Search

In this paper, we investigate the method of solving partial differential equations (PDEs) using artificial neural network (ANN) structures, which have been actively applied in artificial intelligence models. The ANN model for solving PDEs offers the advantage of providing explicit and continuous solutions. However, the ANN model for solving PDEs cannot construct a conventionally solvable linear system with known matrix solvers; thus, computational speed could be a significant concern. We study the implementation of the multigrid method, developing a general concept for a coarse-grid correction method to be integrated into the ANN-PDE architecture, with the goal of enhancing computational efficiency. By developing a reduced form of the multigrid method for ANN, we demonstrate that it can be interpreted as an equivalent representation of the classic multigrid expansion. We validated the applicability of the proposed method through rigorous experiments, which included analyzing loss decay and the number of iterations along with improvements in terms of accuracy, speed, and complexity. We accomplished this by employing the gradient descent method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to update the gradients while solving the given ANN systems of PDEs. [ABSTRACT FROM AUTHOR]

The BFGS method is a common and effective method for solving unconstrained optimization problems in quasi-Newton algorithm. However, many scholars have proven that the algorithm may fail in some cases for nonconvex problems under Wolfe conditions. In this paper, an adaptive projection BFGS algorithm is proposed naturally which can solve nonconvex problems, and the following properties are shown in this algorithm: ➀ The generation of the step size α j satisfies the popular Wolfe conditions; ➁ a specific condition is proposed which has sufficient descent property, and if the current point satisfies this condition, the ordinary BFGS iteration process proceeds as usual; ➂ otherwise, the next iteration point x j + 1 is generated by the proposed adaptive projection method. This algorithm is globally convergent for nonconvex problems under the weak-Wolfe-Powell (WWP) conditions and has a superlinear convergence rate, which can be regarded as an extension of projection BFGS method proposed by Yuan et al. (J. Comput. Appl. Math. 327:274-294, 2018). Furthermore, the final numerical results and the application of the algorithm to the Muskingum model demonstrate the feasibility and competitiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this article, we propose an adaptation of the Hyperbolic Augmented Lagrangian algorithm to solve multiobjective optimization problems. After conducting a literature review, we provide a definition of the Hyperbolic Augmented Lagrange method. Under clearly defined assumptions, we demonstrate that any limit point of a sequence generated by the proposed approach is feasible and constitutes a Pareto-optimal solution. To assess the effectiveness of the proposed algorithm, we incorporate two variants of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, thereby offering two approaches for solving well-known tests problems from the literature. We employ metrics to evaluate the performance of these two approaches. [ABSTRACT FROM AUTHOR]

We present two quasi-Newton algorithms for solving unconstrained optimization problems based on two modified secant relations to get reliable approximations of the Hessian matrices of the objective function. The proposed methods make use of both gradient and function values, and utilize information from the two most recent steps in contrast to the the usual secant relation using only the latest step. We show that the modified BFGS methods based on the new secant relations are globally convergent and have a local superlinear rate of convergence. Computational experiments are made on problems from the CUTEst library. Comparative numerical results show competitiveness of the proposed methods in the sense of the Dolan–Moré performance profiles. [ABSTRACT FROM AUTHOR]

It is known that many tensor data have symmetric or partially symmetric structure and structural tensors have structure preserving Candecomp/Parafac (CP) decompositions. However, the well-known alternating least squares (ALS) method cannot realize structure preserving CP decompositions of tensors. Hence, in this paper, we consider numerical problems of structure preserving rank-R approximation and structure preserving CP decomposition of partially symmetric tensors. For the problem of structure preserving rank-R approximation, we derive the gradient formula of the objective function, obtain BFGS iterative formulas, propose a BFGS algorithm for positive partially symmetric rank-R approximation, and discuss the convergence of the algorithm. For the problem of structure preserving CP decomposition, we give a necessary condition for partially symmetric tensors with even orders to have positive partially symmetric CP decompositions, and design a general partially symmetric rank-R approximation algorithm. Finally, some numerical examples are given. Through numerical examples, we find that if a tensor has a positive partially symmetric CP decomposition then its partially symmetric rank CP decomposition must be a positive CP decomposition. In addition, we compare the BFGS algorithm proposed in this paper with the standard CP-ALS method. Numerical examples show that the BFGS algorithm has better stability and faster computing speed than CP-ALS algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a modification of the q-BFGS method for nonlinear unconstrained optimization problems. For this modification, we use a simple symmetric positive definite matrix and propose a new q-quasi-Newton equation, which is close to the ordinary q-quasi-Newton equation in the limiting case. This method uses only first order q-derivatives to build an approximate q-Hessian over a number of iterations. The q-Armijo-Wolfe line search condition is used to calculate step length, which guarantees that the objective function value is decreasing. This modified q-BFGS method preserves the global convergence properties of the q-BFGS method, without the convexity assumption on the objective function. Numerical results on some test problems are presented, which show that an improvement has been achieved. Moreover, we depict the numerical results through the performance profiles. [ABSTRACT FROM AUTHOR]

This work implements the BFGS (Broyden-Fletcher-Goldfarb-Shanno) optimization method for training the type-1 and singleton fuzzy logic system applied to solve binary classification problems. The BFGS is a quasi-Newton method that approximates the second-order information using the gradient of the cost function. Additionally, the Golden Section method is used to obtain the step size for each line search in a descent direction. The effectiveness of the proposed method is demonstrated by using well-established classification metrics evaluated in popular datasets from the literature. Comparisons between the proposed approach and well-known gradient-based methods available are also provided, showing that the BFGS achieves improved performance in terms of accuracy, mean squared error, and the number of epoch demanded during the training phase. [ABSTRACT FROM AUTHOR]

The BFGS method, which has great numerical stability, is one of the quasi-Newton line search methods. However, the global convergence of the BFGS method with a Wolfe line search is still an open problem for general functions. Recently, Yuan et al. (Appl. Math. Mode. 47:811–825, 2017) presented a modified weak Wolfe-Powell (WWP) line search that globally converges for general functions; however, they only partially solved this open problem. In this paper, a further modified WWP line search is proposed, and the global convergence of the BFGS method is established under suitable conditions for general functions. The numerical results show that the new line search produces more interesting results than the normal line search. Moreover, a fact engineering problem is studied with the Muskingum model to show the performance of the presented algorithm. [ABSTRACT FROM AUTHOR]

We propose a BFGS method with Wolfe line searches for unconstrained multiobjective optimization problems. The algorithm is well defined even for general nonconvex problems. Global convergence and R-linear convergence to a Pareto optimal point are established for strongly convex problems. In the local convergence analysis, if the objective functions are locally strongly convex with Lipschitz continuous Hessians, the rate of convergence is Q-superlinear. In this respect, our method exactly mimics the classical BFGS method for single-criterion optimization. [ABSTRACT FROM AUTHOR]

Abstract Variants of the Newton method are very popular for solving unconstrained optimization problems. The study on global convergence of the BFGS method has also made good progress. The q-gradient reduces to its classical version when q approaches 1. In this paper, we propose a quantum-Broyden–Fletcher–Goldfarb–Shanno algorithm where the Hessian is constructed using the q-gradient and descent direction is found at each iteration. The algorithm presented in this paper is implemented by applying the independent parameter q in the Armijo–Wolfe conditions to compute the step length which guarantees that the objective function value decreases. The global convergence is established without the convexity assumption on the objective function. Further, the proposed method is verified by the numerical test problems and the results are depicted through the performance profiles.

Prediction of pollution in water flow is a very important task. To this end, it is imperative to be able to define the uncertainty in a model prediction. This is the purpose of sensitivity analysis whose role is to identify what uncertainty in the model outputs is attributable to the model inputs (parameters in this case). Traditionally, this is achieved by running the model perturbed by many random samples in the parameter space to determine their impact on the model outputs. It provides information on how much of the output variance is controlled by each parameter of the inputs. The theoretical results related to the procedure based adjoint approach for computing a sensitivity of the response function (RF) to changes in the input source are presented in the paper. It is shown that this approach allows to compute, by one single integration of the adjoint equation over a given time interval, a sensitivity of the RF to any source located in the domain of interest. The proposed approach is applied to the 2D Saint-Venant flow equations for modelling the water pollution problem. A numerical experiment is formulated and implemented for the Thanh Nhan Lake in Hanoi for studying a sensitivity of some RF to observations. The numerical model is constructed by applying the well-known finite-volume method. Two appropriate optimization problems are introduced and solved on the basis of the BFGS algorithm. The numerical results show the efficiency of the proposed method and confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, a class of bilateral free boundary problem is considered. This identification problem is formulated as a shape optimization problem via the definition of a cost functional. The existence of an optimal solution for the optimization problem is proved. An augmented Lagrangian approach is used to facilitate the computation of the shape derivatives. A numerical approach is proposed to solve iteratively the shape optimization problem, also we prove its convergence. Various type of domains are investigated to confirm the validity of the presented approach. For the numerical stability we investigate the shape without and with noisy data. [ABSTRACT FROM AUTHOR]

For general large-scale optimization problems compact representations exist in which recursive quasi-Newton update formulas are represented as compact matrix factorizations. For problems in which the objective function contains additional structure, recent structured quasi-Newton methods exploit available second-derivative information and approximate unavailable second derivatives. This article develops the compact representations of two structured Broyden-Fletcher-Goldfarb-Shanno update formulas. The compact representations enable efficient limited memory and initialization strategies. Two limited memory line search algorithms are described for which extensive numerical results demonstrate the efficacy of the algorithms, including comparisons to IPOPT on large machine learning problems, and to L-BFGS on a real world large scale ptychographic imaging application. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Computer-Aided Design & Computer Graphics / Jisuanji Fuzhu Sheji Yu Tuxingxue Xuebao is the property of Gai Kan Bian Wei Hui and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider Broyden class updates for large scale optimization problems in n dimensions, restricting attention to the case when the initial second derivative approximation is the identity matrix. Under this assumption we present an implementation of the Broyden class based on a coordinate transformation on each iteration. It requires only 2 n k + O (k 2) + O (n) multiplications on the kth iteration and stores n K + O (K 2) + O (n) numbers, where K is the total number of iterations. We investigate a modification of this algorithm by a scaling approach and show a substantial improvement in performance over the BFGS method. We also study several adaptations of the new implementation to the limited memory situation, presenting algorithms that work with a fixed amount of storage independent of the number of iterations. We show that one such algorithm retains the property of quadratic termination. The practical performance of the new methods is compared with the performance of Nocedal's (Math Comput 35:773--782, 1980) method, which is considered the benchmark in limited memory algorithms. The tests show that the new algorithms can be significantly more efficient than Nocedal's method. Finally, we show how a scaling technique can significantly improve both Nocedal's method and the new generalized conjugate gradient algorithm. [ABSTRACT FROM AUTHOR]

There are two problems for nonconvex functions under the weak Wolfe–Powell line search in unconstrained optimization problems. The first one is the global convergence of the Polak–Ribière–Polyak conjugate gradient algorithm and the second is the global convergence of the Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method. Many scholars have proven that the two problems do not converge, even under an exact line search. Two circle counterexamples were proposed to generate the nonconvergence of the Polak–Ribière–Polyak algorithm for the nonconvex functions under the exact line search, which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new Polak–Ribière–Polyak algorithm is designed by the following steps. (i) Given the current point and a parabolic surface is designed; (ii) An assistant point is defined based on the current point; (iii) The assistant point is projected onto the surface to generate the next point; (iv) The presented algorithm has the global convergence for nonconvex functions with the weak Wolfe–Powell line search. A similar technique is used for the quasi-Newton method to get a new quasi-Newton algorithm and to establish its global convergence. Numerical results show that the given algorithms are more competitive than other similar algorithms. Meanwhile, the well-known hydrologic engineering application problem, called parameter estimation problem of nonlinear Muskingum model, is also done by the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper is focused on improving global convergence of the modified BFGS algorithm with Yuan-Wei-Lu line search formula. This improvement has been achieved by presenting a different line search approach and it is proved that the BFGS method with this line search converges globally if the function to be minimized has Lipschitz continuous gradients. The performance of the suggested algorithm is investigated via mathematical analysis and a simulation study. [ABSTRACT FROM AUTHOR]

In this paper, we consider an unconstrained optimization problem and propose a new family of modified BFGS methods to solve it. As it is known, classic BFGS method is not always globally convergence for nonconvex functions. To overcome this difficulty, we introduce a new modified weak-Wolfe–Powell line search technique. Under this new technique, we prove global convergence of the new family of modified BFGS methods and the classic BFGS method, for nonconvex functions. Furthermore, all members of this family have at least o (‖ s ‖ 5) error order. Our obtained results from numerical experiments on 77 standard unconstrained problems, indicate that the algorithms developed in this paper are promising and more effective than some similar algorithms. [ABSTRACT FROM AUTHOR]

Unitary symmetric eigenvalues (US-eigenvalues) of symmetric complex tensors and unitary eigenvalues (U-eigenvalues) for non-symmetric complex tensors are very important because of their background of quantum entanglement. US-eigenvalue is a generalization of Z-eigenvalue from the real case to the complex case, which is closely related to the best complex rank-one approximations to higher-order tensors. The problem of finding US-eigenpairs can be converted to an unconstrained nonlinear optimization problem with complex variables, their complex conjugate variables and real variables. However, optimization methods often need a first- or second-order derivative of the objective function, and cannot be applied to real valued functions of complex variables because they are not necessarily analytic in their argument. In this paper, we first establish the first-order complex Taylor series and Wirtinger calculus of complex gradient of real-valued functions with complex variables, their complex conjugate variables and real variables. Based on this theory, we propose a norm descent cautious BFGS method for computing US-eigenpairs of a symmetric complex tensor. Under appropriate conditions, global convergence and superlinear convergence of the proposed method are established. As an application, we give a method to compute U-eigenpairs for a non-symmetric complex tensor by finding the US-eigenpairs of its symmetric embedding. The numerical examples are presented to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of interval-valued multiobjective optimization problems (IVMOP), and formulate a certain equivalent multiobjective optimization problem (MOP). We establish that any Pareto optimal solution of the equivalent MOP is also an effective solution of IVMOP. We introduce two variants of the quasi-Newton method to solve IVMOP when every component of the objective function has second-order partial generalized Hukuhara derivatives. In the first variant, we assume that the objective function of IVMOP is strongly convex, and approximate Hessian matrix of each component of the objective function of MOP by using the well-known BFGS method. The algorithm generated by this variant exhibits superlinear convergence to a locally effective solution of IVMOP. In the second variant, we approximate the Hessian matrix of each component of MOP by applying the Levenberg–Marquardt method. Moreover, this variant can be applied to non-convex functions, and the sequence generated by this variant converges to a locally effective solution of IVMOP as well. Finally, some numerical examples are furnished to illustrate the effectiveness of our developed methodology. [ABSTRACT FROM AUTHOR]

Copyright of NaUKMA Research Papers. Computer Science is the property of National University of Kyiv-Mohyla Academy, Faculty of Humanities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

• Formulate the no-toll equilibrium (NTE) and the congestion and environmental toll (CET) equilibrium problems with heterogeneous users as unconstrained optimization problems. • Modify the BFGS method to solve these problems. • Formulate the optimal congestion toll and CET design problems as bi-level programs. • Develop the double BFGS method to solve the two bi-level programs. We design a congestion and environmental toll (CET) scheme for the morning commute with heterogeneous users in a single OD network with parallel routes. The designed toll scheme relies upon the concept of marginal-cost pricing and is anonymous. The Henderson approach is used to model road congestion and the tolling problem to examine commuter's arrival time and route choice at the CET equilibrium (CETE). Linear interpolation is applied to approximate the emission cost function and the resulting CETE problem is formulated as an unconstrained optimization problem, which is solved by the modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. Unlike the existing approach, this novel approach does not require that the arrival of each group of commuters at the destination at the equilibrium follows a predetermined order, and can handle non-monotone (emission) cost function. As two special cases, no-toll equilibrium (NTE) and the congestion toll equilibrium (CTE) are also examined, and the two resultant equilibrium problems are formulated and solved by the same approach. This approach is shown to be more efficient than the existing approach. Bi-level programming models are proposed to formulate the optimal congestion toll and CET design problems, in which the CTE and CETE problems are the corresponding lower level problem. These models are solved by the double BFGS method, which uses a classical BFGS method to solve the upper level model and the proposed BFGS method to solve the lower level model. Finally, numerical examples are provided to illustrate the properties of the models and the efficiency of the proposed solution algorithms. [ABSTRACT FROM AUTHOR]

We consider some non-smooth functions and investigate the numerical behavior of the Piecewise Linear Hompotopy (PLH) method ([Bozântan, A., An implementation of the piecewise-linear homotopy algorithm for the computation of fixed points, Creat. Math. Inform., 19 (2010), No. 2, 140-148] and [Bozântan, A. and Berinde, V., Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems, Creat. Math. Inform., 22 (2013), No. 1, 41-46]). We compare the PLH method with the BFGS with inexact line search, a quasi-Newton method, having some results reported in [Lewis, A. S. and Overton, M. L., Nonsmooth optimization via BFGS, submitted to SIAM J. Optimiz, (2009)]. For most of the considered cases, the characteristics of the PLH method are quite similar to the BFGS method, that is, the PLH method converges to local minimum values and the convergence rate seems to be linear with respect to the number of function evaluations, but we also identify some issues with the PLH method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a globally convergent BFGS method to solve Variational Inequality Problems (VIPs). In fact, a globalization technique on the basis of the hyperplane projection method is applied to the BFGS method. The technique, which is independent of any merit function, is applicable for pseudo-monotone problems. The proposed method applies the BFGS direction and tries to reduce the distance of iterates to the solution set. This property, called Fejer monotonicity of iterates with respect to the solution set, is the basis of the convergence analysis. The method applied to pseudo-monotone VIP is globally convergent in the sense that subproblems always have unique solutions, and the sequence of iterates converges to a solution to the problem without any regularity assumption. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We propose a new family of distributions called the exponentiated Kumaraswamy-G class with three extra positive parameters, which generalizes the Cordeiro and de Castro's family. Some special distributions in the new classare discussed. We derive some mathematical properties of the proposedclass including explicit expressions for the quantile function, ordinary andincomplete moments, generating function, mean deviations, reliability, Rényientropy and Shannon entropy. The method of maximum likelihood is used to fit the distributions in the proposed class. Simulations are performed in orderto assess the asymptotic behavior of the maximum likelihood estimates.We illustrate its potentiality with applications to two real data sets whichshow that the extended Weibull model in the new class provides a better fit than other generalized Weibull distributions. [ABSTRACT FROM AUTHOR]

We provide a maximum entropy derivation of a new family of BFGS-like methods. Similar results are then derived for block BFGS methods. This also yields an independent proof of a result of Fletcher 1991 and its generalization to the block case. [ABSTRACT FROM AUTHOR]

A new adaptive scaled Broyden-Fletcher-Goldfarb-Shanno (BFGS) method for unconstrained optimization is presented. The third term in the standard BFGS update formula is scaled in order to reduce the large eigenvalues of the approximation to the Hessian of the minimizing function. Under the inexact Wolfe line search conditions, the global convergence of the adaptive scaled BFGS method is proved in very general conditions without assuming the convexity of the minimizing function. Using 80 unconstrained optimization test functions with a medium number of variables, the preliminary numerical experiments show that this variant of the scaled BFGS method is more efficient than the standard BFGS update or than some other scaled BFGS methods. [ABSTRACT FROM AUTHOR]

In this paper, a Riemannian BFGS method for minimizing a smooth function on a Riemannian manifold is defined, based on a Riemannian generalization of a cautious update and a weak line search condition. It is proven that the Riemannian BFGS method converges (i) globally to stationary points without assuming the objective function to be convex and (ii) superlinearly to a nondegenerate minimizer. Using the weak line search condition removes the need for information from differentiated retraction. The joint matrix diagonalization problem is chosen to demonstrate the performance of the algorithms with various parameters, line search conditions, and pairs of retraction and vector transport. A preliminary version can be found in [Numerical Mathematics and Advanced Applications: ENUMATH 2015, Lect. Notes Comput. Sci. Eng. 112, Springer, New York, 2016, pp. 627{634]. [ABSTRACT FROM AUTHOR]

The standard BFGS method plays an important role among the quasi-Newton algorithms for constrained/un-constrained optimization problems. However, Dai (2003) constructed a counterexample to demonstrate that this method may fail for non-convex functions with inexact Wolfe line searches, and Mascarenhas (2004) showed the nonconvergence of this method and other methods in the Broyden family, even with exact line search techniques. These works motivate us to try to find another way to obtain another quasi-Newton method with better convergence based on the standard BFGS formula. In this paper, four approaches are used in the designed algorithm: (1) a modified weak Wolfe–Powell line search technique is introduced; (2) if one defined condition is satisfied, then the search direction and its associated step length are accepted, and the next point is designed; (3) otherwise, a parabolic will be presented, and it is regarded as the projection surface to avoid using the failed direction, and the next point x k + 1 is generated by a projection technique; and (4) to easily obtain the global convergence of the proposed algorithm, the projection point is not used in the current BFGS update but rather for all the next iterations. The new algorithm possesses global convergence for general functions under the inexact modified weak Wolfe–Powell line search technique, and it is shown that other methods in the Broyden class also have this property. Numerical results are reported to show the performance of the given algorithm and other similar methods. [ABSTRACT FROM AUTHOR]

Phase equilibrium calculations are one of the main steps in the compositional simulation of hydrocarbon reservoirs, well tubing, oil and gas pipelines as well as separation units. The main phases in all hydrocarbon production systems from a reservoir to a stock tank are gas, oil, and water. In most studies, the aqueous phase is considered bulk and phase equilibrium calculations are conducted only on the oil and gas phases. In some conditions such as thermal processes, the solubility of water in other phases and also hydrocarbon components in the water phase cannot be ignored. The robustness of equilibrium calculations for a wide range of compositions, temperatures, and pressures, including near critical area and saddle point curve, is essential for rapid and stable simulation studies. In this study, we propose the three-phase equilibrium calculations in detail for water, oil and gas phases. All the handling steps and strategies to establish a robust and efficient phase stability testing and flash calculation are explained. Moreover, appropriate tolerance values for decision-makings in the algorithm are defined. We successfully present the effectiveness of our algorithm for some three-phase samples covering a wide range of composition and production conditions. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The BFGS method is one of the most effective quasi-Newton algorithms for optimization problems. However, its global convergence for general functions is still open. In this paper, under a new line search technique, this problem is solved, and it is shown that other methods in the Broyden class also possess this property. Moreover, the global convergence of the PRP method is established in the case of this new line search. Numerical results are reported to show that the new line search technique is competitive to that of the normal line search. [ABSTRACT FROM AUTHOR]

We have developed a multilevel and adaption parametric strategies solved by optimization algorithms which require only the availability of objective function values but no derivative information. The key success of these hierarchical strategies refer to the quality of the downward and upward transfers of information. In this paper, we extend our approach when using a derivative-based optimization algorithms. The aim is to better re-initialize the Hessian and the gradient during the optimization process based on our construction of the downward and upward operators. The efficiency of this proposed approach is demonstrated by numerical experiments on an inverse shape model.