Extending the inverse sequential quasi-Newton method for on-line monitoring and controlling of process conditions in the solidification of alloys

Tourn, Benjamín A.; Hostos, Juan C. Álvarez; Fachinotti, Víctor D.

doi:10.1016/j.icheatmasstransfer.2023.106647

Cited by 7 publications

(6 citation statements)

References 62 publications

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“…Property 1 follows from the fact that movement to the point c k+1 is carried out along the normal to the hyperplane <z k , c> = yk, that is, along the shortest path (Figure 1). Movement to other points on the hyperplane, for example to point A, satisfy only the condition in (14). □ Let us denote the residual as r k = c k − c*.…”

Section: Matrix Learning Algorithms In Quasi-newton Methodsmentioning

confidence: 99%

“…Thus, the Kaczmarz algorithm provides a solution to the equality in (14) for the last observation, while it implements a local learning strategy, i.e., a strategy for iteratively improving the approximation quality from a functional (15) point of view. If the learning vectors are orthogonal, the solution is found in no more than n iterations.…”

Section: Matrix Learning Algorithms In Quasi-newton Methodsmentioning

confidence: 99%

“…Quasi-Newton minimization methods are effective tools of solving smooth minimization problems when the function level curves have a high degree of elongation [4][5][6][7]. QNMs are commonly applied in a wide range of areas, such as biology [8], image processing [9], technics [10][11][12][13][14][15], and deep learning [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Machine Learning in Quasi-Newton Methods

Krutikov,

Tovbis,

Stanimirović

et al. 2024

Axioms

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In this article, we consider the correction of metric matrices in quasi-Newton methods (QNM) from the perspective of machine learning theory. Based on training information for estimating the matrix of the second derivatives of a function, we formulate a quality functional and minimize it by using gradient machine learning algorithms. We demonstrate that this approach leads us to the well-known ways of updating metric matrices used in QNM. The learning algorithm for finding metric matrices performs minimization along a system of directions, the orthogonality of which determines the convergence rate of the learning process. The degree of learning vectors’ orthogonality can be increased both by choosing a QNM and by using additional orthogonalization methods. It has been shown theoretically that the orthogonality degree of learning vectors in the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is higher than in the Davidon–Fletcher–Powell (DFP) method, which determines the advantage of the BFGS method. In our paper, we discuss some orthogonalization techniques. One of them is to include iterations with orthogonalization or an exact one-dimensional descent. As a result, it is theoretically possible to detect the cumulative effect of reducing the optimization space on quadratic functions. Another way to increase the orthogonality degree of learning vectors at the initial stages of the QNM is a special choice of initial metric matrices. Our computational experiments on problems with a high degree of conditionality have confirmed the stated theoretical assumptions.

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Section: Matrix Learning Algorithms In Quasi-newton Methodsmentioning

confidence: 99%

Section: Matrix Learning Algorithms In Quasi-newton Methodsmentioning

confidence: 99%

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Machine Learning in Quasi-Newton Methods

Krutikov,

Tovbis,

Stanimirović

et al. 2024

Axioms

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show abstract

“…In modeling the spread of infections, QN is useful for the identification of the unknown model coefficients [5]. QN methods are also useful for the modeling of complex crack propagation [6], fluid-structure interaction [7][8][9], melting and solidification of alloys [10], heat transfer systems [11], etc.…”

Section: Introductionmentioning

confidence: 99%

On the Convergence Rate of Quasi-Newton Methods on Strongly Convex Functions with Lipschitz Gradient

Krutikov,

Tovbis,

Stanimirović

et al. 2023

Mathematics

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The main results of the study of the convergence rate of quasi-Newton minimization methods were obtained under the assumption that the method operates in the region of the extremum of the function, where there is a stable quadratic representation of the function. Methods based on the quadratic model of the function in the extremum area show significant advantages over classical gradient methods. When solving a specific problem using the quasi-Newton method, a huge number of iterations occur outside the extremum area, unless there is a stable quadratic approximation of the function. In this paper, we study the convergence rate of quasi-Newton-type methods on strongly convex functions with a Lipschitz gradient, without using local quadratic approximations of a function based on the properties of its Hessian. We proved that quasi-Newton methods converge on strongly convex functions with a Lipschitz gradient with the rate of a geometric progression, while the estimate of the convergence rate improves with the increasing number of iterations, which reflects the fact that the learning (adaptation) effect accumulates as the method operates. Another important fact discovered during the theoretical study is the ability of quasi-Newton methods to eliminate the background that slows down the convergence rate. This elimination is achieved through a certain linear transformation that normalizes the elongation of function level surfaces in different directions. All studies were carried out without any assumptions regarding the matrix of second derivatives of the function being minimized.

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“…Stabilization techniques for inverse problems have two main categories: gradientbased methods and stochastic-based methods. Gradient-based methods include the Levenberg-Marquardt method [12,[14][15][16], the function specification method [17], and the conjugate gradient method [18][19][20]. Stochastic-based methods encompass the Bayesian method [21], the fuzzy inference method [22], and the deep neural network algorithms [23].…”

Section: Introductionmentioning

confidence: 99%

Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques

Zhang,

Zou,

Xiao

2023

Metals

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A two-dimensional transient inverse heat-conduction problem (2DIHCP) was established to determine the mold heat flux using observed temperatures. The sequential regularization method (SRM) was used with zeroth-, first-, and second-order spatial regularization to solve the 2DIHCP. The accuracy of the 2DIHCP was investigated under two strict test conditions (Case 1: heat flux with time-spatial periodically varying, and Case 2: that with sharp variations). The effects of the number of future time steps, regularization parameters, order of regularization, discrete grids, and time step size on the accuracy of the 2DIHCP were analyzed. The results showed that the minimum relative error (epred) of the predicted Case 1 heat flux is 5.05%, 5.39%, and 5.88% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 6.31%, 6.30%, and 6.36%. Notably, zeroth- and first-order spatial regularization had higher accuracy than second-order spatial regularization, while zeroth-order spatial regularization was comparable to first-order. Additionally, first-order spatial regularization was more accurate in reconstructing heat flux containing sharp spatial variations. The CPU time of the predicted Case 2 heat flux is 1.71, 1.71, and 1.70 s for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 1 heat flux are 6.18, 6.15, and 6.17 s. It is noteworthy that the choice of spatial regularization order does not significantly impact the required computing time. Lastly, the minimum epred of Case 2 heat flux with zeroth-order spatial regularization is 7.96%, 6.42%, and 7.87% for time step sizes of 1/fs, 1/2fs, and 1/5fs, respectively. The accuracy of the inverse analysis displays an initial improvement followed by degradation as the time step size decreases. A recommended time step size is 1/2fs, where fs denotes the temperature-sampling rate.

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Extending the inverse sequential quasi-Newton method for on-line monitoring and controlling of process conditions in the solidification of alloys

Cited by 7 publications

References 62 publications

Machine Learning in Quasi-Newton Methods

Machine Learning in Quasi-Newton Methods

On the Convergence Rate of Quasi-Newton Methods on Strongly Convex Functions with Lipschitz Gradient

Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques

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