Academic literature on the topic 'Hessian damping'

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.

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.

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.

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.

Motivé par l'omniprésence de l'optimisation dans de nombreux domaines de la science et de l'ingénierie, en particulier dans la science des données, ce manuscrit de thèse exploite le lien étroit entre les systèmes dynamiques dissipatifs à temps continu et les algorithmes d'optimisation pour fournir une analyse systématique du comportement global et local de plusieurs systèmes du premier et du second ordre, en se concentrant sur le cadre convexe, stochastique et en dimension infinie d'une part, et le cadre non convexe, déterministe et en dimension finie d'autre part. Pour les problèmes de minimisation convexe stochastique dans des espaces de Hilbert réels séparables de dimension infinie, notre proposition clé est de les analyser à travers le prisme des équations différentielles stochastiques (EDS) et des inclusions différentielles stochastiques (IDS), ainsi que de leurs variantes inertielles. Nous considérons d'abord les problèmes convexes différentiables lisses et les EDS du premier ordre, en démontrant une convergence faible presque sûre vers les minimiseurs sous hypothèse d'intégrabilité du bruit et en fournissant une analyse globale et locale complète de la complexité. Nous étudions également des problèmes convexes non lisses composites utilisant des IDS du premier ordre et montrons que, sous des conditions d'intégrabilité du bruit, la convergence faible presque sûre des trajectoires vers les minimiseurs, et avec la régularisation de Tikhonov la convergence forte presque sûre des trajectoires vers la solution de norme minimale. Nous développons ensuite un cadre mathématique unifié pour analyser la dynamique inertielle stochastique du second ordre via la reparamétrisation temporelle et le moyennage de la dynamique stochastique du premier ordre, ce qui permet d'obtenir une convergence faible presque sûre des trajectoires vers les minimiseurs et une convergence rapide des valeurs et des gradients. Ces résultats sont étendus à des EDS plus générales du second ordre avec un amortissement visqueux et Hessien, en utilisant une analyse de Lyapunov spécifique pour prouver la convergence et établir de nouveaux taux de convergence. Enfin, nous étudions des problèmes d'optimisation déterministes non convexes et proposons plusieurs algorithmes inertiels pour les résoudre, dérivés d'équations différentielles ordinaires (EDO) du second ordre combinant à la fois un amortissement visqueux sans vanité et un amortissement géométrique piloté par le Hessien, sous des formes explicites et implicites. Nous prouvons d'abord la convergence des trajectoires en temps continu des EDO vers un point critique pour des objectives vérifiant la propriété de Kurdyka-Lojasiewicz (KL) avec des taux explicites, et génériquement vers un minimum local si l'objective est Morse. De plus, nous proposons des schémas algorithmiques par une discrétisation appropriée de ces EDO et montrons que toutes les propriétés précédentes des trajectoires en temps continu sont toujours valables dans le cadre discret sous réserve d'un choix approprié de la taille du pas
Motivated by the ubiquity of optimization in many areas of science and engineering, particularly in data science, this thesis exploits the close link between continuous-time dissipative dynamical systems and optimization algorithms to provide a systematic analysis of the global and local behavior of several first- and second-order systems, focusing on convex, stochastic, and infinite-dimensional settings on the one hand, and non-convex, deterministic, and finite-dimensional settings on the other hand. For stochastic convex minimization problems in infinite-dimensional separable real Hilbert spaces, our key proposal is to analyze them through the lens of stochastic differential equations (SDEs) and inclusions (SDIs), as well as their inertial variants. We first consider smooth differentiable convex problems and first-order SDEs, demonstrating almost sure weak convergence towards minimizers under integrability of the noise and providing a comprehensive global and local complexity analysis. We also study composite non-smooth convex problems using first-order SDIs, and show under integrability conditions on the noise, almost sure weak convergence of the trajectory towards a minimizer, with Tikhonov regularization almost sure strong convergence of trajectory to the minimal norm solution. We then turn to developing a unified mathematical framework for analyzing second-order stochastic inertial dynamics via time scaling and averaging of stochastic first-order dynamics, achieving almost sure weak convergence of trajectories towards minimizers and fast convergence of values and gradients. These results are extended to more general second-order SDEs with viscous and Hessian-driven damping, utilizing a dedicated Lyapunov analysis to prove convergence and establish new convergence rates. Finally, we study deterministic non-convex optimization problems and propose several inertial algorithms to solve them derived from second-order ordinary differential equations (ODEs) combining both non-vanishing viscous damping and geometric Hessian-driven damping in explicit and implicit forms. We first prove convergence of the continuous-time trajectories of the ODEs to a critical point under the Kurdyka-Lojasiewicz (KL) property with explicit rates, and generically to a local minimum under a Morse condition. Moreover, we propose algorithmic schemes by appropriate discretization of these ODEs and show that all previous properties of the continuous-time trajectories still hold in the discrete setting under a proper choice of the stepsize

This paper aims at optimizing some critical characteristics of a fixed-tank biaxial vehicle, with respect to the lateral performance of the installed tank. For the description of the fixed-tank vehicle, a linear half car model with six degrees of freedom is implemented, subject to many types of road irregularities. The relative position of the tank mountings, with respect to the vehicle frame, as well as their corresponding stiffness and damping characteristics are optimized, such that the maximum values of vertical and rotational acceleration of the tank are minimized, under the geometrical constraints of the vehicle. For the optimization tasks, the BFGS quasi-Newton and the (μ+λ)-Evolution Strategy methods have been implemented. The former outperforms conventional Newton’s methods, due to the secant approximation of the Hessian, while the latter has been shown to perform better in many engineering applications, compared to other categories of EA.