Numerical Methods for Grid Equations

The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE-equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of O(h −2 ) algebraic complex-

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“…The theorem follows from Theorem 2 and standard results on preconditioned conjugate gradient algorithms, see [13,37,38] for more details.…”

Section: Iterative Solver and Its Parallelizationmentioning

confidence: 87%

Fast parallel solvers for symmetric boundary element domain decomposition equations

Carstensen

Kuhn

Langer

1998

Numerische Mathematik

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“…Trigono metric and Fourier polynomials [5] are little used for interpolation of non equidistant series because of identi cal problems of instability of sampling.…”

Section: Methods Of Interpolation and Approximation Of Non Equi Distamentioning

confidence: 99%

“…Mean square error of CRV spectral parameters (MSESP) is calculated as: (5) where N is number of integral parameters (Table 1, N = 4).…”

Section: Rhythmocardiogram Approximation Methods For Calculation Of Smentioning

confidence: 99%

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Rhythmocardiogram Approximation Methods for Calculation of Spectral Parameters of Cardiac Rhythm Variability

Kukushkin¹,

Maistrov

Bogomolov³

2010

Biomed Eng

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IntroductionDiagnosis of human functional systems can be based on the cardiac rhythm variability (CRV) spectrum [1 4]. The CRV spectrum is calculated from spectral function of power (SFP). SFP can be correctly estimated only for signals with uniform temporal quantization.In the human body RCG is determined heteroge neously in time: the abscissa is RR interval time (hetero geneous) and the ordinate is RR interval time. RCG spectral analysis (SFP values in different frequency ranges) requires RCG sampling and equidistant signal evaluation. Most such methods are based on equidistant signal evaluation and subsequent uniform temporal sam pling.RCG can be approximated using interpolation methods. In contrast to other approximation methods, interpolation methods require approximation of all experimental points. RR intervals are often approximat ed excluding artifacts. As a result, RCG contains gaps or excess intervals, which are excluded using interpolation methods.The literature emphasizes the importance of sam pling methods [1,6,13,16,17]. CRV accuracy depends on these methods. There is no standard method of RCG sampling. This standard should provide RCG approxima tion and CRV spectral analysis. The goal of this work was to implement rhythmocardiogram approximation meth ods for calculation of spectral parameters of CRV. RCG Signal Approximation Quality SubstantiationExisting methods of CRV spectral analysis have revealed that approximation and equidistant sampling provides preliminary SFP function evaluation and RCG analysis. RCG signal approximation should provide coin cidence between SFP and RCG signal. RCG signal approximation methods should also meet the stability Rhythmocardiogram (RCG) approximation methods have been considered as non equidistant time series for cal culation of spectral parameters of cardiac rhythm variability (CRV). Model RCG signals provided a basis for a transition function of an interpolator. Experimental and theoretically computed transition functions of the interpolator were comparatively analyzed. Stability and accuracy parameters of CRV are discussed for model signals and actual experimental data. Correction of the transition function of the interpolator was implemented using lin ear and spline interpolation. Its efficiency was tested using model signals and actual experimental signals. lower, f 0k , Hz 0.15 0.04 0.015 0.003 upper, f mk , Hz min(0.4; HR/2) 0.15 0.04 HR/2 TABLE 1. Spectral CRV parameters Boundary frequency Spectral parameter SFP of RCG signal in high frequency (HF) range, msec 2 /Hz SFP of RCG signal in low frequency (LF) range, msec 2 /Hz SFP of RCG signal in very low frequency (VLF) range, msec 2 /Hz Total SFP of RCG signal (TR), msec 2 /Hz

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“…The solution of problem (2.14) is well known (cf., e.g., [3]). If we denote the Chebyshev polynomial of the second kind by U i (t)…”

Section: The Two-frequency Decompositionmentioning

confidence: 99%

Two-frequency decomposition

Buzdin

Wittum

2004

Numerische Mathematik

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In this paper, we present a preconditioner for large systems of linear equations based on the block decomposition for block-tridiagonal matrices. This decomposition is in many respects similar to the frequency-filtering method of Wittum [8] and also to the frequency-filtering decomposition of Wagner [4]- [6]. In contrast to these methods, our approach requires no pointwise filtering conditions but, as in [1], only averaged ones; this simplifies the implementation without any loss of efficiency. Theoretical analysis of the model problem leads to the convergence rate 1 − O(h 1 3 ). Numerical experiments demonstrate similar convergence behaviour for a wider class of problems. (2000): 65F10, 65N22 Mathematics Subject Classification

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Numerical Methods for Grid Equations

Cited by 104 publications

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Fast parallel solvers for symmetric boundary element domain decomposition equations

Fast parallel solvers for symmetric boundary element domain decomposition equations

Rhythmocardiogram Approximation Methods for Calculation of Spectral Parameters of Cardiac Rhythm Variability

Two-frequency decomposition

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