioner on the GPU (MICPCGA) is proposed. In our proposed method, for this case, we overcome the drawback that the MIC preconditioner is generally difficult to parallelise on the GPU owing to the forward/backward substitutions, and thus present an efficient parallel implementation method on the GPU (GPUFBS). In addition, the vector operations are optimised by grouping several vector operations into a single kernel, and an inner-product kernel is suggested, and a kernel for the sparse matrix-vector multiplication in the CUSPARSE library is adopted. Numerical results show that our proposed GPUFBS and MICPCGA both can achieve a significant speedup, and compared to an approximate inverse SSOR preconditioned conjugate gradient algorithm (SSORPCGA), our proposed MICPCGA not only obtains a bigger speedup, but also has higher precision in solving the 3D parabolic equation.

Preconditioned Krylov Subspace Methods for Sampling Multivariate Gaussian Distributions

A common problem in statistics is to compute sample vectors from a multivariate Gaussian distribution with zero mean and a given covariance matrix $A$. A canonical approach to the problem is to compute vectors of the form $y = S z$, where $S$ is the Cholesky factor or square root of $A$, and $z$ is a standard normal vector. When $A$ is large, such an approach becomes computationally expensive. This paper considers preconditioned Krylov subspace methods to perform this task. The Lanczos process provides a means to approximate $A^{1/2} z$ for any vector $z$ from an $m$-dimensional Krylov subspace. The main contribution of this paper is to show how to enhance the convergence of the process via preconditioning. Both incomplete Cholesky preconditioners and approximate inverse preconditioners are discussed. It is argued that the latter class of preconditioners has an advantage in the context of sampling. Numerical tests, performed with stationary covariance matrices used to model Gaussian processes, illustrate the dramatic improvement in computation time that can result from preconditioning.

SIMPLE METHODS TO RAISE THE ROBUSTNESS AND EFFICIENCY OF THE INCOMPLETE CHOLESKY PRECONDITIONERS FOR FEM SIMULATION OF ELECTROMAGNETIC PROBLEMS

In this paper, the flnite element method (FEM) is applied to the analysis of three- dimensional (3D) electromagnetic structures. The incomplete Cholesky (IC) preconditioner based on shifted operators is used to solve the flnite element linear systems. Several strategies are adopted to raise the e-ciency and robustness of the preconditioner. Numerical experiments for several microwave devices demonstrate the superior numerical convergence and robustness of the proposed preocnditioner.

Research on the conjugate gradient algorithm with a modified incomplete Cholesky preconditioner on GPU

In this study, we discover the parallelism of the forward/backward substitutions (FBS) for two cases and thus propose an efficient preconditioned conjugate gradient algorithm with the modified incomplete Cholesky preconditioner on the GPU (GPUMICPCGA). For our proposed GPUMICPCGA, the following are distinct characteristics: (1) the vector operations are optimized by grouping several vector operations into single kernels, (2) a new kernel of inner product and a new kernel of the sparse matrix–vector multiplication with high optimization are presented, and (3) an efficient parallel implementation of FBS on the GPU (GPUFBS) for two cases are suggested. Numerical results show that our proposed kernels outperform the corresponding ones presented in CUBLAS or CUSPARSE, and GPUFBS is almost 3 times faster than the implementation of FBS using the CUSPARSE library. Furthermore, GPUMICPCGA has better behavior than its counterpart implemented by the CUBLAS and CUSPARSE libraries.

Automatic Determination of Acceleration Factor Based on Residual and Functional in Shifted ICCG Method for 3-D Electromagnetic Field Analyses

This paper describes new strategy for determining an optimal acceleration factor in the shifted incomplete Cholesky conjugate gradient (ICCG) method. Although a useful method focused on the maximum diagonal entry in the incomplete Cholesky (IC) preconditioner was already proposed, it cannot always provide the optimal acceleration factor. In this paper, we propose new automatic determination methods based on the minimization of the norms of the remainder matrix, the residual, and the functional in the shifted ICCG method and discuss its effectiveness in the 3-D electromagnetic field analyses using the edge-based finite element method. It is concluded that the proposed methods based on the minimization of the norm of the residual or the functional can determine an appropriate acceleration factor within the acceptable computational cost and improve the convergence property of the shifted ICCG method.