On Variable Metric Methods of Minimization

Pearson, John D.

doi:10.21236/ad0666700

Cited by 48 publications

(20 citation statements)

References 4 publications

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“…Examples of rank-one update methods are Broyden's first and second method [22], the (inverse) column-updating method [125,126], the symmetric rankone update method [39], Pearson's method [146] and McCormick's method [130]. However, Pearson's method and McCormick's method are only applicable to problems where R is symmetric positive definite at the solution of R(x) = 0.…”

Section: Algorithm 51mentioning

confidence: 99%

Partitioned Simulation of Fluid-Structure Interaction

Degroote¹

2013

Arch Computat Methods Eng

123

107

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In this review article, the focus is on partitioned simulation techniques for strongly coupled fluid-structure interaction problems, especially on techniques which use at least one of the solvers as a black box. First, a number of analyses are reviewed to explain why Gauss-Seidel coupling iterations converge slowly or not at all for fluid-structure interaction problems with strong coupling. This provides the theoretical basis for the fast convergence of quasi-Newton and multi-level techniques. Second, several partitioned techniques that couple two black-box solvers are compared with respect to implementation and performance. Furthermore, performance comparisons between partitioned and monolithic techniques are examined. Subsequently, two similar techniques to couple a black-box solver with an accessible solver are analyzed. In addition, several other techniques for fluid-structure interaction simulations are studied and various methods to take into account deforming fluid domains are discussed.

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Section: Algorithm 51mentioning

confidence: 99%

Partitioned Simulation of Fluid-Structure Interaction

Degroote¹

2013

Arch Computat Methods Eng

123

107

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“…where U ranges from .1 to 1 in steps of .1; Rosenbrock's function with the initial estimates suggested by Leon [8], and defined by Wood's function as documented by Pearson [9], and defined by Box's three-parameter exponential problem was also tried, but nonuniqueness of the optimum caused different methods to converge to different optima, invalidating comparisons.…”

Section: Generation Of Approximating Matricesmentioning

confidence: 99%

Conditioning of Quasi-Newton Methods for Function Minimization

Shanno¹

1970

Mathematics of Computation

787

761

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Abstract. Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.

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“…In these methods, the problem (1.1) is assumed to have an associated unknown natural scaling, and the norm on "" used to determine each least-change secant update is itself updated at each iteration to reflect current information about the natural scaling. Examples of rescaled least-change secant update methods are a single-rank update method due to Pearson [23], [7], which is obtained with "", and the Davidon-Fletcher-Powell (DFP) method [8], [17], [7], [12], in which is the subspace of symmetric matrices in ,n.…”

mentioning

confidence: 99%

“…Examples of such methods, in which C(x)=-O, Yk --F(Xk/)--F(Xk) and Wk Sk, are a single-rank update method due to G. McCormick (see Pearson [23]), obtained by taking 4 -"", and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [4], [5], [16], [18], [28], [12], [26], which results when 4 is the subspace of symmetric matrices in "". In both of these methods, the norm on R"" after k iterations is taken to be weighted Frobenius norm I1' []w, where W is a positive-definite, symmetric matrix satisfying Wyk Sk (SO again, g ).…”

mentioning

confidence: 99%

“…After Returning to the nonlinear least-squares example of 2, one sees that if C and y are given as before and if v =F(x/)-F(x)=R'(x/)TR(x+)-R'(x)TR(x), then (4.4) is the DFP analogue of (2.20), i.e., A/ is the DFP augmentation of the Gauss-Newton Hessian R'(x)rR'(x) considered in [13]. Also, (4.4) reduces to (2.20) [23], [7] and the DavidonFletcher-Powell method, but also the nonlinear least-squares methods of [13] employing the update (4.4), when v F(x+)-F(x) and y is given by (3.3).…”

mentioning

confidence: 99%

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Convergence Theorems for Least-Change Secant Update Methods

Dennis¹,

Walker²

1981

SIAM J. Numer. Anal.

125

105

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Abstract. The purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means. many readers might feel that it is easier to understand. The theorems here readily imply local and q-superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of Marwil and Toint and the nonlinear least-squares method of DennisGay-Welsch.

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On Variable Metric Methods of Minimization

Cited by 48 publications

References 4 publications

Partitioned Simulation of Fluid-Structure Interaction

Partitioned Simulation of Fluid-Structure Interaction

Conditioning of Quasi-Newton Methods for Function Minimization

Convergence Theorems for Least-Change Secant Update Methods

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