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

Parallel diagonally implicit Runge-Kutta-Nyström methods

Parallel Iterative Methods with Factored Preconditioning Matrices for Solving Discrete Elliptic Equations

A pair of parallel sequences $\{ u_k \} \to u,\{ v_k \} \to 0$ (1.3) is generated to solve the $n \times n$ linear system $Au = (I_n - B)u = f$. Convergence depends only on the geometry or shape of $\sigma (B)$, the set of eigenvalues of B. The parallel method is applied to the singularly perturbed convection-diffusion equation (6.1), when the Reynolds number in the direction of flow is large. Numerical comparisons with known results are given. Our theory also applies to the class of possibly nonsymmetric A with real spectrum (cf. Theorem 5.1) and to several other classes of systems as well. Computations to generate the sequences are relatively straightforward, as is indicated in our main result, Theorem 4.1. In fact, the parameters of the embracing ellipse for $\sigma (B)^2 $ (4.6) completely determine (i) the coefficients for the parallel sequences $\{ u_k \} \to u$ and $\{ v_k \} \to 0$ and (ii) the spectral radius (4.4), which characterizes their asymptotic convergence rate (2.4). Figure 5.1 illustrates some geometries for $\sigma (B)$ that are accommodated by our theory and Figure 7.1 shows the eigenvalue bowtie region arising from the convection-diffusion equation with large Reynolds number.

In this paper we show the results of some research carried out on parallel iterative methods to solve equations. In particular we study general classes of one point parallel methods and multipoint ones without memory, and we point out the convergence order of these methods and the conditions which are both necessary and sufficient for them to be optimal. In addition we prove that the convergence order for multipoint parallel procedures without memory cannot be more thenr(r+)m−1, wherer indicates the number of the parallel processor used andm the number of the functions and eventual derivatives, calculated not simultaneously in every iteration.

It is widely accepted that the asynchronous parallel methods are more suitable than the synchronous ones on a grid architecture. Indeed, they outperform the synchronous methods, because they overlap the communications of the synchronous methods with computations. However, they also usually execute more iterations than the synchronous ones and thus consume more energy. To reduce the energy consumption of the CPUs executing such methods, the Dynamic voltage and frequency scaling technique can be used. It lowers the frequency of a CPU to reduce its energy consumption, but it also decreases its computing power. Therefore, the frequency that gives the best trade-off between energy consumption and performance must be selected. This paper presents a new online frequency selecting algorithm for parallel iterative asynchronous methods running over grids. It selects a vector of frequencies that gives the best trade-off between energy consumption and performance. New energy and performance models were used in this algorithm to predict the execution time and the energy consumption of synchronous, asynchronous, or hybrid iterative applications running over grids. The proposed algorithm was evaluated on the SimGrid simulator. The experiments showed that synchronously applying the proposed algorithm to the asynchronous version of the application reduces on average its energy consumption by 22% and speeds it up by 5.72%. Finally, the proposed algorithm was also compared to a method that uses the well-known energy and delay product and the comparison results showed that it outperforms this method in terms of energy consumption and performance.

State‐of‐the‐art computational methods for linear acoustics are reviewed. The equations of linear acoustics are summarized and then transformed to the frequency domain for time‐harmonic waves governed by the Helmholtz equation. Three major current challenges in the field are specifically addressed: numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales and requiring a large computational effort; the effective treatment of unbounded domains by domain‐based methods; and parallel iterative methods for large complex, ill‐conditioned and possibly indefinite equation systems for high‐frequency problems. A priori error estimates, including both dispersion (phase error) and global pollution effects for moderate to large wave numbers are discussed. Stabilized and other wave‐based discretization methods are reviewed. Domain‐based methods for modeling exterior domains are described including Dirichlet‐to‐Neumann (DtN) methods, absorbing boundary conditions, infinite elements, and the perfectly matched layer (PML). Efficient equation‐solving methods for the resulting complex‐symmetric (non‐Hermitian) matrix systems are discussed including parallel iterative methods and domain decomposition methods. Numerical methods for direct solution of the acoustic wave equation in the time domain are also reviewed.

This paper presents two partition methods that speed up iterative search methods applied to vehicle routing problems including a large number of vehicles. Indeed, using a simple implementation of taboo search as an iterative search method, every best‐known solution to classical problems was found. The first partition method (based on a partition into polar regions) is appropriate for Euclidean problems whose cities are regularly distributed around a central depot. The second partition method is suitable for any problem and is based on the arborescence built from the shortest paths from any city to the depot. Finally, solutions that are believed to be optimum are given for problems generated on a grid. © 1993 by John Wiley & Sons, Inc.

The article is devoted to the analysis of methods for solving systems of multivalued logical equations by iteration methods. Iterative methods for solving such systems of equations are a mathematical description of the main process of functional-logical simulation, which is used at the stage of designing digital systems for objects control to verify the correctness of the design. Consideration of multi-valued values of logical signals at the outputs of blocks and elements of digital systems is explained by the fact that in some cases, to analyze the correctness of time relationships when simulating the hardware of digital systems, a several valued representation of logical signals is used, as well as that recently, logical elements are being developed that implement four or more valued logic. Based on the analysis of the structure of the system of logical equations used in digital hardware simulation, using graph and logical models, an analysis is made of the existence of solutions and their number. Iterative methods of a simple and generalized iteration are analyzed, a relationship is shown between the number of solutions of the system of equations and its graph representation, which reflects a given circuit of connecting elements of the hardware of a digital control system. For the generalized iteration method, options with a different structure of the iteration trace are considered, in particular, it is shown that, with a certain structure of the iteration trace, the generalized iteration turns into a simple iteration or Seidel iteration. It is shown that the generalized iteration most adequately describes the process of simulating the switching of logical signals in a simulated circuit of digital control systems hardware. The correspondence between various options of functional-logical simulation of digital systems and the used methods of iterative solution of systems of logical equations is shown.

A mixed elastohydrodrynamic (EHL) model for journal bearings considering an axial flow due to shaft axial motion and misalignment is developed for lubrication performance evaluation. A new, faster mixed EHL computing technology utilizing the odd–even successive overrelaxation (OESOR) parallel numerical iterative method is proposed based on the red–black successive overrelaxation (RBSOR) method to minimize the execution time prolonged by the complexity caused by the axial flow and misalignments. The multithreaded computing scheme conducted by the OpenMP directive using different meshes and threads suggests that the OESOR method exhibits better efficiency. A series of transient analyses was conducted to solve the mixed EHL model with the parallel OESOR method. The results show that the axial flow and misalignments significantly affect the average pressure, hydrodynamic and asperity contact pressure, elastic deformation, and other characteristics of the journal bearings.

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

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.

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.

We consider an ill-posed equation in a Hilbert space with a noisy operator and a noisy right-hand side. The noise level information is given in a general form, as a norm of a certain operator applied to the noise. We derive the monotone error rule (ME-rule) for the choice of the regularization parameter in many methods, giving parameter such that the error is monotonically increasing for larger parameters in the Tikhonov method and for smaller stopping indices in iteration methods. Regularization methods considered include -scale regularization in (iterated) Tikhonov method and in iteration methods (Landweber method, CG type methods, semi-iterative methods). We also consider modifications of the ME-rule and show in numerical experiments (test problems from Hansen’s Regularization Toolbox, including the sideways heat equation) their advantages over the discrepancy principle.

In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

Assessment of Tikhonov-type regularization methods for solving atmospheric inverse problems