Parallel diagonally implicit Runge-Kutta-Nyström methods

This paper examines diagonally implicit iteration methods for solving implicit Runge–Kutta methods with high stage order on parallel computers. These iteration methods are such that after a finite number of m iterations, the iterated Runge–Kutta method belongs to the class of diagonally implicit Runge–Kutta methods (DIRK methods) using $mk$ implicit stages where k is the number of stages of the generating implicit Runge–Kutta method (corrector method). However, a large number of the stages of this DIRK method can be computed in parallel, so that the number of stages that have to be computed sequentially is only m. The iteration parameters of the method are tuned in such a way that fast convergence to the stability characteristics of the corrector method is achieved. By means of numerical experiments it is also shown that the solution produced by the resulting iteration method converges rapidly to the corrector solution so that both stability and accuracy characteristics are comparable with those of the corrector. This implies that the reduced accuracy often shown when integrating stiff problems by means of DIRK methods already available in the literature (which is caused by a low stage order) is not shown by the DIRK methods developed in this paper, provided that the corrector method has a sufficiently high stage order.

We study nonlinear inverse problems of the form F ( x ) = y , and their stable solution via iterative regularization methods, in particular by Newton-type methods, which are well-known for their fast convergence for well-posed problems. A basic step of Newton’s method consists of calculating the update Δ x k via solution of a linearized equation , which will in general be ill-posed if the nonlinear problem is. Thus, for ill-posed problems, the linearized equations have to be solved by some regularization method. In particular for large scale problems, e.g., inverse problems in partial differential equations, where F ( x ) is only defined implicitly via the solution of a PDE, iterative methods have to be used for this purpose. In order to keep the overall effort, i.e., the overall number of iterations, as small as possible, appropriate preconditioning has to be applied. We propose and analyse a general preconditioning strategy in Hilbert scales, and show that the overall number of iterations can be reduced to about the square root by preconditioning. Moreover, in many examples differential operators can be used as preconditioners, and thus preconditioning is almost for free. The theoretical results are illustrated in numerical examples. A comparison with preconditioned Landweber iteration shows that the iteration numbers can be further reduced, if fast iterative methods, e.g., the v -methods, are used for the solution of the linearized problems.

Background and Objectives: Numerical iterative methods are widely used to compute reachability probabilities and expected rewards in probabilistic model checking of Markov decision processes. Several approaches have been proposed to improve the performance of these iterative methods. Reducing the total number of iterations is an important goal that is followed by several techniques in this paper.Methods: In this paper, we consider MDPs with different levels of non-determinism. We show that modified policy iteration performs better than the other standard iterative methods when the degree of non-determinism increases. We propose some novel methods to improve the performance of the modified policy iteration method. Our approach is to define several criteria to avoid useless iterations and updates of the modified policy iteration method. The first criterion considers the graphical structure of the model to define the number of iterations after each policy modification. The second criterion is a dynamic one to determine when a policy should be updated. The third proposed approach in this work is to use several priority heaps to select states for value updates.Results: Our proposed methods are implemented in the PRISM model checker and applied on several standard case studies. The results of experiments show that the running times of our approaches are less than the running time of the standard and previous methods for most case studies. The main reason for these results is that the total numbers of iterations and updates are reduced by our approaches, which results in an improvement in the performance of the method. While the running times are reduced by our approaches, the precision of computations are kept in most cases.Conclusion: The proposed techniques are able to reduce the number of iterations and accelerate the convergence to the optimal policies. The also prioritize the commutations to reduce the total number of updates.

Parallel iterative regularization methods for solving systems of ill-posed equations

Introduction: Three-dimensional (3D)-printed custom pelvic implants have become a clinically viable option for patients undergoing pelvic cancer surgery with resection of the hip joint. However, increased clinical utilization has also necessitated improved implant durability, especially with regard to the compression screws used to secure the implant to remaining pelvic bone. This study evaluated six different finite element (FE) screw modeling methods for predicting compression screw pullout and fatigue failure in a custom pelvic implant secured to bone using nine compression screws. Methods: Three modeling methods (tied constraints (TIE), bolt load with constant force (BL-CF), and bolt load with constant length (BL-CL)) generated screw axial forces using functionality built into Abaqus FE software; while the remaining three modeling methods (isotropic pseudo-thermal field (ISO), orthotropic pseudo-thermal field (ORT), and equal-and-opposite force field (FOR)) generated screw axial forces using iterative physics-based relationships that can be implemented in any FE software. The ability of all six modeling methods to match specified screw pretension forces and predict screw pullout and fatigue failure was evaluated using an FE model of a custom pelvic implant with total hip replacement. The applied hip contact forces in the FE model were estimated at two locations in a gait cycle. For each of the nine screws in the custom implant FE model, likelihood of screw pullout failure was predicted using maximum screw axial force, while likelihood of screw fatigue failure was predicted using maximum von Mises stress. Results: The three iterative physics-based modeling methods and the non-iterative Abaqus BL-CL method produced nearly identical predictions for likelihood of screw pullout and fatigue failure, while the other two built-in Abaqus modeling methods yielded vastly different predictions. However, the Abaqus BL-CL method required the least computation time, largely because an iterative process was not needed to induce specified screw pretension forces. Of the three iterative methods, FOR required the fewest iterations and thus the least computation time. Discussion: These findings suggest that the BL-CL screw modeling method is the best option when Abaqus is used for predicting screw pullout and fatigue failure in custom pelvis prostheses, while the iterative physics-based FOR method is the best option if FE software other than Abaqus is used.

Hardware trends have motivated the development of mixed precision algorithms in numerical linear algebra, which aim to decrease runtime while maintaining acceptable accuracy. One recent development is the development of an adaptive precision sparse matrix–vector produce routine, which may be used to accelerate the solution of sparse linear systems by iterative methods. This approach is also applicable to the application of inexact preconditioners, such as sparse approximate inverse preconditioners used in Krylov subspace methods. In this work, we develop an adaptive precision sparse approximate inverse preconditioner and demonstrate its use within a five-precision GMRES-based iterative refinement method. We call this algorithm variant BSPAI-GMRES-IR. We then analyze the conditions for the convergence of BSPAI-GMRES-IR, and determine settings under which BSPAI-GMRES-IR will produce similar backward and forward errors as the existing SPAI-GMRES-IR method, the latter of which does not use adaptive precision in preconditioning. Our numerical experiments show that this approach can potentially lead to a reduction in the cost of storing and applying sparse approximate inverse preconditioners, although a significant reduction in cost may comes at the expense of increasing the number of GMRES iterations required for convergence.

Recently, acoustic analysis techniques are used in the design of acoustic environments both inside and outside of rooms containing a sound source. However, when the analysis area is expanded or the frequency is increased, unknowns of a solving liner equation increase, and large-scale analyses becomes necessary. Therefore, in this study, a large-scale acoustic analysis method has been developed based on a parallel finite element method. In the proposed large-scale analysis method, an iterative method is applied to the interface problem as the iterative domain decomposition method, which is known to be an effective parallelization method. To improve the convergence of the iterative method, balancing domain decomposition has been applied as a preconditioner step, and its effectiveness in large-scale acoustic analysis is demonstrated. The acoustic analysis code is developed in this study that is shown to be capable of the large-scale analysis of finite element models on the order of tens of millions of elements with high accuracy.

Numerical Linear Algebra with ApplicationsVolume 6, Issue 1 p. 1-1 EditorialFree Access Czech–US workshop on iterative methods and parallel computing Ronold W. Freund, Corresponding Author Ronold W. Freund Murray Hill, New Jersey, U.S.AMurray Hill, New Jersey, U.S.A===Search for more papers by this authorIvo Marek, Ivo Marek Prague, Czech RepublicSearch for more papers by this author Ronold W. Freund, Corresponding Author Ronold W. Freund Murray Hill, New Jersey, U.S.AMurray Hill, New Jersey, U.S.A===Search for more papers by this authorIvo Marek, Ivo Marek Prague, Czech RepublicSearch for more papers by this author First published: 22 June 1999 https://doi.org/10.1002/(SICI)1099-1506(199901/02)6:1<1::AID-NLA165>3.0.CO;2-DCitations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume6, Issue1Special Issue: Czech–US Workshop on Iterative Methods and Parallel ComputingJanuary/February 1999Pages 1-1 RelatedInformation

Atmospheric tomography, i.e., the reconstruction of the turbulence profile in the atmosphere, is a challenging task for adaptive optics (AO) systems for the next generation of extremely large telescopes. Within the AO community, the solver of first choice is the so-called Matrix Vector Multiplication (MVM) method, which directly applies the (regularized) generalized inverse of the system operator to the data. For small telescopes this approach is feasible, however, for larger systems such as the European Extremely Large Telescope (ELT), the atmospheric tomography problem is considerably more complex, and the computational efficiency becomes an issue. Iterative methods such as the Finite Element Wavelet Hybrid Algorithm (FEWHA) are a promising alternative. FEWHA is a wavelet-based reconstructor that uses the well-known iterative preconditioned conjugate gradient (PCG) method as a solver. The number of floating point operations and the memory usage are decreased significantly by using a matrix-free representation of the forward operator. A crucial indicator for the real-time performance are the number of PCG iterations. In this paper, we propose an augmented version of FEWHA, where the number of iterations is decreased by 50% using an augmented Krylov subspace method. We demonstrate that a parallel implementation of augmented FEWHA allows the fulfilment of the real-time requirements of the ELT.

An iterative method is introduced for solving noisy, ill-conditioned inverse problems. Analysis of the semi-convergence behavior identifies three error components iteration error, noise error, and initial guess error. A derived expression explains how the three errors are related to each other relative to the number of iterations. The Standard Tikhonov regularization method is just the first iteration of the iterative method and the derived noise damping filter is a generalization of the Standard Tikhonov filter. The derived filter is a function two parameters, a regularization parameter and the iteration number parameter. The new method is tested on image reconstruction from projections simulated data set.

A myriad of methods, including many optimization and estimation algorithms, require the sensitivities of a dynamic system governed by a set of ordinary differential equations (ODEs). In this paper, the decoupled direct method (DDM) is derived for the calculation of first- and second-order state transition matrices (STMs) using fixed-step-size Runge-Kutta (RK) ODE solvers. Implications for both explicit and implicit RK methods are discussed. Emphasis is placed on the implicit class because of recent interest in the astrodynamics community and because the decoupling of the state propagation and STM calculation eliminates the need to evaluate the Jacobian and Hessian iteratively at each propagation step. Thus, significant computational savings are achievable compared to the traditional propagation of the variational equations, particularly for systems with expensive dynamics. In the current work, two implementations of the DDM are extended both to second-order STMs and to the double-integrator form of the RK equations. One of the strategies presented in this paper, newly derived to include second-order sensitivities, takes advantage of the structure of the system, allowing for the STMs of nonlinear systems to be calculated using efficient linear-algebra techniques rather than iterative methods. This linear-algebra-based approach is shown to be more efficient than the second alternative, and is also more efficient than the propagation of the variational equations for computationally intensive dynamical systems. The effects of important implicit RK customizations, such as variable-fidelity dynamics models and parallelization, are also examined in two examples: a simple, proof-of-concept forced harmonic oscillator, and an Earth-orbiting space object subject to either a low- or high-computational-cost geopotential model.

In this paper, we construct a Grunwald implicit approximation equation of one-dimensional (1D) time-fractional parabolic equations by applying the discretization process based the second-order implicit finite difference scheme and Grunwald fractional derivative operator. Then this approximation equation leads the generated linear system. We also consider using the Modified Kaudd Successive Over Relaxation (MKSOR) iterative method for solving linear system iteratively. For comparison purposes, three numerical examples are presented to illustrate the efficiency of MKSOR iterative method over another two existing iterative methods. Based on the findings of recorded numerical results, it has revealed that the MKSOR method involves less number of iterations and the computation time during the the implementation of iteration process.

ADAPTIVE MODELING FOR BIT RATE CONTROL

Avoiding Distortion in Corporate Valuation Litigation: An Application of Discounted Cash Flow

In this paper, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem using the projection technique. This alternative equivalent formulation is used to suggest and analyze some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also considered. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.