Integral Equation Problems Research Articles

Fast Integral Equation Solution by Multilevel Green's Function Interpolation Combined With Multilevel Fast Multipole Method

A fast wideband integral equation (IE) solver combining the multilevel interpolatory fast Fourier transform accelerated approach (MLIPFFT) with the multilevel fast multipole method (MLFMM) is discussed. On electrically fine levels within an oct-tree multilevel structure, coupling computations are performed by MLIPFFT. This method is based on a 3D Lagrange factorization of the pertinent Green's functions with a smooth approximation error in space and it does not suffer a low frequency breakdown as known from MLFMM. For high frequency integral equation problems, MLIPFFT has decreased computational efficiency as the Nyquist theorem requires increasing numbers of samples in 3 dimensions. Due to a transition from the interpolation point based MLIPFFT source/receive formulation towards an appropriate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -space representation at a certain level within the oct-tree, the high frequency efficient MLFMM can be employed for coarse levels. The hybrid algorithm is hence well suited for fast wideband integral equation solutions. Both, mixed-potential and direct-field formulations are considered. Furthermore, a method for MLIPFFT extrapolation error reduction based on fine level interpolation domain spreading is introduced. In several numerical examples, the performance of the proposed algorithm is demonstrated.

Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity

AbstractBased on potential theory, Steklov eigensolutions of elastic problems can be converted into eigenvalue problems of boundary integral equations (BIEs). The kernels of these BIEs are characterized by logarithmic and Hilbert singularities. In this article, the Nyström methods are presented for obtaining eigensolutions (λ(i),u(i)), which have to deal with the two kinds of singularities simultaneously. The solutions possess high accuracy orders O(h3) and an asymptotic error expansion with odd powers. Using h3 ‐Richardson extrapolation algorithms, we can greatly improve the accuracy orders to O(h5). Furthermore, a generalized Fourier series is constructed by the eigensolutions, and then solving the elasticity displacement and traction problems involves just calculating the coefficients of the series. A class of elasticity problems with boundary Γ is solved with high convergence rate O(h5). The efficiency is illustrated by a numerical example. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

Integral Equation Method for Free Vibration of Annular Plates with Elastic Supports

Annular plates are commonly found in the fields of engineering. The present study is concerned with the integral equation method for the free vibrations of annular plates with elastic supports. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first and the second kind is used to construct the Green's function of annular plates. The eigenvalue problem of free vibration of annular plates with Elastic Supports is transformed into the eigenvalue problem of integral equation. And then, the problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical example shows the significant advantages of the present method.

Biomolecular electrostatics using a fast multipole BEM on up to 512 gpus and a billion unknowns

We present teraflop-scale calculations of biomolecular electrostatics enabled by the combination of algorithmic and hardware acceleration. The algorithmic acceleration is achieved with the fast multipole method ( fmm) in conjunction with a boundary element method ( bem) formulation of the continuum electrostatic model, as well as the bibee approximation to bem. The hardware acceleration is achieved through graphics processors, gpus. We demonstrate the power of our algorithms and software for the calculation of the electrostatic interactions between biological molecules in solution. The applications demonstrated include the electrostatics of protein–drug binding and several multi-million atom systems consisting of hundreds to thousands of copies of lysozyme molecules. The parallel scalability of the software was studied in a cluster at the Nagasaki Advanced Computing Center, using 128 nodes, each with 4 gpus. Delicate tuning has resulted in strong scaling with parallel efficiency of 0.8 for 256 and 0.5 for 512 gpus. The largest application run, with over 20 million atoms and one billion unknowns, required only one minute on 512 gpus. We are currently adapting our bem software to solve the linearized Poisson–Boltzmann equation for dilute ionic solutions, and it is also designed to be flexible enough to be extended for a variety of integral equation problems, ranging from Poisson problems to Helmholtz problems in electromagnetics and acoustics to high Reynolds number flow.

Higher order singular integral equations on complex hypersphere

A theory of a class of higher order singular integral under the operator ( L f ) ( u ) = 1 u ¯ 1 [ u 1 ∂ f ∂ u 1 ( u ) - u ¯ 1 ∂ f ∂ u ¯ 1 ( u ) + f ( u ) ] is given. We transform the higher order singular integral to a usual Cauchy integral, extend the permutation formula of the higher order singular integral deduced by Qian and Zhong in [4] to a general case, and discuss the regularization problem of the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere.

Iterative Near-Field Preconditioner for the Multilevel Fast Multipole Algorithm

For iterative solutions of large and difficult integral-equation problems in computational electromagnetics using the multilevel fast multipole algorithm (MLFMA), preconditioners are usually built from the available sparse near-field matrix. The exact solution of the near-field system for the preconditioning operation is infeasible because the LU factors lose their sparsity during the factorization. To prevent this, incomplete factors or approximate inverses can be generated so that the sparsity is preserved, but at the expense of losing some information stored in the near-field matrix. As an alternative strategy, the entire near-field matrix can be used in an iterative solver for preconditioning purposes. This can be accomplished with low cost and complexity since Krylov subspace solvers merely require matrix-vector multiplications and the near-field matrix is sparse. Therefore, the preconditioning solution can be obtained by another iterative process, nested in the outer solver, provided that the outer Krylov subspace solver is flexible. With this strategy, we propose using the iterative solution of the near-field system as a preconditioner for the original system, which is also solved iteratively. Furthermore, we use a fixed preconditioner obtained from the near-field matrix as a preconditioner to the inner iterative solver. MLFMA solutions of several model problems establish the effectiveness of the proposed nested iterative near-field preconditioner, allowing us to report the efficient solution of electric-field and combined-field integral-equation problems involving difficult geometries and millions of unknowns.

Application of Periodized Harmonic Wavelets towards Solution of Eigenvalue Problems for Integral Equations

This article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated.

Functions with prescribed best linear approximations

A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames.

Cascadic multilevel methods for ill-posed problems

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error. This paper shows that cascadic multilevel methods with a conjugate gradient-type method as basic iterative scheme are regularization methods. The iterations are terminated by a stopping rule based on the discrepancy principle.

Incremental single shooting—A robust method for the estimation of parameters in dynamical systems

Estimating the parameters of a dynamical system based on measurement data is an important but complex task in industrial and scientific practice [Schittkowski, K. (2002). Numerical data fitting in dynamical systems: A practical introduction with applications and software. Kluwer Academic Publishers]. Due to its importance, different approaches to solve this kind of problem have been developed. The most established ones are single shooting [Bard, Y. (1974). Nonlinear parameter estimation. New York: Academic Press], multiple shooting [Bock, H. G. (1983). Recent advances in parameter identification techniques for ODE. In P. Deuflhard, & E. Hairer (Eds.), Numerical treatment of inverse problems in differential and integral equations (pp. 95–121). Boston: Birkhäuser] and full discretization techniques [Biegler, L. T. (1984). Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Computers & Chemical Engineering, 8, 243–248]. Single shooting is the most natural and simple approach to the problem, directly combining numerical integration and optimization techniques. However, for unstable or singular systems the numerical integration of the dynamic model may fail. This problem is especially severe in the framework of parameter estimation, since the dynamic model has to be integrated many times with different parameter estimates. Multiple shooting and full discretization aim at overcoming these deficiencies but suffer from other drawbacks. Therefore single shooting is still widely applied in industry and academia. In this work we present a novel method called Incremental Single Shooting or ISS for short, which aims at overcoming the deficiencies of the classical single shooting approach while keeping its advantages.

On estimation algorithms for ordinary differential equations

This article addresses the problem of estimating the parameters of a system of ordinary differential equations given data derived from noisy observations on the state variables. This problem is important in a range of applications in areas such as adaptive, real time control. There are two main classes of method for attacking this problem, and their equivalence and effectiveness (consistency) are discussed. Recent rate of convergence results for the major implementation techniques are summarized, and some matters requiring further consideration indicated. References H. G. Bock. Recent advances in parameter identification techniques in {ODE}. In P. Deuflhard and E. Hairer, editors, Numerical Treatment of Inverse Problems in Differential and Integral Equations , pages 95--121. Birkhauser, 1983. J. Nocedal and S. J. Wright. Numerical Optimization . Springer--Verlag, 1999. M. R. Osborne. Numerical questions in ODE boundary value problems. In Wayne Read, Jay W. Larson, and A. J. Roberts, editors, Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006 , volume 48 of ANZIAM J. , pages C899--C926, February 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/79 [February 11, 2008]. M. R. Osborne. Fisher's method of scoring. Int. Stat. Rev. , 86:271--286, 1992. M. R. Osborne. The bock iteration for the ode estimation problem. 2008. in preparation. I. Tjoa and L. T. Biegler. Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic systems. Ind. Eng. Chem. Res. , 30:376--385, 1991.

Renormalization-group symmetries for solutions of nonlinear boundary value problems

About 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov renormalization group treated as a Lie group of continuous transformations. Overwhelmingly dominating practical quantum field theory calculations, the renormalization-group method formed the basis for the discovery of the asymptotic freedom of strong nuclear interactions and underlies the Grand Unification scenario. This paper draws on lectures delivered at the XIII School for Nonlinear Waves, Nizhnii Novgorod, Russia, 1–7 March 2006 [see V F Kovalev, D V Shirkov "Renormalization group symmetry for solutions of boundary value problems" in Nonlinear Waves 2006 (Ed. by A V Gaponov-Grekhov) (N. Novgorod: IAP RAS, 2007) p. 433] to describe the logical framework of a new algorithm based on the modern theory of transformation groups and to present the most interesting results of application of the method to differential and/or integral equation problems and to problems that involve linear functionals of solutions. Examples from nonlinear optics, kinetic theory, and plasma dynamics are given, where new analytic solutions obtained with this algorithm have allowed describing the singularity structure for self-focusing of a laser beam in a nonlinear medium, studying generation of harmonics in weakly inhomogeneous plasma, and investigating the energy spectra of accelerated ions in expanding plasma bunches.

Efficient Parallelization of the Multilevel Fast Multipole Algorithm for the Solution of Large-Scale Scattering Problems

We present fast and accurate solutions of large-scale scattering problems involving three-dimensional closed conductors with arbitrary shapes using the multilevel fast multipole algorithm (MLFMA). With an efficient parallelization of MLFMA, scattering problems that are discretized with tens of millions of unknowns are easily solved on a cluster of computers. We extensively investigate the parallelization of MLFMA, identify the bottlenecks, and provide remedial procedures to improve the efficiency of the implementations. The accuracy of the solutions is demonstrated on a scattering problem involving a sphere of radius discretized with 41 883 638 unknowns, the largest integral-equation problem solved to date. In addition to canonical problems, we also present the solution of real-life problems involving complicated targets with large dimensions.

Regularization for Higher Order Singular Integral Equations

By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations.

Boundary element method for spatially‐periodic potential problems

AbstractIn this paper, we propose a boundary element method for two‐dimensional potential problems with one‐dimensional spatial periodicity, which have been difficult to be solved by the ordinary boundary element method. In the presented method, we reduce the potential problems with Dirichlet and Neumann boundary conditions to integral equation problems with the periodic fundamental solution of the Laplace operator and, then, obtain approximate solutions by solving linear systems given by discretizing the integral equations. Numerical examples are also included. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Gedanken Experiments to Understand the Internal Resonance Problems of Electromagnetic Scattering

The internal resonance problem of integral equations is a problem that has plagued solutions of scalar and vector integral equations. A lucid way to explain the occurrence of the internal resonance problem for integral equations of scattering is given in this paper. The physical insight is gotten by performing Gedanken experiments for MFIE for PEC, and for penetrable scatterers when EFIE and MFIE alone are applied. These proofs provide with physical clarity the reasons of the internal resonance problem, as well as the physical character of the internal resonance problem. The connection of the uniqueness of CFIE and the cavity resonance problem is reported.

Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns

The solution of extremely large scattering problems that are formulated by integral equations and discretised with tens of millions of unknowns is reported. Accurate and efficient solutions are performed by employing a parallel implementation of the multilevel fast multipole algorithm. The effectiveness of the implementation is demonstrated on a sphere problem containing more than 33 million unknowns, which is the largest integral-equation problem ever solved to our knowledge.

A class of concave operators with applications

In this paper we provide some properties of a class of concave operators and apply these results in discussing three-point boundary value problems for differential equations and nonlinear Volterra integral equations.

Linearization method for solving nonlinear integral equations

The objective of this paper is to assess both the applicability and the accuracy of linearization method in several problems of general nonlinear integral equations. This method provides piecewise linear integral equations which can be easily integrated. It is shown that the accuracy of linearization method can be substantially improved by employing variable steps which adjust themselves to the solution. This approach can reveal that, under this method, the nonlinear integral equations can be transformed into the linear integral equations which may be integrated using classical methods. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.