Chapter 6: Finite-Dimensional Hilbert Spaces and the Matrix Formulation of the Conjugate Gradient Method

The conjugate gradient method using vectorial descent parameters is used to solve the inverse problem of simultaneously estimating one boundary condition and one thermo-physical property of an heterogeneous material. The advantage of the conjugate gradient method lies in that no a priori information is needed on the variation of the unknown quantities. The heterogeneous material model under consideration consists of a matrix in imperfect contact with embedded separated particles. The first unknown of interest in this inverse problem is the constant coupling parameter Ψ which characterizes the thermal contact (conductance) behavior between the matrix and the particles. The second unknown is the time dependent applied heat flux on one external side of the matrix. As a result, this problem is concerned with a combined parameter and function estimation at the same time. Thus two descent parameters, each one corresponding to each unknown function parameter, are derived and used in the iterative process. The developed inverse analysis is based on the transient temperature measurements taken from some sensors implanted inside the matrix only during the process of heating. Several numerical test cases were performed and show that the developed method provides an accurate estimation of thermo-physical properties and boundary condition in a very short practical time.

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An adaptive Hager-Zhang conjugate gradient method

Filomat ◽

10.2298/fil1614715b ◽

2016 ◽

Vol 30 (14) ◽

pp. 3715-3723

Author(s):

Saman Babaie-Kafaki ◽

Reza Ghanbari

Keyword(s):

Conjugate Gradient Method ◽

Condition Number ◽

Conjugate Gradient ◽

Gradient Method ◽

Test Problems ◽

Objective Functions ◽

Lower And Upper Bounds ◽

Comparative Testing ◽

The Matrix ◽

Adaptive Switch

Based on a singular value study, lower and upper bounds for the condition number of the matrix which generates search directions of the Hager-Zhang conjugate gradient method are obtained. Then, based on the insight gained by our analysis, a modified version of the Hager-Zhang method is proposed, using an adaptive switch form the Hager-Zhang method to the Hestenes-Stiefel method when the mentioned condition number is large. A brief global convergence analysis is made for the uniformly convex objective functions. Numerical comparisons between the implementations of the proposed method and the Hager-Zhang method are made on a set of unconstrained optimization test problems of the CUTEr collection, using the performance profile introduced by Dolan and Mor?. Comparative testing results are reported.

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Chapter 5: Conjugate Gradient Method in Hilbert Spaces

Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs ◽

10.1137/1.9781611973846.ch5 ◽

2014 ◽

pp. 35-46

Keyword(s):

Conjugate Gradient Method ◽

Hilbert Spaces ◽

Conjugate Gradient ◽

Gradient Method

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Tomographic inversion via the conjugate gradient method

Geophysics ◽

10.1190/1.1442293 ◽

1987 ◽

Vol 52 (2) ◽

pp. 179-185 ◽

Cited By ~ 127

Author(s):

John A. Scales

Keyword(s):

Conjugate Gradient Method ◽

Conjugate Gradient ◽

Gradient Method ◽

Sparse Matrices ◽

Conjugate Gradient Algorithm ◽

Gradient Algorithm ◽

Algebraic Equations ◽

Tomographic Inversion ◽

Linear Algebraic Equations ◽

The Matrix

Tomographic inversion of seismic traveltime residuals is now an established and widely used technique for imaging the Earth’s interior. This inversion procedure results in large, but sparse, rectangular systems of linear algebraic equations; in practice there may be tens or even hundreds of thousands of simultaneous equations. This paper applies the classic conjugate gradient algorithm of Hestenes and Stiefel to the least‐squares solution of large, sparse systems of traveltime equations. The conjugate gradient method is fast, accurate, and easily adapted to take advantage of the sparsity of the matrix. The techniques necessary for manipulating sparse matrices are outlined in the Appendix. In addition, the results of the conjugate gradient algorithm are compared to results from two of the more widely used tomographic inversion algorithms.

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A restart scheme for the Dai–Liao conjugate gradient method by ignoring a direction of maximum magnification by the search direction matrix

RAIRO - Operations Research ◽

10.1051/ro/2019045 ◽

2020 ◽

Vol 54 (4) ◽

pp. 981-991

Author(s):

Zohre Aminifard ◽

Saman Babaie-Kafaki

Keyword(s):

Conjugate Gradient Method ◽

Conjugate Gradient ◽

Gradient Method ◽

Numerical Stability ◽

Gradient Methods ◽

Singular Value ◽

Search Direction ◽

Value Analysis ◽

The Matrix ◽

Computational Errors

As known, finding an effective restart procedure for the conjugate gradient methods has been considered as an open problem. Here, we aim to study the problem for the Dai–Liao conjugate gradient method. In this context, based on a singular value analysis conducted on the Dai–Liao search direction matrix, it is shown that when the gradient approximately lies in the direction of the maximum magnification by the matrix, the method may get into some computational errors as well as it may converge hardly. In such situation, ignoring the Dai–Liao search direction in the sense of performing a restart may enhance the numerical stability as well as may accelerate the convergence. Numerical results are reported; they demonstrate effectiveness of the suggested restart procedure in the sense of the Dolan–Moré performance profile.

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