Journal articles: 'Hessian damping'

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Gressman, Philip T. "Damping oscillatory integrals by the Hessian determinant via Schrödinger." Mathematical Research Letters 23, no. 2 (2016): 405–30. http://dx.doi.org/10.4310/mrl.2016.v23.n2.a6.

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Attouch, Hedy, Juan Peypouquet, and Patrick Redont. "Fast convex optimization via inertial dynamics with Hessian driven damping." Journal of Differential Equations 261, no. 10 (November 2016): 5734–83. http://dx.doi.org/10.1016/j.jde.2016.08.020.

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Niederländer, Simon K. "Second-Order Dynamics with Hessian-Driven Damping for Linearly Constrained Convex Minimization." SIAM Journal on Control and Optimization 59, no. 5 (January 2021): 3708–36. http://dx.doi.org/10.1137/20m1323679.

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Alvarez, F., H. Attouch, J. Bolte, and P. Redont. "A second-order gradient-like dissipative dynamical system with Hessian-driven damping." Journal de Mathématiques Pures et Appliquées 81, no. 8 (2002): 747–79. http://dx.doi.org/10.1016/s0021-7824(01)01253-3.

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Sologubov, A., and I. Kirpichnikova. "MULTIVARIABLE CONTROL OF SOLAR BATTERY POWER: ELECTROTECHNICAL COMPLEX AS OBJECT WITH HESSIAN-DRIVEN GRADIENT FLOWS." Bulletin of the South Ural State University series "Power Engineering" 21, no. 3 (2021): 57–65. http://dx.doi.org/10.14529/power210307.

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The paper presents certain development results for the novel extremum seeking controller based on Nesterov’s gradient flows for solar tracking systems. It achieves convergence to an arbitrarily small neighborhood of the set of the cost function optimizers. Our results evident ate that for arbitrarily large compact sets of initial conditions, and arbitrarily small neighborhoods of the optimizer, the controller can be tuned to guarantee convergence taking into account the influence of the Hessian, as well as with tuning parameters that have a fairly clear physical meaning. The influence of the Hessian as a vector field, which is a reflection of the distortion of transient processes in the system, and taking it into account is an urgent task, since it allows for a more flexible impact on the speed of transient processes, and by endowing the system with some damping and smoothing, also for its improved quality.

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Adly, Samir, and Hedy Attouch. "Finite Convergence of Proximal-Gradient Inertial Algorithms Combining Dry Friction with Hessian-Driven Damping." SIAM Journal on Optimization 30, no. 3 (January 2020): 2134–62. http://dx.doi.org/10.1137/19m1307779.

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Valenciano, Alejandro A., Biondo L. Biondi, and Robert G. Clapp. "Imaging by target-oriented wave-equation inversion." GEOPHYSICS 74, no. 6 (November 2009): WCA109—WCA120. http://dx.doi.org/10.1190/1.3250267.

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Wave-equation inversion is a powerful technique able to build higher-resolution images with balanced amplitudes in complex subsurface areas relative to migration alone. Wave-equation inversion can be performed in image space without making velocity-model or acquisition-geometry approximations. Our method explicitly computes the least-squares Hessian matrix, defined from the modeling/migration operators, and uses a linear solver to find the solution of the resulting system of equations. One important advantage of the explicit computation of the Hessian, compared to iterative modeling/migration operations schemes, is that most of the work (precomputing the Hessian) is done up front; afterward, different inversion parameters or schemes can be tried at lower cost. Another advantage is that the method canhandle 3D data in a target-oriented fashion. The inversion in the presence of a complex overburden leads to an ill-conditioned system of equations that must be regularized to obtain a stable numerical solution. Regularization can be implemented in the poststack-image domain (zero subsurface offset), where the options for a regularization operator are limited to a customary damping, or in the prestack-image domain (subsurface offset), where a physically inspired regularization operator (differential semblance) can be applied. Though the prestack-image-domain inversion is more expensive than the poststack-image-domain inversion, it can improve the reflectors' continuity into the shadow zones with an enhanced signal-to-noise ratio. Improved subsalt-sediment images in the Sigsbee2b synthetic model and a 3D Gulf of Mexico field data set confirm the benefits of the inversion.

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Namala, Dheeraj Kumar, and V. Surendranath. "Parameter Estimation of Spring-Damping System using Unconstrained Optimization by the Quasi-Newton Methods using Line Search Techniques." Advanced Journal of Graduate Research 5, no. 1 (September 9, 2018): 1–7. http://dx.doi.org/10.21467/ajgr.5.1.1-7.

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Optimization is the basic tools to study the behaviour of many complicated mechanical systems by having the knowledge of differential equations which determine the system. The basis of this paper was to present a method to estimate the parameters such as spring constant and damping coefficient of the spring damped system by unconstrained optimization using derivative methods Such as quasi-newton method by Broyden-Fletcher-Goldfarb-Shanno and davidon-Fletcher-Powell hessian updating method by using backtracking line search methods along with Armijo’s condition.it uses the output error approximation procedure. It shows the convergence of different methods which are used to estimate the parameters and how accurately they are measured.

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Attouch, Hedy, Paul-Emile Maingé, and Patrick Redont. "A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws." Differential Equations & Applications, no. 1 (2012): 27–65. http://dx.doi.org/10.7153/dea-04-04.

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Washio, Takumi, Akihiro Fujii, and Toshiaki Hisada. "On random force correction for large time steps in semi-implicitly discretized overdamped Langevin equations." AIMS Mathematics 9, no. 8 (2024): 20793–810. http://dx.doi.org/10.3934/math.20241011.

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In this study, we focused on the treatment of random forces in a semi-implicitly discretized overdamped Langevin (OL) equation with large time steps. In the usual implicit approach for a nonstochastic mechanical equation, the product of the time interval and Hessian matrix was added to the friction matrix to construct the coefficient matrix for solution updates, which were performed using Newton iteration. When large time steps were used, the additional term, which could be regarded as an artificial friction term, prevented the amplification of oscillations associated with large eigenvalues of the Hessian matrix. In this case, the damping of the high-frequency terms did not cause any discrepancy because they were outside of our interest. However, in OL equations, the friction coefficient was coupled to the random force; therefore, excessive artificial friction may have obscured the effects caused by the stochastic properties of the fluctuations. Consequently, we modified the random force in the proposed semi-implicit scheme so that the total random force was consistent with the friction including the additional artificial term. By deriving a discrete Fokker-Planck (FP) equation from the discretized OL equation, we showed how our modification improved the distribution of the numerical solutions of discrete stochastic processes. Finally, we confirmed the validity of our approach in numerical simulations of a freely jointed chain.

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Boţ, Radu Ioan, and Ernö Robert Csetnek. "A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities." Optimization 68, no. 7 (March 21, 2018): 1265–77. http://dx.doi.org/10.1080/02331934.2018.1452922.

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Chang, Dick Mun, Hong Seng Sim, Yong Kheng Goh, Sing Yee Chua, and Wah June Leong. "Diagonal Variable Matrix Method in Solving Inverse Problem in Image Processing." ITM Web of Conferences 67 (2024): 01039. http://dx.doi.org/10.1051/itmconf/20246701039.

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In this paper, we introduce a new gradient method called the Diagonal Variable Matrix method. Our proposed method is aimed to minimize Hk+1 over the log-determinant norm subject to weak secant relation. The derived diagonal matrix Hk+1 is the approximation of the inverse Hessian matrix, which enables the calculation of the search direction, dk = −Hk+1gk, where gk denotes the gradient of the objective function. The proposed method is coupled with the backtracking Armijo line search. The proposed method is specifically designed to reduce the number of iterations and training duration, particularly in the context of solving large-dimensional problems. Finally, as a practical illustration, the proposed method is applied to solve the image deblurring problem, and its performance is analyzed using image quality metrics. The results demonstrate that the proposed method outperforms various conjugate gradient (CG) methods and multiple damping gradient method.

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Boţ, Radu Ioan, and Ernö Robert Csetnek. "Approaching Nonsmooth Nonconvex Optimization Problems Through First Order Dynamical Systems with Hidden Acceleration and Hessian Driven Damping Terms." Set-Valued and Variational Analysis 26, no. 2 (May 31, 2017): 227–45. http://dx.doi.org/10.1007/s11228-017-0411-1.

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Zhai, Chongpu, Eric B. Herbold, and Ryan C. Hurley. "The influence of packing structure and interparticle forces on ultrasound transmission in granular media." Proceedings of the National Academy of Sciences 117, no. 28 (June 29, 2020): 16234–42. http://dx.doi.org/10.1073/pnas.2004356117.

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Ultrasound propagation through externally stressed, disordered granular materials was experimentally and numerically investigated. Experiments employed piezoelectric transducers to excite and detect longitudinal ultrasound waves of various frequencies traveling through randomly packed sapphire spheres subjected to uniaxial compression. The experiments featured in situ X-ray tomography and diffraction measurements of contact fabric, particle kinematics, average per-particle stress tensors, and interparticle forces. The experimentally measured packing configuration and inferred interparticle forces at different sample stresses were used to construct spring networks characterized by Hessian and damping matrices. The ultrasound responses of these network were simulated to investigate the origins of wave velocity, acoustic paths, dispersion, and attenuation. Results revealed that both packing structure and interparticle force heterogeneity played an important role in controlling wave velocity and dispersion, while packing structure alone quantitatively explained most of the observed wave attenuation. This research provides insight into time- and frequency-domain features of wave propagation in randomly packed granular materials, shedding light on the fundamental mechanisms controlling wave velocities, dispersion, and attenuation in such systems.

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Song, Shu Ni, Jing Yi Liu, and Jin Qian. "Application of an Improved Trust-Region Method to Rigid-Plastic Finite Element Analysis in Strip Rolling." Materials Science Forum 704-705 (December 2011): 216–22. http://dx.doi.org/10.4028/www.scientific.net/msf.704-705.216.

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Rigid-plastic finite element analysis (RPFEA) is an efficient and practical method to calculate rolling parameters in the strip rolling process. To solve the system of simulations equations involved in the RPFEA, a numerous of numerical methods, including the standard Newton-Raphson method, the modified Newton-Raphson method, and etc., have been proposed by different researchers. However, the computational time of the existed numerical methods can not meet the requirement of the online application. By tracking the computational time consumption for the main components in the standard Newton-Raphson method used in finite element analysis, it was found that linear search of damping factor occupies the most of the computational time. Thus, more efforts should be put on the linear search of damping factor to speed up the solving procedure, so that the online application of RPFEA is possible. In this paper, an improved trust-region method is developed to speed up the solving procedure, in which the Hessian matrix is forced to positive definite so as to improve the condition number of matrix. The numerical experiments are carried out to compare the proposed method with the standard Newton-Raphson method based on the practical data collected from a steel company in China. The numerical results demonstrate that the computational time of the proposed method outperforms that of the standard Newton-Raphson method and can meet the requirement of online application. Meanwhile the computational values of rolling force obtained by the proposed method are in good agreement with experimental values, which verifies the validity and stability of the proposed method.

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Liu, Jinyue, and Yanghua Wang. "Seismic simultaneous inversion using a multidamped subspace method." GEOPHYSICS 85, no. 1 (November 22, 2019): R1—R10. http://dx.doi.org/10.1190/geo2018-0470.1.

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Seismic inversion of amplitude variation with offset (AVO) plays a key role in seismic interpretation and reservoir characterization. The AVO inversion should be a simultaneous inversion that inverts for three elastic parameters simultaneously: the P-wave impedance, S-wave impedance, and density. Using only seismic P-wave reflection data with a limited source-receiver offset range, the AVO simultaneous inversion can obtain two elastic parameters reliably, but it is difficult to invert for the third parameter, usually the density term. To address this difficulty in the AVO simultaneous inversion, we used a subspace inversion method in which we partitioned the elastic parameters into different subspaces. We parameterized each single elastic parameter with a truncated Fourier series and inverted for the Fourier coefficients. Because the Fourier coefficients of different wavenumber components have different sensitivities, we grouped the Fourier coefficients of low-, medium-, and high-wavenumber components into different subspaces. We further assigned different damping factors to the Hessian matrix corresponding to different wavenumber components within each subspace. This inversion scheme is referred to as a multidamped subspace method. Synthetic and field seismic data examples confirmed that the AVO simultaneous inversion with this multidamped subspace method is capable of producing reliable estimation of the three elastic parameters simultaneously.

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Cristo, Rafael Abreu, and Milton Porsani. "MULTISCALES FULL WAVEFORM INVERSION USING A FILTERING APPROACH AND GRADIENT PRECONDITIONING." Revista Brasileira de Geofísica 35, no. 4 (April 18, 2018): 247. http://dx.doi.org/10.22564/rbgf.v35i4.899.

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ABSTRACT. The FWI multiscale approach in data domain produces better results because the problem gets closer to the overall minimum avoiding the local minima. The method works in different scales, avoiding the initial velocity model choice as well as the cycle skipping. Regarding to multiscale approach, it was done choosing frequencies band performed by Wiener filter and a SVD filter trace by trace both in data domain. The trace by trace SVD filter works taking each trace of the gradient and assembles on the shifted matrix traces and do the decomposition from low to high frequency. In addition this multiscale approach in data domain was compared to another multiscale approach using damping filters on the objective function (MDFOF). Due to the problem of geometrical spreading, during the propagation of the wave field, the deeper regions of the model are not well illuminated, hence the preconditioning of the objective function gradient was done in order to eliminate this problem and allow the deeper regions to be compared. Keywords: SVD filter; Full waveform inversion; Gradient preconditioning; Pseudo Hessian diagonal. RESUMO. A abordagem multiescala no problema da FWI, produz melhores resultados pois o problema consegue convergir para o mínimo global, evitando o problema do mínimo local. O método funciona em diferentes escalas, evitando o problema da escolha no modelo inicial de velocidade bem como o problema de salto de ciclo. Em relação à abordagem multiescala, o mesmo foi realizado escolhendo bandas de frequências usando o filtro de Wiener e o filtro SVD traço a traço. O filtro SVD traço a traço funciona tomando cada traço do gradiente e da matriz de traços deslocados a faz a decomposição das baixas às altas frequências. Além dessa, abordagem multiescala no domínio do dado, outra abordagem multiescala usando filtros atenuantes foi comparada com a abordagem multiescala no domínio do dado. Devido ao problema de divergência esférica, durante a propagação da onda, as regiões mais profundas do modelo não são corretamente imageadas, portanto faz-se necessário o precondicionamento do gradiente foi feito com intuito de eliminar esse problema e permitir a comparação das duas abordagens nas regiões mais profundas do modelo. Palavras-chave: Filtro SVD; Inversion complete da forma de onda; Precondicionamento do gradiente; Diagonal da Pseudo Hessiana.

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Gao, Guohua, and Albert C. Reynolds. "An Improved Implementation of the LBFGS Algorithm for Automatic History Matching." SPE Journal 11, no. 01 (March 1, 2006): 5–17. http://dx.doi.org/10.2118/90058-pa.

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Summary For large scale history matching problems, where it is not feasible to compute individual sensitivity coefficients, the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) is an efficient optimization algorithm, (Zhang and Reynolds, 2002; Zhang, 2002). However, computational experiments reveal that application of the original implementation of LBFGS may encounter the following problems:converge to a model which gives an unacceptable match of production data;generate a bad search direction that either leads to false convergence or a restart with the steepest descent direction which radically reduces the convergence rate;exhibit overshooting and undershooting, i.e., converge to a vector of model parameters which contains some abnormally high or low values of model parameters which are physically unreasonable. Overshooting and undershooting can occur even though all history matching problems are formulated in a Bayesian framework with a prior model providing regularization. We show that the rate of convergence and the robustness of the algorithm can be significantly improved by:a more robust line search algorithm motivated by the theoretical result that the Wolfe conditions should be satisfied;an application of a data damping procedure at early iterations orenforcing constraints on the model parameters. Computational experiments also indicate thata simple rescaling of model parameters prior to application of the optimization algorithm can improve the convergence properties of the algorithm although the scaling procedure used can not be theoretically validated. Introduction Minimization of a smooth objective function is customarily done using a gradient based optimization algorithm such as the Gauss- Newton (GN) method or Levenberg-Marquardt (LM) algorithm. The standard implementations of these algorithms (Tan and Kalogerakis, 1991; Wu et al., 1999; Li et al., 2003), however, require the computation of all sensitivity coefficients in order to formulate the Hessian matrix. We are interested in history matching problems where the number of data to be matched ranges from a few hundred to several thousand and the number of reservoir variables or model parameters to be estimated or simulated ranges from a few hundred to a hundred thousand or more. For the larger problems in this range, the computer resources required to compute all sensitivity coefficients would prohibit the use of the standard Gauss- Newton and Levenberg-Marquardt algorithms. Even for the smallest problems in this range, computation of all sensitivity coefficients may not be feasible as the resulting GN and LM algorithms may require the equivalent of several hundred simulation runs. The relative computational efficiency of GN, LM, nonlinear conjugate gradient and quasi-Newton methods have been discussed in some detail by Zhang and Reynolds (2002) and Zhang (2002).

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Attouch, Hedy, Jalal Fadili, and Vyacheslav Kungurtsev. "On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping." Evolution Equations and Control Theory, 2022, 0. http://dx.doi.org/10.3934/eect.2022022.

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<p style='text-indent:20px;'>Second-order continuous-time dissipative dynamical systems with viscous and Hessian driven damping have inspired effective first-order algorithms for solving convex optimization problems. While preserving the fast convergence properties of the Nesterov-type acceleration, the Hessian driven damping makes it possible to significantly attenuate the oscillations. To study the stability of these algorithms with respect to perturbations, we analyze the behaviour of the corresponding continuous systems when the gradient computation is subject to exogenous additive errors. We provide a quantitative analysis of the asymptotic behaviour of two types of systems, those with implicit and explicit Hessian driven damping. We consider convex, strongly convex, and non-smooth objective functions defined on a real Hilbert space and show that, depending on the formulation, different integrability conditions on the perturbations are sufficient to maintain the convergence rates of the systems. We highlight the differences between the implicit and explicit Hessian damping, and in particular point out that the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case.</p>

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Maulén, Juan José, and Juan Peypouquet. "A Speed Restart Scheme for a Dynamics with Hessian-Driven Damping." Journal of Optimization Theory and Applications, September 5, 2023. http://dx.doi.org/10.1007/s10957-023-02290-5.

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AbstractIn this paper, we analyze a speed restarting scheme for the inertial dynamics with Hessian-driven damping, introduced by Attouch et al. (J Differ Equ 261(10):5734–5783, 2016). We establish a linear convergence rate for the function values along the restarted trajectories. Numerical experiments suggest that the Hessian-driven damping and the restarting scheme together improve the performance of the dynamics and corresponding iterative algorithms in practice.

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Heydari, Abbas, and Li Li. "Dependency of critical damping on various parameters of tapered bidirectional graded circular plates rested on Hetenyi medium." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, August 24, 2020, 095440622095249. http://dx.doi.org/10.1177/0954406220952498.

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In industrial machinery and automobiles the critical damping plates are used as shock absorbers to dampen system and return rest position, in the shortest period of time. The paper is focused on the effects of various parameters on critical damping of a complicated system including a tapered bidirectional graded circular plate subjected to radially variable in-plane pre-load and rested on visco-Hetenyi elastic medium with semi-rigid restraint. By employing a Chebyshev-Ritz method, dependency of critical damping is investigated for the first time on various parameters including bidirectional arbitrary material gradation, variable in-plane pre-load, Hetenyi elastic medium parameters, tapering, semi-rigid restraint and Poisson’s ratio. For the Chebyshev-Ritz method developed here, the proper scale factor and orthogonal shifted Chebyshev polynomials of the first kind without auxiliary function requirement are used. The conventional rule of mixture and Mori-Tanaka homogenization scheme are used to model transverse gradation. The characteristic equation is calculated by vanishing determinant of the total potential energy Hessian. An equivalent ABAQUS model is introduced to eliminate necessity of complicated modeling of original structure.

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Attouch, Hedy, Zaki Chbani, Jalal Fadili, and Hassan Riahi. "First-order optimization algorithms via inertial systems with Hessian driven damping." Mathematical Programming, November 16, 2020. http://dx.doi.org/10.1007/s10107-020-01591-1.

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Assis, Carlos A. M., Hervé Chauris, François Audebert, and Paul Williamson. "Investigating Hessian-based inversion velocity analysis." GEOPHYSICS, December 13, 2023, 1–139. http://dx.doi.org/10.1190/geo2022-0689.1.

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Inversion velocity analysis (IVA) is an image domain method built upon the spatial scale separation of the model. Accordingly, the IVA method is performed with an iterative process composed of two minimization steps consisting of migration (inner loop) and tomography (outer loop), respectively, with each step accounting for its Hessian or not. The migration part provides the common image gathers (CIGs) with extension in the horizontal subsurface offset. Then, the differential semblance optimization (DSO) misfit measures the focusing of the events in the CIGs which indicates the quality of the velocity model. Commonly, the velocity updates are obtained from the DSO gradient. IVA is a modified version where the approximate inverse replaces the adjoint of the inner loop process: in that case, the migration Hessian is approximately diagonal in the high-frequency regime. In this work, we report the implementation of the tomographic Hessian (i.e., the second derivative of the DSO misfit with respect to the background model) for the estimation of the background velocity model. We apply the second-order adjoint-state method to obtain the application of the tomographic Hessian on a vector. Then, we use the truncated-Newton method to obtain the update directions by computing approximately the application of the inverse of the tomographic Hessian on the descent direction. We also make a theoretical comparison between the tomography in the IVA and full-waveform inversion contexts. Two numerical examples are used to compare, in terms of geophysical results and computational costs, the truncated-Newton method with different gradient-based optimization methods applied to IVA. A small model allows us to evaluate the eigenvalues of the tomographic Hessian which explains the large damping needed in the truncated-Newton case.

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Boţ, Radu Ioan, Ernö Robert Csetnek, and Szilárd Csaba László. "Tikhonov regularization of a second order dynamical system with Hessian driven damping." Mathematical Programming, June 11, 2020. http://dx.doi.org/10.1007/s10107-020-01528-8.

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He, Xin, Feng Tian, An-qi Li, and Ya-Ping Fang. "Convergence rates of mixed primal-dual dynamical systems with Hessian driven damping." Optimization, September 1, 2023, 1–26. http://dx.doi.org/10.1080/02331934.2023.2253813.

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Aujol, Jean-François, Charles Dossal, Van Hao Hoàng, Hippolyte Labarrière, and Aude Rondepierre. "Fast Convergence of Inertial Dynamics with Hessian-Driven Damping Under Geometry Assumptions." Applied Mathematics & Optimization 88, no. 3 (September 20, 2023). http://dx.doi.org/10.1007/s00245-023-10058-6.

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Vu, Bao Anh, David Gunawan, and Andrew Zammit-Mangion. "R-VGAL: a sequential variational Bayes algorithm for generalised linear mixed models." Statistics and Computing 34, no. 3 (April 6, 2024). http://dx.doi.org/10.1007/s11222-024-10422-8.

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AbstractModels with random effects, such as generalised linear mixed models (GLMMs), are often used for analysing clustered data. Parameter inference with these models is difficult because of the presence of cluster-specific random effects, which must be integrated out when evaluating the likelihood function. Here, we propose a sequential variational Bayes algorithm, called Recursive Variational Gaussian Approximation for Latent variable models (R-VGAL), for estimating parameters in GLMMs. The R-VGAL algorithm operates on the data sequentially, requires only a single pass through the data, and can provide parameter updates as new data are collected without the need of re-processing the previous data. At each update, the R-VGAL algorithm requires the gradient and Hessian of a “partial” log-likelihood function evaluated at the new observation, which are generally not available in closed form for GLMMs. To circumvent this issue, we propose using an importance-sampling-based approach for estimating the gradient and Hessian via Fisher’s and Louis’ identities. We find that R-VGAL can be unstable when traversing the first few data points, but that this issue can be mitigated by introducing a damping factor in the initial steps of the algorithm. Through illustrations on both simulated and real datasets, we show that R-VGAL provides good approximations to posterior distributions, that it can be made robust through damping, and that it is computationally efficient.

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László, Szilárd Csaba. "Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization." Computational Optimization and Applications, October 19, 2024. http://dx.doi.org/10.1007/s10589-024-00620-5.

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AbstractThis paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $$\mathop {\text {argmin}}$$ argmin set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.

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Adly, Samir, Hedy Attouch, and Van Nam Vo. "Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms." Fixed Point Theory and Algorithms for Sciences and Engineering 2021, no. 1 (October 18, 2021). http://dx.doi.org/10.1186/s13663-021-00702-7.

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AbstractIn a Hilbert space $\mathcal{H}$ H , we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator $A= \nabla f +B $ A = ∇ f + B , where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as $t\to +\infty $ t → + ∞ of the generated trajectories towards the zeros of $\nabla f +B$ ∇ f + B . The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

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Attouch, Hedy, Aïcha Balhag, Zaki Chbani, and Hassan Riahi. "Accelerated Gradient Methods Combining Tikhonov Regularization with Geometric Damping Driven by the Hessian." Applied Mathematics & Optimization 88, no. 2 (May 31, 2023). http://dx.doi.org/10.1007/s00245-023-09997-x.

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Zhong, Gangfan, Xiaozhe Hu, Ming Tang, and Liuqiang Zhong. "Fast Convex Optimization via Differential Equation with Hessian-Driven Damping and Tikhonov Regularization." Journal of Optimization Theory and Applications, May 30, 2024. http://dx.doi.org/10.1007/s10957-024-02462-x.

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Attouch, Hedy, Zaki Chbani, Jalal Fadili, and Hassan Riahi. "Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping." Optimization, December 3, 2021, 1–40. http://dx.doi.org/10.1080/02331934.2021.2009828.

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Attouch, Hedy, Aïcha Balhag, Zaki Chbani, and Hassan Riahi. "Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling." Evolution Equations & Control Theory, 2021, 0. http://dx.doi.org/10.3934/eect.2021010.

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34

Boţ, Radu Ioan, and Mikhail A. Karapetyants. "A fast continuous time approach with time scaling for nonsmooth convex optimization." Advances in Continuous and Discrete Models 2022, no. 1 (December 16, 2022). http://dx.doi.org/10.1186/s13662-022-03744-2.

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AbstractIn a Hilbert setting, we study the convergence properties of the second order in time dynamical system combining viscous and Hessian-driven damping with time scaling in relation to the minimization of a nonsmooth and convex function. The system is formulated in terms of the gradient of the Moreau envelope of the objective function with a time-dependent parameter. We show fast convergence rates for the Moreau envelope, its gradient along the trajectory, and also for the system velocity. From here, we derive fast convergence rates for the objective function along a path which is the image of the trajectory of the system through the proximal operator of the first. Moreover, we prove the weak convergence of the trajectory of the system to a global minimizer of the objective function. Finally, we provide multiple numerical examples illustrating the theoretical results.

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35

Karapetyants, Mikhail A. "A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique." Computational Optimization and Applications, October 25, 2023. http://dx.doi.org/10.1007/s10589-023-00536-6.

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AbstractIn this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.

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36

Muhammad, Lawal, Mohammad Y. Waziri, Ibrahim Mohammed Sulaiman, Issam A. R. Moghrabi, and Aceng Sambas. "An improved preconditioned conjugate gradient method for unconstrained optimization problem with application in Robot arm control." Engineering Reports, August 6, 2024. http://dx.doi.org/10.1002/eng2.12968.

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AbstractThis work suggests improved conjugate gradient methods for enhancing the efficiency and robustness of the classical conjugate gradient methods. The study modifies the diagonal of the inverse Hessian approximation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi‐Newton update in order to build a preconditioner for nonlinear conjugate gradient (NCG) methods applied to large‐scale unconstrained optimization problems. Damping techniques were embedded into the algorithm to impose the positive definiteness of the diagonal approximation. This made the methods easy to use and presented a viable way to increase the effectiveness of unconstrained optimization techniques. Experimental findings from a collection of benchmark problems demonstrate the efficiency and robustness of the proposed method when compared to five other existing NCG algorithms. Moreover, the successful application of the algorithm to manipulate robotic planar motion control systems with 3 degrees of freedom has been demonstrated, highlighting the practicality of the proposed approach.

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37

Attouch, Hedy, Radu Ioan Boţ, and Dang-Khoa Nguyen. "Fast Convex Optimization via Time Scale and Averaging of the Steepest Descent." Mathematics of Operations Research, October 16, 2024. http://dx.doi.org/10.1287/moor.2023.0186.

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In a Hilbert setting, we develop a gradient-based dynamic approach for fast solving convex optimization problems. By applying time scaling, averaging, and perturbation techniques to the continuous steepest descent (SD), we obtain high-resolution ordinary differential equations of the Nesterov and Ravine methods. These dynamics involve asymptotically vanishing viscous damping and Hessian-driven damping (either in explicit or implicit form). Mathematical analysis does not require developing a Lyapunov analysis for inertial systems. We simply exploit classical convergence results for SD and its external perturbation version, then use tools of differential and integral calculus, including Jensen’s inequality. The method is flexible, and by way of illustration, we show how it applies starting from other important dynamics in optimization. We consider the case in which the initial dynamic is the regularized Newton method, then the case in which the starting dynamic is the differential inclusion associated with a convex lower semicontinuous potential, and finally we show that the technique can be naturally extended to the case of a monotone cocoercive operator. Our approach leads to parallel algorithmic results, which we study in the case of fast gradient and proximal algorithms. Our averaging technique shows new links between the Nesterov and Ravine methods. Funding: The research of R.I. Boţ and D.-K. Nguyen was supported by the Austrian Science Fund (FWF), projects W 1260 and P 34922-N.

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38

Bagy, Akram Chahid, Zaki Chbani, and Hassan Riahi. "Strong Convergence of Trajectories via Inertial Dynamics Combining Hessian-Driven Damping and Tikhonov Regularization for General Convex Minimizations." Numerical Functional Analysis and Optimization, October 17, 2023, 1–29. http://dx.doi.org/10.1080/01630563.2023.2262828.

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39

Csetnek, Ernö Robert, and Mikhail A. Karapetyants. "Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling." Journal of Optimization Theory and Applications, August 21, 2024. http://dx.doi.org/10.1007/s10957-024-02500-8.

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AbstractIn a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a setting where the newly introduced system preserves and even improves the well-known fast convergence properties of the function and Moreau envelope along the trajectories and also of the gradient of Moreau envelope due to the presence of time scaling. Moreover, in a different setting we prove strong convergence of the trajectories to the element of minimal norm from the set of all minimizers of the objective. The manuscript concludes with various numerical results.

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40

Aksenov, Vitalii, Alexey Vasyukov, and Katerina Beklemysheva. "Acquiring elastic properties of thin composite structure from vibrational testing data." Journal of Inverse and Ill-posed Problems, January 16, 2024. http://dx.doi.org/10.1515/jiip-2022-0081.

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Abstract The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments.

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41

"On the strong convergence of an inertial proximal algorithm with a time scale, Hessian-driven damping, and a Tikhonov regularization term." Journal of Applied and Numerical Optimization 6, no. 2 (2024). http://dx.doi.org/10.23952/jano.6.2024.2.06.

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42

Boţ, Radu Ioan, Ernö Robert Csetnek, and Dang-Khoa Nguyen. "Fast Optimistic Gradient Descent Ascent (OGDA) Method in Continuous and Discrete Time." Foundations of Computational Mathematics, November 29, 2023. http://dx.doi.org/10.1007/s10208-023-09636-5.

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AbstractIn the framework of real Hilbert spaces, we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V. The starting point of our investigations is a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates of order $$o \left( \frac{1}{t\beta (t)} \right) $$ o 1 t β ( t ) for $$\Vert V(z(t))\Vert $$ ‖ V ( z ( t ) ) ‖ , where $$z(\cdot )$$ z ( · ) denotes the generated trajectory and $$\beta (\cdot )$$ β ( · ) is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of V. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that the generated sequence of iterates $$(z_k)_{k \ge 0}$$ ( z k ) k ≥ 0 shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order $$o \left( \frac{1}{k\beta _k} \right) $$ o 1 k β k for $$\Vert V(z^k)\Vert $$ ‖ V ( z k ) ‖ and the restricted gap function, where $$(\beta _k)_{k \ge 0}$$ ( β k ) k ≥ 0 is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm, we show by additionally assuming that the operator V is Lipschitz continuous convergence rates of order $$o \left( \frac{1}{k} \right) $$ o 1 k for $$\Vert V(z^k)\Vert $$ ‖ V ( z k ) ‖ and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition to these, we prove for both algorithms the convergence of the iterates to a zero of V. To our knowledge, our study exhibits the best-known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods designed to solve monotone equations governed by monotone and Lipschitz continuous operators.

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43

Espindola-Carmona, Armando, Ridvan Örsvuran, P. Martin Mai, Ebru Bozdağ, and Daniel B. Peter. "Resolution and trade-offs in global anelastic full-waveform inversion." Geophysical Journal International, November 29, 2023. http://dx.doi.org/10.1093/gji/ggad462.

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Summary Improving the resolution of seismic anelastic models is critical for a better understanding of the Earth’s subsurface structure and dynamics. Seismic attenuation plays a crucial role in estimating water content, partial melting, and temperature variations in the Earth’s crust and mantle. However, compared to seismic wavespeed models, seismic attenuation tomography models tend to be less resolved. This is due to the complexity of amplitude measurements and the challenge of isolating the effect of attenuation in the data from other parameters. Physical dispersion caused by attenuation also affects seismic wavespeeds, and neglecting scattering/defocusing effects in classical anelastic models can lead to biased results. To overcome these challenges, it is essential to account for the full 3D complexity of seismic wave propagation. Although various synthetic tests have been conducted to validate anelastic Full-Waveform Inversion (FWI), there is still a lack of understanding regarding the trade-off between elastic and anelastic parameters, as well as the variable influence of different parameter classes on the data. In this context, we present a synthetic study to explore different strategies for global anelastic inversions. To assess the resolution and sensitivity for different misfit functions, we first perform mono-parameter inversions by inverting only for attenuation. Then, to study trade-offs between parameters and resolution, we test two different inversion strategies (simultaneous and sequential) to jointly constrain the elastic and anelastic parameters. We found that a sequential inversion strategy performs better for imaging attenuation than a simultaneous inversion. We also demonstrate the dominance of seismic wavespeeds over attenuation, underscoring the importance of determining a good approximation of the Hessian matrix and suitable damping factors for each parameter class.

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44

Adly, Samir, and Hedy Attouch. "Accelerated optimization through time-scale analysis of inertial dynamics with asymptotic vanishing and Hessian-driven dampings." Optimization, July 3, 2024, 1–38. http://dx.doi.org/10.1080/02331934.2024.2359540.

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Journal articles on the topic 'Hessian damping'

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