Solution techniques for the p‐version of the adaptive finite element method

A novel sine transform based preconditioned conjugate gradient (PCG) method for signal restoration is presented. If the bandwidth 2? +1 in the model matrix is a constant, then the asymptotic complexity for the new algorithm is O(N) per iterative step, where Nis the order of the model. This complexity is reduced by an order of magnitude in comparison with the known fast Fourier transform (FFT) technique proposed recently in the literature. Moreover, if ? has a size similar to N, then the new PCG method is accomplished by the fast sine transform (FST) and the fast Hartley transform (FHT). The computational and storage cost per preconditioning step for the new PCG method is reduced by 50% as compared to the FFT approach. To stabilize the computation, a special Tikhonov regularization is introduced. Numerical experimentations show that the new PCG method is of a speedy convergence rate. The new PCG methods maintain less computational complexity per iterative step and have a better convergence rate than the other known PCG methods.

This paper presents a novel hardware solution for solving sparse matrices using emulation technology. The proposed solution introduces a bio-inspired technique, which is Genetic Algorithm (GA) to solve the sparse system of linear equations (SLEs). The proposed solution can be used with any physical system that can be modeled as SLEs. Computations involved in Finite Element Method (FEM) consume too much time, which affects the final time-to-market value. Profiling shows that the most time-consuming part in the simulation process is the solver part which is responsible for solving the resultant SLEs generated from the FEM. The total number of equations may reach thousands or millions of linear equations. Hardware implementation of GA and making a good use of parallelism in implementing the architecture accelerate the time taken by the solver part. Solving the system by classical numerical methods, such as conjugate gradient (CG) implemented in software is performed sequentially, here the architecture allows the parallelism in performing GA operators, such as selection, crossover, and mutation. The parallelism in the proposed architecture is the superiority point which we rely on. The results show that our proposed method to solve the SLEs outperforms the classical CG software implementation on run time and number of equations.

A projection method for solving nonsymmetric linear systems on multiprocessors

Many of the fast methods for factoring integers and computing discrete logarithms require the solution of large sparse linear systems of equations over finite fields. This paper presents the results of implementations of several linear algebra algorithms. It shows that very large sparse systems can be solved efficiently by using combinations of structured Gaussian elimination and the conjugate gradient, Lanczos, and Wiedemann methods.

In this study, we analyze the real-time solution of a series of augmented least squares problems, which are generated by adding information to an original least squares model repetitively. Instead of solving the least squares problems directly, we transform them into a batch of saddle point linear systems and subsequently solve the linear systems using restrictively preconditioned conjugate gradient (RPCG) methods. Approximation of the new Schur complement is generated effectively based on a previously approximated Schur complement. Owing to the variations of the preconditioned conjugate gradient method, the proposed methods generate convergence results similar to the conjugate gradient method and achieve a very fast convergent iterative sequence when the coefficient matrix is well preconditioned. Numerical tests show that the new methods are more effective than some standard Krylov subspace methods. Updated RPCG methods meet the requirement of real-time computing successfully for multifactor models.

Gauss-Seidel preconditioned conjugate gradient (GS-PCG) method based on element-by-element (EBE) is derived and the calculation process of finite element method (FEM) is presented in this paper. The corresponding FEM program are developed and implemented on the magnetic field calculations which produced by straight conductor carrying current and 3-phases induction motor respectively. The validity and accuracy of the method have been verified by comparing the result with the analytical solution. Compared with Jacobi preconditioned conjugate gradient (J-PCG) method based on EBE-FEM, less computational time and cost are needed for the proposed method with same accuracy. The proposed method can be applied to electromagnetic field analysis of transformer and motor, which may refer large scale numerical computation.

We present a parallel domain decomposition method based on a hybrid direct-iterative solver for 3D frequency-domain modelling of visco-acoustic waves. Frequency-domain seismic modelling reduces to the solution of a large and sparse system of linear equations for each frequency. The hybrid directiterative approach aims to overcome the memory requirement and the limited scalability of direct-solver approaches, on the one hand, and to iteratively solve better-conditioned system than in global iterative approaches on the other hand. The domain decomposition is based upon the algebraic Schur complement method. The reduced Schur complement system is solved with the global minimum residual method (GMRES) and is preconditioned by an algebraic additive Schwarz preconditioner. The MUMPS direct solver is used to factorize local impedance matrices defined on each subdomain. Simulations in the salt models for frequencies up to 12.5 Hz show that the number of iterations increases linearly with frequencies when the grid interval is matched to the frequency and the size of the subdomains is kept constant over frequencies. This makes the time complexity of the hybrid approach similar to that of global iterative solvers. Possible improvements of the method for multi-source simulation involve the use of block iterative solver and two levels of parallelism.

During my visit to LLNL during the summer of 2010, I worked on algebraic multilevel solvers for large sparse systems of linear equations arising from discretizations of partial differential equations. The particular solver of interest is based on ILU decomposition. The setup phase for this AMG solve is just the single ILU decomposition, and its corresponding error matrix. Because the ILU uses a minimum degree or similar sparse matrix ordering, most of the fill-in, and hence most of the error, is concentrated in the lower right corner of the factored matrix. All of the major multigrid components - the smoother, the coarse level correction matrices, and the fine-to-coarse and coarse-to-fine rectangular transfer matrices, are defined in terms of various blocks of the ILU factorization. Although such a strategy is not likely to be optimal in terms of convergence properties, it has a relatively low setup cost, and therefore is useful in situations where setup costs for more traditional AMG approaches cannot be amortized over the solution of many linear systems using the same matrix. Such a situation arises in adaptive methods, where often just one linear system is solved at each step of an adaptive feedback loop, or in solving nonlinear equations by approximate Newton methods, where the approximate Jacobian might change substantially from iteration to iteration. In general terms, coarse levels are defined in terms of successively smaller lower right blocks of the matrix, typically decreasing geometrically in order. The most difficult issue was the coarse grid correction matrix. The preconditioner/smoother for a given level is just the corresponding lower right blocks of the ILU factorization. The coarse level matrix itself is just the Schur complement; this matrix is not known exactly using just the ILU decomposition in the setup phase. Thus we approximate this matrix using various combinations of the preconditioning matrix and the error matrix. During my visit, several approximations of this type were implemented and tested. While some improved the convergence rate of the overall method, these gains had to be balanced against the additional costs involved in creating and applying these matrices. By this more stringent criterion, none of the improved approximations could be characterized as an unqualified success.

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

Data Flow Analysis of Orthogonal Properties on the Conjugate Gradient and the Lanczos Algorithm

High-resolution global ocean general circulation models (OGCMs) play a key role in accurate ocean forecasting. However, the models of the operational forecasting systems are still not in high resolution due to the subsequent high demand for large computation, as well as the low parallel efficiency barrier. Good scalability is an important index of parallel efficiency and is still a challenge for OGCMs. We found that the communication cost in a barotropic solver, namely, the preconditioned conjugate gradient (PCG) method, is the key bottleneck for scalability due to the high frequency of the global reductions. In this work, we developed a new algorithm—a communication-avoiding Krylov subspace method with a PCG (CA-PCG)—to improve scalability and then applied it to the Nucleus for European Modelling of the Ocean (NEMO) as an example. For PCG, inner product operations with global communication were needed in every iteration, while for CA-PCG, inner product operations were only needed every eight iterations. Therefore, the global communication cost decreased from more than 94.5% of the total execution time with PCG to less than 63.4% with CA-PCG. As a result, the execution time of the barotropic modes decreased from more than 17,000 s with PCG to less than 6000 s with CA-PCG, and the total execution time decreased from more than 18,000 s with PCG to less than 6200 s with CA-PCG. Besides, the ratio of the speedup can also be increased from 3.7 to 4.6. In summary, the high process count scalability when using CA-PCG was effectively improved from that using the PCG method, providing a highly effective solution for accurate ocean simulation.

A preconditioned Chebyshev basis communication-avoiding conjugate gradient method (P-CBCG) is applied to the pressure Poisson equation in a multiphase thermal-hydraulic CFD code JUPITER, and its computational performance and convergence properties are compared against a preconditioned conjugate gradient (P-CG) method and a preconditioned communication-avoiding conjugate gradient (P-CACG) method on the Oakforest-PACS, which consists of 8,208 KNLs. The P-CBCG method reduces the number of collective communications with keeping the robustness of convergence properties. Compared with the P-CACG method, an order of magnitude larger communication-avoiding steps are enabled by the improved robustness. It is shown that the P-CBCG method is 1.38 $$\times $$ and 1.17 $$\times $$ faster than the P-CG and P-CACG methods at 2,000 processors, respectively.

Computer subroutines for solving large sparse systems of linear equations usually involve fairly complicated data structures and storage management. In most cases, the user of such subroutines simply wants top solve his problem and does not have to understand how the storage management is done, or how the matrix components are stored. Thus software for solving sparse systems of linear equations should be designed in such a way that it can insulate the user from these considerations, while still allowing the software to be used in a variety of ways.

Iterative solvers allow for a trade-off between speed and accuracy. We propose an iterative method for the estimation of the internal states of a given discrete-time linear state-space model from a series of noisy measurements. In particular we identify the MAP estimate of those states as being the solution of a sparse system of linear equations and derive an iterative solver based on the conjugate gradient method. We derive convergence results to quantify the trade-off between speed and accuracy and finally apply the method to channel estimation where it is shown to outperform Kalman smoothing complexity-wise.

We present a geometric framework for surface denoising using graph signal processing, which is an emerging field that aims to develop new tools for processing and analyzing graph-structured data. The proposed approach is formulated as a constrained optimization problem whose objective function consists of a fidelity term specified by a noise model and a regularization term associated with prior data. Both terms are weighted by a normalized mesh Laplacian, which is defined in terms of a data-adaptive kernel similarity matrix in conjunction with matrix balancing. Minimizing the objective function reduces it to iteratively solve a sparse system of linear equations via the conjugate gradient method. Extensive experiments on noisy carpal bone surfaces demonstrate the effectiveness of our approach in comparison with existing methods. We perform both qualitative and quantitative comparisons using various evaluation metrics.