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A unified approach to identifying an unknown spacewise dependent source in a variable coefficient parabolic equation from final and integral overdeterminations

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APPLIED NUMERICAL MATHEMATICS 2014年 78卷 49-67页

作者： Hasanov, Alemdar Pektas, Burhan Izmir Univ Dept Math & Comp Sci TR-35350 Izmir Turkey

An adjoint problem approach with subsequent conjugate gradient algorithm (CGA) for a class of problems of identification of an unknown spacewise dependent source in a variable coefficient parabolic equation u(t) = (k(x)u(x))(x) + F(x)H(t), (x, t) is an element of(0, l) x (0, T] is proposed. The cases of final time and time-average, i.e. integral type, temperature observations are considered. We use well-known Tikhonov regularization method and show that the adjoint problems, corresponding to inverse problems ISPF1 and ISPF2 can uniquely be derived by the Lagrange multiplier method. This result allows us to obtain representation formula for the unique solutions of each regularized inverse problem. Using standard Fourier analysis, we show that series solutions for the case in which the governing parabolic equation has constant coefficient, coincide with the Picard's singular value decomposition. It is shown that use of these series solutions in CGA as an initial guess substantially reduces the number of iterations. A comparative numerical analysis between the proposed version of CGA and the Fourier method is performed using typical classes of sources, including oscillating and discontinuous functions. Numerical experiments for variable coefficient parabolic equation with different smoothness properties show the effectiveness of the proposed version of CGA. (C) 2013 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Inverse source problem Parabolic equation Final and time-average temperature overdeterminations Integral representation formula Filter function conjugate gradient algorithm

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Krylov subspace algorithms and circulant-embedding method for efficient wideband single-carrier equalization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2008年第6期56卷 2483-2495页

作者： Guan, Raymond Strohmer, Thomas Univ Calif Davis Dept Math Davis CA 95616 USA

Wider bandwidth allows higher data rate by transmitting narrower pulses. However, doing so would also increase the discrete channel memory length. For single-carrier communication systems this results in higher computational burden at the receiver. We are concerned with single-carrier nonblock transmission schemes with receiver oversampling, as they can provide higher spectral efficiency than block transmission schemes in the presence of large delay spreads. We first propose a simple finite-impulse-response (FIR) equalizer that is based on the circulant-embedding (CE) method and analyze its performance by investigating the relationship between solutions of various finite-dimensional models and the original infinite-dimensional problem. We show that under proper conditions the CE FIR equalizer converges exponentially fast to the IIR equalizer. We then focus on the conjugate gradient (CG) algorithm as an efficient means for equalization that is specifically well suited for dealing with large-delay-spread channels. We discuss the importance of stopping the iterations for the CG algorithm at the right time in the presence of noise and present several reliable low-cost stopping criteria. It turns out that the CG algorithm equipped with appropriate stopping criteria can outperform MMSE equalizers. Since both the CE and the CG methods can be efficiently implemented via fast Fourier transforms, equalization complexity is only in the order of N log (N) for N data symbols. Several numerical experiments demonstrate the performance of the proposed methods.

关键词： adaptive signal processing conjugate gradient algorithm FIR equalizer single-carrier equalization

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Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method

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IMA JOURNAL OF NUMERICAL ANALYSIS 2017年第4期37卷 1961-1985页

作者： Smears, Iain INRIA Paris 2 Rue Simone Iff F-75012 Paris France

The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. Drawing on parabolic inf-sup theory, we first construct a left preconditioner that transforms the linear system to a symmetric positive definite problem to be solved by the preconditioned conjugate gradient algorithm. We then prove that the transformed system can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number bounded by 4 for any time-step size, any approximation order and any positive-definite self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate preconditioners, and show the feasibility of the high-order solution of large problems.

关键词： discontinuous Galerkin time discretizations parabolic PDE preconditioning conjugate gradient algorithm

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Love-wave waveform inversion in time domain for shallow shear-wave velocity

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GEOPHYSICS 2016年第1期81卷 R1-R14页

作者： Pan, Yudi Xia, Jianghai Xu, Yixian Gao, Lingli Xu, Zongbo China Univ Geosci Inst Geophys & Geomat Subsurface Imaging & Sensing Lab Wuhan 430074 Hubei Peoples R China China Univ Geosci Inst Geophys & Geomat Hubei Subsurface Multiscale Imaging Key Lab Wuhan 430074 Hubei Peoples R China

High-frequency surface-wave techniques are widely used to estimate S-wave velocity of near-surface materials. Surface-wave methods based on inversions of dispersion curves are only suitable to laterally homogeneous or smoothly laterally varying heterogeneous earth models due to the layered-model assumption during calculation of dispersion curves. Waveform inversion directly fits the waveform of observed data, and it can be applied to any kinds of earth models. We have used the Love-wave waveform inversion in the time domain to estimate near-surface S-wave velocity. We used the finite-difference method as the forward modeling method. The source effect was removed by the deconvolution technique, which made our method independent of the source wavelet. We defined the difference between the deconvolved observed and calculated waveform as the misfit function. We divided the model into different sizes of blocks depending on the resolution of the Love waves, and we updated the S-wave velocity of each block via a conjugate gradient algorithm. We used two synthetic models to test the effectiveness of our method. A real-world case verified the validity of our method.

关键词： Love wave waveform inversion shear-wave velocity conjugate gradient algorithm

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A comparative study on low-memory iterative solvers for FFT-based homogenization of periodic media

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JOURNAL OF COMPUTATIONAL PHYSICS 2016年 321卷 151-168页

作者： Mishra, Nachiketa Vondrejc, Jaroslav Zeman, Jan Czech Tech Univ Fac Civil Engn Dept Mech Thakurova 7 Prague 16629 6 Czech Republic Tech Univ Carolo Wilhelmina Braunschweig Inst Comp Sci Muhlenpfordtstr 23 D-38106 Braunschweig Germany

In this paper, we assess the performance of four iterative algorithms for solving non-symmetric rank-deficient linear systems arising in the FFT-based homogenization of heterogeneous materials defined by digital images. Our framework is based on the Fourier-Galerkin method with exact and approximate integrations that has recently been shown to generalize the Lippmann-Schwinger setting of the original work by Moulinec and Suquet from 1994. It follows from this variational format that the ensuing system of linear equations can be solved by general-purpose iterative algorithms for symmetric positive-definite systems, such as the Richardson, the conjugate gradient, and the Chebyshev algorithms, that are compared here to the Eyre-Milton scheme - the most efficient specialized method currently available. Our numerical experiments, carried out for two-dimensional elliptic problems, reveal that the conjugate gradient algorithm is the most efficient option, while the Eyre-Milton method performs comparably to the Chebyshev semi-iteration. The Richardson algorithm, equivalent to the still widely used original Moulinec-Suquet solver, exhibits the slowest convergence. Besides this, we hope that our study highlights the potential of the well-established techniques of numerical linear algebra to further increase the efficiency of FFT-based homogenization methods. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Numerical homogenization Fourier-Galerkin method Fast Fourier Transform Guaranteed bounds Richardson iteration conjugate gradient algorithm Chebyshev semi-iterative method Eyre-Milton scheme

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A MULTILOOP ROBUST CONTROLLER-DESIGN STUDY USING SINGULAR VALUE gradientS

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 1985年第4期8卷 514-519页

作者： NEWSOM, JR MUKHOPADHYAY, V GEORGE WASHINGTON UNIV WASHINGTONDC 20052 JOINT INST ADV FLIGHT SCI HAMPTONVA

A method for designing robust feedback controllers for multiloop systems is presented. Robustness is characterized in terms of the minimum singular value of the system return difference matrix at the plant input. Analytical gradients of the singular values with respect to design variables in the controller are derived. A cumulative measure of the singular values and their gradients with respect to the design variables are used with a numerical optimization technique to increase the system's robustness. Both unconstrained and constrained optimization techniques are evaluated. Numerical results are presented for a two-input/two-output drone flight control system. © 1985 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.

关键词： Attitude Control System Single Input Single Output System Closed Loop System Elevon Linear Quadratic Gaussian Numerical Optimization Frequency Domain Computing Sideslip Angle conjugate gradient algorithm

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algorithmS FOR THE COMPUTATION OF WEIGHTS IN BOUNDED INFLUENCE REGRESSION

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 1988年第3期6卷 251-276页

作者： MARAZZI, A UNIV LAUSANNE INST MED SOCIALE & PREVENT DIV STAT & INFORMAT CH-1005 LAUSANNE SWITZERLAND

A preliminary step for the computation of bounded influence regression is the determination of weights for the points x i in the factor space. This can be achieved through the computation of a robust affinely invariant covariance matrix (with fixed location) and the computation of robust distances of the x i to the origin, in the metric defined by the covariance matrix. For a well known fixed point algorithm convergence has been proved. However, its speed is not very satisfactory in practice. Newton-like and conjugate gradient approaches have been suggested by P.J. Huber. This paper reviews the various proposals, discusses their implementations, some of their convergence properties and compares empirically their performances.

关键词： Robust regression Bounded influence regression Robust covariances Fixed point algorithm Newton algorithm conjugate gradient algorithm

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Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient

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ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK 2005年第3期85卷 155-162页

作者： Bouby, C Fortuné, D Pietraszkiewicz, W Vallée, C Polish Acad Sci Inst Fluid Flow Mach PL-80952 Gdansk Poland Univ Lille 1 LML F-59655 Villeneuve Dascq France LMS F-86962 Futuroscope France

We develop a new effective method of determining the rotation R and the stretches U and V in the polar decomposition F RU = VR of the deformation gradient. The method is based on a minimum property of R to have the smallest "distance" from F in the Euclidean norm. The proposed method does not require to perform any square root and/or inverse operations. With each F having nine independent components we associate a 4 x 4 symmetric traceless matrix Q. The rotation is described by quaternion parameters from which a quadrivector X is formed. It is shown that X corresponding to R maximizes X(T)QX over all X satisfying (XX)-X-T = 1. We prefer to equivalently maximize the Rayleigh quotient Y(T)QY/(YY)-Y-T over all non-vanishing Y and to deduce X by subsequent normalization X = Y/root(YY)-Y-T. The maximization of the Rayleigh quotient is performed by a conjugate gradient algorithm with all iterative steps carried out by explicit closed formulae. Efficiency and accuracy of the method is illustrated by a numerical example.

关键词： polar decomposition rotation quaternion Rayleigh quotient conjugate gradient algorithm continuum mechanics

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Time-domain hyperbolic Radon transform for separation of P-P and P-SV wavefields

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STUDIA GEOPHYSICA ET GEODAETICA 2016年第1期60卷 91-111页

作者： Jiang, Xiaoxue Zheng, Fan Jia, Haiqing Lin, Jun Yang, Hongyuan Jilin Univ Coll Instrumentat & Elect Engn Changchun 130026 Peoples R China

A time-domain hyperbolic Radon transform based method for separating multicomponent seismic data into P-P and P-SV wavefields is presented. This wavefield separation method isolates P-P and P-SV wavefields in the Radon panel due to their differences in slowness, and an inverse transform of only part of the data leads to separated wavefields. A problem of hyperbolic Radon transform is that it works in the time domain entailing the inversion of large operators which is prohibitively time-consuming. By applying the conjugate gradient algorithm during the inversion of hyperbolic Radon transform, the computational cost can be kept reasonably low for practical application. Synthetic data examples prove that P-P and P-SV wavefield separation by hyperbolic Radon transform produces more accurate separated wavefields compared with separation by high-resolution parabolic Radon transform, and the feasibility of the proposed separation scheme is also verified by a real field data example.

关键词： time-domain hyperbolic Radon transform multicomponent data wavefield separation conjugate gradient algorithm

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STATISTICAL-ANALYSIS OF STATIC SHAPE CONTROL IN SPACE STRUCTURES

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AIAA JOURNAL 1990年第8期28卷 1504-1508页

作者： BURDISSO, RA HAFTKA, RT Virginia Polytechnic Institute and State University Blacksburg Virginia

The article addresses the problem of efficient analysis of the statistics of initial and corrected shape distortions in space structures. Two approaches for improving efficiency are considered. One is an adjoint technique for calculating distortion shapes;the second is a modal expansion of distortion shapes in terms of pseudo-vibration modes. The two techniques are applied to the problem of optimizing actuator locations on a 55 m radiometer antenna. The adjoint analysis technique is used with a discrete-variable optimization method. The modal approximation technique is coupled with a standard conjugate-gradient continuous optimization method. The agreement between the two sets of results is good, validating both the approximate analysis and optimally of the results. © 1990 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.

关键词： Large Space Structures Statistical Analysis Actuators Antennas Shape Control Finite Element Analysis Trusses conjugate gradient algorithm Monte Carlo Simulation Structural Response

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