Academic literature on the topic 'Hessian matrices'
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Journal articles on the topic "Hessian matrices"
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Li, Xiang, Shusen Wang, and Zhihua Zhang. "Do Subsampled Newton Methods Work for High-Dimensional Data?" Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 4723–30. http://dx.doi.org/10.1609/aaai.v34i04.5905.
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Subsampled Newton methods approximate Hessian matrices through subsampling techniques to alleviate the per-iteration cost. Previous results require Ω (d) samples to approximate Hessians, where d is the dimension of data points, making it less practical for high-dimensional data. The situation is deteriorated when d is comparably as large as the number of data points n, which requires to take the whole dataset into account, making subsampling not useful. This paper theoretically justifies the effectiveness of subsampled Newton methods on strongly convex empirical risk minimization with high dimensional data. Specifically, we provably require only Θ˜(deffγ) samples for approximating the Hessian matrices, where deffγ is the γ-ridge leverage and can be much smaller than d as long as nγ ≫ 1. Our theories work for three types of Newton methods: subsampled Netwon, distributed Newton, and proximal Newton.
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Ek, David, and Anders Forsgren. "Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function." Computational Optimization and Applications 79, no. 3 (April 28, 2021): 789–816. http://dx.doi.org/10.1007/s10589-021-00277-4.
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AbstractThe main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.
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Coleman, Thomas F., Burton S. Garbow, and Jorge J. Moré. "Software for estimating sparse Hessian matrices." ACM Transactions on Mathematical Software 11, no. 4 (December 1985): 363–77. http://dx.doi.org/10.1145/6187.6190.
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Lu, Yi, Yan Shi, and Jianping Yu. "Kinematic analysis of limited-dof parallel manipulators based on translational/rotational Jacobian and Hessian matrices." Robotica 27, no. 7 (February 27, 2009): 971–80. http://dx.doi.org/10.1017/s0263574709005396.
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SUMMARYThis paper proposes an approach for solving the velocity and acceleration of the limited-dof (dof n < 6) parallel kinematic machines with linear active legs by means of translational/rotational Jacobian and Hessian matrices. First, based on the established or derived constraint and displacement equations, the translational/rotational Jacobian and Hessian matrices are derived. Second, the formulae for solving inverse/forward velocities and accelerations are derived from translational and rotational Jacobian/Hessian matrices. Third, a 2SPR + UPU PKM and a 2SPS + RPRR PKM are illustrated for explaining how to use this method. This approach is simple because it needs neither to eliminate 6-n rows of an n × 6 Jacobian matrix nor to determine the screw or pose of the constrained wrench.
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Ma, W. J., T. M. Wu, J. Hsieh, and S. L. Chang. "Level statistics of Hessian matrices: random matrices with conservation constraints." Physica A: Statistical Mechanics and its Applications 321, no. 1-2 (April 2003): 364–68. http://dx.doi.org/10.1016/s0378-4371(02)01796-x.
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FONSECA, IRENE, GIOVANNI LEONI, and ROBERTO PARONI. "ON HESSIAN MATRICES IN THE SPACE BH." Communications in Contemporary Mathematics 07, no. 04 (August 2005): 401–20. http://dx.doi.org/10.1142/s0219199705001805.
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An extension of Alberti's result to second order derivatives is obtained. Precisely, if Ω is an open subset of ℝN and if f ∈ L1(Ω;ℝN × N) is symmetric-valued, then there exist u ∈ W1,1 (Ω) with ∇u ∈ BV(Ω;ℝN) and a constant C > 0 depending only on N such that [Formula: see text] and [Formula: see text]
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Erleben, Kenny, and Sheldon Andrews. "Solving inverse kinematics using exact Hessian matrices." Computers & Graphics 78 (February 2019): 1–11. http://dx.doi.org/10.1016/j.cag.2018.10.012.
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Tuan, Nguyen Thai Minh, Chung Thanh Pham, Do Dang Khoa, and Phan Dang Phong. "KINEMATIC AND DYNAMIC ANALYSIS OF MULTIBODY SYSTEMS USING THE KRONECKER PRODUCT." Vietnam Journal of Science and Technology 57, no. 1 (February 18, 2019): 112. http://dx.doi.org/10.15625/2525-2518/57/1/12285.
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This paper employ Khang’s definition of the partial derivative of a matrix with respect to a vector and the Kronecker product to define translational and rotational Hessian matrices. With these definitions, the generalized velocities in the expression of a linear acceleration or an angular acceleration are collected into a quadratic term. The relations of Jacobian and Hessian matrices in relative motion are then established. A new matrix form of Lagrange’s equations showing clearly the quadratic term of generalized velocities is also introduced.
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Tůma, Miroslav. "A note on direct methods for approximations of sparse Hessian matrices." Applications of Mathematics 33, no. 3 (1988): 171–76. http://dx.doi.org/10.21136/am.1988.104300.
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Jiang, Bowu, and Jianfeng Zhang. "Least-squares migration with a blockwise Hessian matrix: A prestack time-migration approach." GEOPHYSICS 84, no. 4 (July 1, 2019): R625—R640. http://dx.doi.org/10.1190/geo2018-0533.1.
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We have developed an explicit inverse approach with a Hessian matrix for the least-squares (LS) implementation of prestack time migration (PSTM). A full Hessian matrix is divided into a series of computationally tractable small-sized matrices using a localized approach, thus significantly reducing the size of the inversion. The scheme is implemented by dividing the imaging volume into a series of subvolumes related to the blockwise Hessian matrices that govern the mapping relationship between the migrated result and corresponding reflectivity. The proposed blockwise LS-PSTM can be implemented in a target-oriented fashion. The localized approach that we use to modify the Hessian matrix can eliminate the boundary effects originating from the blockwise implementation. We derive the explicit formula of the offset-dependent Hessian matrix using the deconvolution imaging condition with an analytical Green’s function of PSTM. This avoids the challenging task of estimating the source wavelet. Moreover, migrated gathers can be generated with the proposed scheme. The smaller size of the blockwise Hessian matrix makes it possible to incorporate the total-variation regularization into the inversion, thus attenuating noises significantly. We revealed the proposed blockwise LS-PSTM with synthetic and field data sets. Higher quality common-reflection-point gathers of the field data are obtained.
Dissertations / Theses on the topic "Hessian matrices"
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Medeiros, Rainelly Cunha de. "Degenerations of classical square matrices and their determinantal structure." Universidade Federal da Paraíba, 2017. http://tede.biblioteca.ufpb.br:8080/handle/tede/9318.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In thisthesis,westudycertaindegenerations/specializationsofthegenericsquare matrix overa eld k of characteristiczeroalongitsmainrelatedstructures,suchthe determinantofthematrix,theidealgeneratedbyitspartialderivatives,thepolarmap de ned bythesederivatives,theHessianmatrixandtheidealofsubmaximalminorsof the matrix.Thedegenerationtypesofthegenericsquarematrixconsideredhereare: (1) degenerationby\cloning"(repeating)avariable;(2)replacingasubsetofentriesby zeros, inastrategiclayout;(3)furtherdegenerationsoftheabovetypesstartingfrom certain specializationsofthegenericsquarematrix,suchasthegenericsymmetric matrix andthegenericsquareHankelmatrix.Thefocusinallthesedegenerations is intheinvariantsdescribedabove,highlightingonthehomaloidalbehaviorofthe determinantofthematrix.Forthis,weemploytoolscomingfromcommutativealgebra, with emphasisonidealtheoryandsyzygytheory.
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Gower, Robert Mansel. "Diferenciação automática de matrizes Hessianas." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306026.
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Orientador: Margarida Pinheiro MelloDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Dentro do contexto de programação não linear, vários algoritmos resumem-se à aplicação do método de Netwon aos sistemas constituídos pelas condições de primeira ordem de Lagrange. Nesta classe de métodos é necessário calcular a matriz hessiana. Nosso foco é o cálculo exato, dentro da precisão da máquina, de matrizes hessianas usando diferenciação automática. Para esse fim, exploramos o cálculo da matriz hessiana sob dois pontos de vista. O primeiro é um modelo de grafo que foca nas simetrias que ocorrem no processo do cálculo da hessiana. Este ângulo propicia a intuição de como deve ser calculada a hessiana e leva ao desenvolvimento de um novo método de modo reverso para o cálculo de matrizes hessianas denominado edge pushing. O segundo ponto de vista é uma representação puramente algébrica que reduz o cálculo da hessiana à avaliação de uma expressão. Esta expressão pode ser usada para demonstrar algoritmos já existentes e projetar novos. Para ilustrar, deduzimos dois novos algoritmos, edge pushing e um novo algoritmo de modo direto, e uma série de outros métodos conhecidos [1], [20, p.157] e [9]. Apresentamos estudos teóricos e empíricos sobre o algoritmo edge pushing. Analisamos sua complexidade temporal e de uso de memória. Implementamos o algoritmo como um driver do pacote ADOL-C [19] e efetuamos testes computacionais, comparando sua performance com à de dois outros drivers em dezesseis problemas da coleção CUTE [5]. Os resultados indicam que o novo algoritmo é muito promissor. Pequenas modificações em edge pushing produzem um novo algoritmo, edge pushing sp, para o cálculo da esparsidade de matrizes hessianas, um passo necessário de uma classe de métodos que calculam a matriz hessiana usando colorações de grafos, [14, 19, 30]. Estudos de complexidade e testes numéricos são realizados comparando o novo método contra um outro recentemente desenvolvido [30] e os testes favorecem o novo algoritmo edge pushing sp. No capítulo final, motivado pela disponibilidade crescente de computadores com multiprocesadores, investigamos o processamento em paralelo do cálculo de matrizes hessianas. Examinamos o cálculo em paralelo de matrizes hessianas de funções parcialmente separáveis. Apresentamos uma abordagem desenvolvida para o cômputo em paralelo que pode ser usando em conjunto com qualquer método de cálculo de hessiana e outra estratégia específica para métodos de modo reverso. Testes são executados em um computador com memória compartilhada usando a interface de programação de aplicativo OpenMP
Abstract: In the context of nonlinear programming, many algorithms boil down to the application of Newton's method to the system constituted by the first order Lagrangian conditions. The calculation of Hessian matrices is necessary in this class of solvers. Our focus is on the exact calculation, within machine precision, of Hessian matrices through automatic differentiation. To this end, we detail the calculations of the Hessian matrix under two points of view. The first is an intuitive graph model that focuses on what symmetries occur throughout the Hessian calculation. This provides insight on how one should calculate the Hessian matrix, and we use this enlightened perspective to deduce a new reverse Hessian algorithm called edge pushing. The second viewpoint is a purely algebraic representation of the Hessian calculation via a closed formula. This formula can be used to demonstrate existing algorithms and design new ones. In order to illustrate, we deduce two new algorithms, edge pushing and a new forward algorithm, and a series of other known Hessian methods [1], [20, p.157] and [9]. We present theoretical and empirical studies of the edge pushing algorithm, establishing memory and temporal bounds, and comparing the performance of its computer implementation against that of two algorithms available as drivers of the software ADOL-C [14, 19, 30] on sixteen functions from the CUTE collection [5]. Test results indicate that the new algorithm is very promising. As a by-product of the edge pushing algorithm, we obtain an efficient algorithm, edge pushing sp, for automatically obtaining the sparsity pattern of Hessian matrices, a necessary step in a class of methods used for computing Hessian matrices via graph coloring, [14, 19, 30]. Complexity bounds are developed and numerical tests are carried out comparing the new sparsity detection algorithm against a recently developed method [30] and the results favor the new edge pushing sp algorithm. In the final chapter, motivated by the increasing commercial availability of multiprocessors, we investigate the implementation of parallel versions of the edge pushing algorithm. We address the concurrent calculation of Hessian matrices of partially separable functions. This includes a general approach to be used in conjunction with any Hessian software, and a strategy specific to reverse Hessian methods. Tests are carried out on a shared memory computer using the OpenMP paradigm
Mestrado
Analise Numerica
Mestre em Matemática Aplicada
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Chen, Zhangchi. "Differential invariants of parabolic surfaces and of CR hypersurfaces; Directed harmonic currents near non-hyperbolic linearized singularities; Hartogs’ type extension of holomorphic line bundles; (Non-)invertible circulant matrices On differential invariants of parabolic surfaces A counterexample to Hartogs’ type extension of holomorphic line bundles Directed harmonic currents near non-hyperbolic linearized singularities Affine Homogeneous Surfaces with Hessian rank 2 and Algebras of Differential Invariants On nonsingularity of circulant matrices." Thesis, université Paris-Saclay, 2021. http://www.theses.fr/2021UPASM005.
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La thèse se compose de 6 articles. (1) Nous calculons les générateurs des SA₃(ℝ)-invariants pour les surfaces paraboliques. (2) Nous calculons les invariants rigides relatifs pour les hypersurfaces rigides 2-non-dégénérées de rang de Levi constant 1 dans ℂ³: V₀, I₀, Q₀ ayant 11, 52, 824 monômes au numérateur. (3) Nous organisons tous les modèles affinement homogènes non-dégénérés dans ℂ³ en branches inéquivalentes. (4) Pour un courant harmonique dirigé autour d'une singularité linéarisée non-hyperbolique qui ne charge pas les séparatrices triviales dont l'extension triviale à travers 0 est ddc-fermée, nous démontrons que le nombre de Lelong en 0 est : 4.1) strictement positif si λ>0 ; 4.2) nul si λ est rationnel et négatif ; 4.3) nul si λ est négatif et si T est invariant sous l'action d'un sous-groupe cofini du groupe de monodromie. (5) Nous construisons des fibrés holomorphes en droites en toute dimension n>=2 non-prolongeables au sens de Hartogs. (6) Nous montrons que les matrices circulantes ayant k entrées 1 et k+1 entrées 0 dans leur première rangée sont toujours non singulières lorsque 2k+1 est soit une puissance d'un nombre premier, soit un produit de deux nombres premiers distincts. Pour tout autre entier 2k+1, nous exhibons une matrice circulante singulièreThe thesis consists of 6 papers. (1) We calculate the generators of SA₃(ℝ)-invariants for parabolic surfaces. (2) We calculate rigid relative invariants for rigid constant Levi-rank 1 and 2-non-degenerate hypersurfaces in ℂ³: V₀, I₀, Q₀ having 11, 52, 824 monomials in their numerators. (3) We organize all affinely homogeneous nondegenerate surfaces in ℂ³ in inequivalent branches. (4) For a directed harmonic current near a non-hyperbolic linearized singularity which does not give mass to any of the trivial separatrices and whose trivial extension across 0 is ddc-closed, we show that the Lelong number at 0 is: 4.1) strictly positive if the eigenvalue λ>0; 4.2) zero if λ is a negative rational number; 4.3) zero if λ<0 and if T is invariant under the action of some cofinite subgroup of the monodromy group. (5) We construct non-extendable, in the sense of Hartogs, holomorphic line bundles in any dimension n>=2. (6) We show that circulant matrices having k ones and k+1 zeros in the first row are always nonsingular when 2k+1 is either a power of a prime, or a product of two distinct primes. For any other integer 2k+1 we exhibit a singular circulant matrix
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Quandt, Joana B. O. "Minimização de funções com restrições canalizadas utilizando falsas hessianas de banda." reponame:Repositório Institucional da UFSC, 1996. https://repositorio.ufsc.br/xmlui/handle/123456789/158013.
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Tese (doutorado) - Universidade Federal de Santa Catarina, Centro TecnologicoMade available in DSpace on 2016-01-08T20:29:53Z (GMT). No. of bitstreams: 1 105390.pdf: 1488361 bytes, checksum: ad6b7b6f2f4ba202017cd97c1c9597a7 (MD5) Previous issue date: 1996
Foi proposto um método para minimização de funções não lineares com restrições canalizadas. Como caso particular foi obtido um método de minimização irrestrita. O método apresentado é do tipo região de confiança, e sua característica principal é que não são utilizadas matrizes Hessianas verdadeiras, mas aproximações do tipo banda para as Hessianas. Essas matrizes de aproximação são também simétricas, e são obtidas por técnicas secantes. Esse tipo de estrutura prefixada permite grande economia de memória computacional, permitindo o uso do algoritmo para problemas de grande porte. Foram apresentados resultados computacionais, quando se utiliza aproximações diagonais, tridiagonas ou pentadiagonais.
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Esteve, yague Carlos. "Étude qualitative de trois problèmes paraboliques non-linéaires." Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCD033.
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Cette thèse est consacrée à l'étude de trois problèmes paraboliques non linéaires : Premièrement, nous considérons un modèle de systèmes micro-électro-mécaniques (MEMS) avec permittivité diélectrique variable. Le modèle est basé sur une équation parabolique avec non-linéarité singulière, qui décrit la déformation dynamique d'une plaque élastique sous les effets d'un potentiel électrostatique. Nous étudions le phénomène de touchdown, ou quenching. Avec le but de contrôler l'ensemble de touchdown, nous donnons des résultats concernant la localisation du touchdown, en termes du profil de permittivité. Dans la deuxième partie de la thèse, nous étudions une équation de Hamilton-Jacobi avec diffusion dans un domaine borné avec conditions de Dirichlet nulles au bord. On analyse l'explosion du gradient (GBU) qui peut avoir lieu sur le bord du domaine. Dans un article précédent, il a été démontré, pour des domaines très particuliers (domaines localement plats et disques), qu'il est possible de construire des solutions pour lesquelles l'ensemble de GBU est réduit à un seul point. Nous démontrons qu'il est possible de construire ce type de solutions pour une large classe de domaines, où la courbure n'est pas forcement constante près du point de GBU.Dans la dernière partie de la thèse, nous étudions le problème d'évolution associé à la j-ème valeur propre de la matrice Hessienne. On démontre tout d'abord l'existence d'une (unique) solution de viscosité, qui peut être approximée par la fonction valeur d'un jeu à deux joueurs et somme nulle, quand la longueur du pas du jeu tend vers 0.On démontre ensuite la convergence exponentielle des solutions du problème d'évolution vers l'unique solution stationnaire. Finalement, pour des cas particuliers (avec données au bord affines), on démontre que la solution coïncide avec la solution stationnaire en temps finiThis thesis is concerned with the study of three nonlinear parabolic problems : We start with a mathematical model for a micro-electro-mechanical system (MEMS) with variable dielectric permittivity. The model is based on a parabolic equation with singular nonlinearity which describes the dynamic deffection of an elastic plate under the effect of an electrostatic potential. We study the touchdown, or quenching, phenomenon. With the aim of controlling the touchdown set, we give results concerning the touchdownl ocalization in terms of the permittivity profile. In the second part of the thesis, we study a diffusive Hamilton-Jacobi equation in a bounded domain with zero Dirichlet boundary conditions. We analyze the gradient blow-up (GBU) that solutions can exhibit on the boundary of the domain. In a previous work, it was shown that single-point GBU solutions can be constructed in very particular domains, namely, locally fat domains and disks. We prove the existence of this kind ofsolutions for a large family of domains, for which the curvature of the domain may be nonconstant near the GBU point. In the last part of the thesis, we study the evolution problem associated to the j-th eigenvalue of the Hessian matrix. First, we show the existence of a (unique) viscosity solution, which can be approximated by the value function of a two-player zero-sumgame as the step length of the game goes to zero. Then, we show that solutions to this evolution problem converge exponentially fast to the unique stationary solution as t goes to ∞. Finally, we show that in some special cases (for affine boundary data) the solution coincides with the stationary solution in finite time
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Nguyen, Thi Van Oanh. "Spectroscopie et stabilité des hydrocarbures aromatiques polycycliques dans les conditions du milieu interstellaire." Paris 11, 2003. http://www.theses.fr/2003PA112299.
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Ce travail porte sur la dynamique intramoléculaire des Hydrocarbures Aromatiques Polycycliques soumis aux conditions du milieu interstellaire (PAH isolé, rotationnellement froid mais vibrationnellement excité). Des études théoriques et expérimentales ont été menées sur leur deux voies de relaxation: émission IR ou fragmentation. Le spectre d'absorption IR a été obtenu par dynamique moléculaire couplée à une méthode semi-classique Adiabatic Switching. La dynamique a été réalisée sur une surface de potentiel semi-empirique Tight-Binding dans le but de simuler tous types de PAHs pouvant dépasser une centaine de carbones. Une paramétrisation du potentiel adaptée aux PAHs a été développée ainsi qu'un modèle donnant la densité d'états anharmonique quantique. La simulation spectrale reproduit les grandes tendances en fonction des variables pertinentes en astrophysique: rôle de l'ionisation (fort changement de l'allure du spectre et de l'intensité totale absolue), et de la température (décalage vers le rouge, élargissement et modification des intensités des bandes), effet d'anharmonicité (énergie de point zéro, fréquences), et de structure (compacité, cycle pentagonal et taille). La cinétique de fragmentation (induite par absorption séquentielle de photons) d'un hydrogène du cation fluorène (ionisation REMPI) a été étudiée à l'aide d'un jet supersonique et d'un spectromètre de masse. Cette méthode expérimentale originale a permis de déterminer la section efficace absolue d'absorption, d'analyser son atténuation avec le nombre de photons absorbés et l'évolution de la constante de dissociation dans un domaine d'énergie relativement large. Une attention particulière a été portée sur les techniques d'analyse des données. Un ajustement libre de la variation de cette constante est proche de celui du modèle statistique PTD mais très différent à basse énergie du modèle RRK. L'énergie d'activation obtenue par ces deux modèles est compatible avec celle déduite du potentiel Tight-BindingThis work deals with the intramolecular dynamics of Polycyclic Aromatic Hydrocarbons submitted to interstellar medium conditions (isolated, rotationally cold but vibrationnally excited PAHs). Theoretical and experimental analyses of the two relaxation channels have been developed: IR emission or fragmentation. The IR absorption spectrum was obtained by classical Molecular Dynamics (on a Tight-Binding semi-empirical potential surface) coupled with the Adiabatic Switching semi-classical method in order to simulate any PAHs (even heavier than a hundred carbons). A new parametrization fitted to these aromatic systems and a quantum anharmonic DOS model were developed. This spectral simulation reproduces the main features as a function of the variables relevant to astrophysics: role of the ionization (strong change of the spectrum profile and of the total absolute intensity), temperature influence (red shift, broadening and modification of the intensity of the peaks), anharmonicity (zero point energy, frequencies), structure effects (compactness, pentagonal cycle and size). The H-loss channel, induced by photon sequential absorption, of the fluorene cation (REMPI ionization) was studied using a supersonic molecular beams and a mass spectrometer. Our original method led to the determination of the absolute absorption cross-section, its attenuation with the number of absorbed photons and of the evolution of the decay rate in a relatively large energetic range. A particular attention was focused on the data analysis techniques. A free fit of this rate is close to the PTD statistical model, but shows a strong discrepancy with the RRK model mainly at low energy. The activation energy obtained through these two models are compatible with that deduced from the Tight-Binding potential
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Saade, Alaa. "Spectral inference methods on sparse graphs : theory and applications." Thesis, Paris Sciences et Lettres (ComUE), 2016. http://www.theses.fr/2016PSLEE024/document.
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Face au déluge actuel de données principalement non structurées, les graphes ont démontré, dans une variété de domaines scientifiques, leur importance croissante comme language abstrait pour décrire des interactions complexes entre des objets complexes. L’un des principaux défis posés par l’étude de ces réseaux est l’inférence de propriétés macroscopiques à grande échelle, affectant un grand nombre d’objets ou d’agents, sur la seule base des interactions microscopiquesqu’entretiennent leurs constituants élémentaires. La physique statistique, créée précisément dans le but d’obtenir les lois macroscopiques de la thermodynamique à partir d’un modèle idéal de particules en interaction, fournit une intuition décisive dans l’étude des réseaux complexes.Dans cette thèse, nous utilisons des méthodes issues de la physique statistique des systèmes désordonnés pour mettre au point et analyser de nouveaux algorithmes d’inférence sur les graphes. Nous nous concentrons sur les méthodes spectrales, utilisant certains vecteurs propres de matrices bien choisies, et sur les graphes parcimonieux, qui contiennent une faible quantité d’information. Nous développons une théorie originale de l’inférence spectrale, fondée sur une relaxation de l’optimisation de certaines énergies libres en champ moyen. Notre approche est donc entièrement probabiliste, et diffère considérablement des motivations plus classiques fondées sur l’optimisation d’une fonction de coût. Nous illustrons l’efficacité de notre approchesur différents problèmes, dont la détection de communautés, la classification non supervisée à partir de similarités mesurées aléatoirement, et la complétion de matricesIn an era of unprecedented deluge of (mostly unstructured) data, graphs are proving more and more useful, across the sciences, as a flexible abstraction to capture complex relationships between complex objects. One of the main challenges arising in the study of such networks is the inference of macroscopic, large-scale properties affecting a large number of objects, based solely on he microscopic interactions between their elementary constituents. Statistical physics, precisely created to recover the macroscopic laws of thermodynamics from an idealized model of interacting particles, provides significant insight to tackle such complex networks.In this dissertation, we use methods derived from the statistical physics of disordered systems to design and study new algorithms for inference on graphs. Our focus is on spectral methods, based on certain eigenvectors of carefully chosen matrices, and sparse graphs, containing only a small amount of information. We develop an original theory of spectral inference based on a relaxation of various meanfield free energy optimizations. Our approach is therefore fully probabilistic, and contrasts with more traditional motivations based on the optimization of a cost function. We illustrate the efficiency of our approach on various problems, including community detection, randomized similarity-based clustering, and matrix completion
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Chahine, Chaza. "Fusion d'informations par la théorie de l'évidence pour la segmentation d'images." Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1030/document.
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La fusion d’informations a été largement étudiée dans le domaine de l’intelligence artificielle. Une information est en général considérée comme imparfaite. Par conséquent, la combinaison de plusieurs sources d’informations (éventuellement hétérogènes) peut conduire à une information plus globale et complète. Dans le domaine de la fusion on distingue généralement les approches probabilistes et non probabilistes dont fait partie la théorie de l’évidence, développée dans les années 70. Cette méthode permet de représenter à la fois, l’incertitude et l’imprécision de l’information, par l’attribution de fonctions de masses qui s’appliquent non pas à une seule hypothèse (ce qui est le cas le plus courant pour les méthodes probabilistes) mais à un ensemble d’hypothèses. Les travaux présentés dans cette thèse concernent la fusion d’informations pour la segmentation d’images.Pour développer cette méthode nous sommes partis de l’algorithme de la « Ligne de Partage des Eaux » (LPE) qui est un des plus utilisés en détection de contours. Intuitivement le principe de la LPE est de considérer l’image comme un relief topographique où la hauteur d’un point correspond à son niveau de gris. On suppose alors que ce relief se remplit d’eau par des sources placées au niveau des minima locaux de l’image, formant ainsi des bassins versants. Les LPE sont alors les barrages construits pour empêcher les eaux provenant de différents bassins de se mélanger. Un problème de cette méthode de détection de contours est que la LPE directement appliquée sur l’image engendre une sur-segmentation, car chaque minimum local engendre une région. Meyer et Beucher ont proposé de résoudre cette question en spécifiant un ensemble de marqueurs qui seront les seules sources d’inondation du relief. L'extraction automatique des marqueurs à partir des images ne conduit pas toujours à un résultat satisfaisant, en particulier dans le cas d'images complexes. Plusieurs méthodes ont été proposées pour déterminer automatiquement ces marqueurs.Nous nous sommes en particulier intéressés à l’approche stochastique d’Angulo et Jeulin qui estiment une fonction de densité de probabilité (fdp) d'un contour (LPE) après M simulations de la segmentation LPE classique. N marqueurs sont choisis aléatoirement pour chaque réalisation. Par conséquent, une valeur de fdp élevée est attribuée aux points de contours correspondant aux fortes réalisations. Mais la décision d’appartenance d’un point à la « classe contour » reste dépendante d’une valeur de seuil. Un résultat unique ne peut donc être obtenu.Pour augmenter la robustesse de cette méthode et l’unicité de sa réponse, nous proposons de combiner des informations grâce à la théorie de l’évidence.La LPE se calcule généralement à partir de l’image gradient, dérivée du premier ordre, qui donne une information globale sur les contours dans l’image. Alors que la matrice Hessienne, matrice des dérivées d’ordre secondaire, donne une information plus locale sur les contours. Notre objectif est donc de combiner ces deux informations de nature complémentaire en utilisant la théorie de l’évidence. Les différentes versions de la fusion sont testées sur des images réelles de la base de données Berkeley. Les résultats sont comparés avec cinq segmentations manuelles fournies, en tant que vérités terrain, avec cette base de données. La qualité des segmentations obtenues par nos méthodes sont fondées sur différentes mesures: l’uniformité, la précision, l’exactitude, la spécificité, la sensibilité ainsi que la distance métrique de HausdorffInformation fusion has been widely studied in the field of artificial intelligence. Information is generally considered imperfect. Therefore, the combination of several sources of information (possibly heterogeneous) can lead to a more comprehensive and complete information. In the field of fusion are generally distinguished probabilistic approaches and non-probabilistic ones which include the theory of evidence, developed in the 70s. This method represents both the uncertainty and imprecision of the information, by assigning masses not only to a hypothesis (which is the most common case for probabilistic methods) but to a set of hypothesis. The work presented in this thesis concerns the fusion of information for image segmentation.To develop this method we start with the algorithm of Watershed which is one of the most used methods for edge detection. Intuitively the principle of the Watershed is to consider the image as a landscape relief where heights of the different points are associated with grey levels. Assuming that the local minima are pierced with holes and the landscape is immersed in a lake, the water filled up from these minima generate the catchment basins, whereas watershed lines are the dams built to prevent mixing waters coming from different basins.The watershed is practically applied to the gradient magnitude, and a region is associated with each minimum. Therefore the fluctuations in the gradient image and the great number of local minima generate a large set of small regions yielding an over segmented result which can hardly be useful. Meyer and Beucher proposed seeded watershed or marked-controlled watershed to surmount this oversegmentation problem. The essential idea of the method is to specify a set of markers (or seeds) to be considered as the only minima to be flooded by water. The number of detected objects is therefore equal to the number of seeds and the result is then markers dependent. The automatic extraction of markers from the images does not lead to a satisfying result especially in the case of complex images. Several methods have been proposed for automatically determining these markers.We are particularly interested in the stochastic approach of Angulo and Jeulin who calculate a probability density function (pdf) of contours after M simulations of segmentation using conventional watershed with N markers randomly selected for each simulation. Therefore, a high pdf value is assigned to strong contour points that are more detected through the process. But the decision that a point belong to the "contour class" remains dependent on a threshold value. A single result cannot be obtained.To increase the robustness of this method and the uniqueness of its response, we propose to combine information with the theory of evidence.The watershed is generally calculated on the gradient image, first order derivative, which gives comprehensive information on the contours in the image.While the Hessian matrix, matrix of second order derivatives, gives more local information on the contours. Our goal is to combine these two complementary information using the theory of evidence. The method is tested on real images from the Berkeley database. The results are compared with five manual segmentation provided as ground truth, with this database. The quality of the segmentation obtained by our methods is tested with different measures: uniformity, precision, recall, specificity, sensitivity and the Hausdorff metric distance
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Safeea, Mohammad. "Des robots manipulateurs collaboratifs sûrs." Thesis, Paris, HESAM, 2020. http://www.theses.fr/2020HESAE036.
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Les manipulateurs industriels collaboratifs ouvrent une nouvelle ère dans la fabrication flexible, où les robots et les humains sont capables de coexister et de travailler ensemble. Cependant, divers défis persistent pour parvenir à une collaboration complète entre les robots et les humains en milieu industriel. Dans cette thèse, deux défis principaux - la sécurité et la collaboration - sont abordés pour atteindre cet objectif. Concernant la sécurité, la thèse présente une méthode d'évitement des collisions en temps réel qui permet au robot d'ajuster les chemins générés hors ligne pour une tâche industrielle, tout en évitant les collisions avec les humains à proximité. En outre, la thèse a présenté une nouvelle méthode pour effectuer un mouvement d'évitement de collision réactif, en utilisant la méthode de Newton du second ordre qui offre divers avantages par rapport aux méthodes traditionnelles utilisées dans la littérature. Sur la collaboration, la thèse présente un mode de guidage manuel précis comme alternative au mode de guidage actuel pour effectuer des opérations de positionnement précis de l'effecteur terminal du robot d’une manière simple et intuitive. La thèse présente également de nouvelles contributions à la formulation mathématique de la dynamique des robots, y compris un algorithme récursif pour calculer la matrice de masse des robots sériels avec un coût minimal du second ordre et un algorithme récursif pour calculer efficacement les symboles de Christoffel. Tous les algorithmes présentés sont validés soit en simulation, soit dans un scénario réelCollaborative industrial manipulators are ushering a new era in flexible manufacturing, where robots and humans are allowed to coexist and work side by side. However, various challenges still persist in achieving full human robot collaboration on the factory floor. In this thesis two main challenges - safety and collaboration - for achieving that goal are addressed. On safety, the thesis presents a real-time collision avoidance method which allows the robot to adjust the offline generated paths of the industrial task in real-time for avoiding collisions with humans nearby. In addition, the thesis presented a new method for performing the reactive collision avoidance motion using second order Newton method which offers various advantages over the traditional methods in the literature. On collaboration, the thesis presents the precision hand-guiding as an alternative to the teach-pendant for performing precise positioning operations of the robot’s end-effector in a simple and intuitive manner. The thesis also presents new contributions into the mathematical formulation of robot dynamics, including a recursive algorithm for calculating the mass matrix of serially linked robots with a minimal second order cost, and a recursive algorithm for calculating Christoffel symbols efficiently. All the presented algorithms are validated either in simulation or in a real-world scenario
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Snežana, Đorđević. "Izbor parametara kod gradijentnih metoda za probleme optimizacije bez ograničenja." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2015. https://www.cris.uns.ac.rs/record.jsf?recordId=94106&source=NDLTD&language=en.
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Posmatra se problem optimizacije bez ograničenja. Za rešavanjeproblema optimizacije bez ograničenja postoji mnoštvo raznovrsnihmetoda. Istraživanje ovde motivisano je potrebom za metodama kojeće brzo konvergirati.Cilj je sistematizacija poznatih rezultata, kao i teorijska i numeričkaanaliza mogućnosti uvođenja parametra u gradijentne metode.Najpre se razmatra problem minimizacije konveksne funkcije višepromenljivih.Problem minimizacije konveksne funkcije više promenljivih ovde serešava bez izračunavanja matrice hesijana, što je naročito aktuelno zasisteme velikih dimenzija, kao i za probleme optimizacije kod kojihne raspolažemo ni tačnom vrednošću funkcije cilja, ni tačnomvrednošću gradijenta. Deo motivacije za istraživanjem ovde leži i upostojanju problema kod kojih je funkcija cilja rezultat simulacija.Numerički rezultati, predstavljeni u Glavi 6, pokazuju da uvođenjeizvesnog parametra može biti korisno, odnosno, dovodi do ubrzanjaodređenog metoda optimizacije.Takođe se predstavlja jedan novi hibridni metod konjugovanoggradijenta, kod koga je parametar konjugovanog gradijentakonveksna kombinacija dva poznata parametra konjugovanoggradijenta.U prvoj glavi opisuje se motivacija kao i osnovni pojmovi potrebni zapraćenje preostalih glava.U drugoj glavi daje se pregled nekih gradijentnih metoda prvog idrugog reda.Četvrta glava sadrži pregled osnovnih pojmova i nekih rezultatavezanih za metode konjugovanih gradijenata.Pomenute glave su tu radi pregleda nekih poznatih rezultata, dok seoriginalni doprinos predstavlja u trećoj, petoj i šestoj glavi.U trećoj glavi se opisuje izvesna modifikacija određenog metoda ukome se koristi multiplikativni parametar, izabran na slučajan način.Dokazuje se linearna konvergencija tako formiranog novog metoda.Peta glava sadrži originalne rezultate koji se odnose na metodekonjugovanih gradijenata. Naime, u ovoj glavi predstavlja se novihibridni metod konjugovanih gradijenata, koji je konveksnakombinacija dva poznata metoda konjugovanih gradijenata.U šestoj glavi se daju rezultati numeričkih eksperimenata, izvršenihna izvesnom skupu test funkcija, koji se odnose na metode iz treće ipete glave. Implementacija svih razmatranih algoritama rađena je upaketu MATHEMATICA. Kriterijum upoređivanja je vreme radacentralne procesorske jedinice.6The problem under consideration is an unconstrained optimizationproblem. There are many different methods made in aim to solve theoptimization problems. The investigation made here is motivated bythe fact that the methods which converge fast are necessary.The main goal is the systematization of some known results and alsotheoretical and numerical analysis of the possibilities to int roducesome parameters within gradient methods.Firstly, the minimization problem is considered, where the objectivefunction is a convex, multivar iable function. This problem is solvedhere without the calculation of Hessian, and such solution is veryimportant, for example, when the big dimension systems are solved,and also for solving optimization problems with unknown values ofthe objective function and its gradient. Partially, this investigation ismotivated by the existence of problems where the objective functionis the result of simulations.Numerical results, presented in Chapter 6, show that the introductionof a parameter is useful, i.e., such introduction results by theacceleration of the known optimization method.Further, one new hybrid conjugate gradient method is presented, inwhich the conjugate gradient parameter is a convex combination oftwo known conjugate gradient parameters.In the first chapter, there is motivation and also the basic co nceptswhich are necessary for the other chapters.The second chapter contains the survey of some first order andsecond order gradient methods.The fourth chapter contains the survey of some basic concepts andresults corresponding to conjugate gradient methods.The first, the second and the fourth chapters are here to help inconsidering of some known results, and the original results arepresented in the chapters 3,5 and 6.In the third chapter, a modification of one unco nstrained optimizationmethod is presented, in which the randomly chosen multiplicativeparameter is used. Also, the linear convergence of such modificationis proved.The fifth chapter contains the original results, corresponding toconjugate gradient methods. Namely, one new hybrid conjugategradient method is presented, and this method is the convexcombination of two known conjugate gradient methods.The sixth chapter consists of the numerical results, performed on a setof test functions, corresponding to methods in the chapters 3 and 5.Implementation of all considered algorithms is made in Mathematica.The comparison criterion is CPU time.
The problem under consideration is an unconstrained optimizationproblem. There are many different methods made in aim to solve theoptimization problems. The investigation made here is motivated bythe fact that the methods which converge fast are necessary.The main goal is the systematization of some known results and alsotheoretical and numerical analysis of the possibilities to int roducesome parameters within gradient methods.Firstly, the minimization problem is considered, where the objectivefunction is a convex, multivar iable function. This problem is solvedhere without the calculation of Hessian, and such solution is veryimportant, for example, when the big dimension systems are solved,and also for solving optimization problems with unknown values ofthe objective function and its gradient. Partially, this investigation ismotivated by the existence of problems where the objective functionis the result of simulations.Numerical results, presented in Chapter 6, show that the introductionof a parameter is useful, i.e., such introduction results by theacceleration of the known optimization method.Further, one new hybrid conjugate gradient method is presented, inwhich the conjugate gradient parameter is a convex combination oftwo known conjugate gradient parameters.In the first chapter, there is motivation and also the basic co nceptswhich are necessary for the other chapters.Key Words Documentation 97The second chapter contains the survey of some first order andsecond order gradient methods.The fourth chapter contains the survey of some basic concepts andresults corresponding to conjugate gradient methods.The first, the second and the fourth chapters are here to help inconsidering of some known results, and the original results arepresented in the chapters 3,5 and 6.In the third chapter, a modification of one unco nstrained optimizationmethod is presented, in which the randomly chosen multiplicativeparameter is used. Also, the linear convergence of such modificationis proved.The fifth chapter contains the original results, corresponding toconjugate gradient methods. Namely, one new hybrid conjugategradient method is presented, and this method is the convexcombination of two known conjugate gradient methods.The sixth chapter consists of the numerical results, performed on a setof test functions, corresponding to methods in the chapters 3 and 5.Implementation of all considered algorithms is made in Mathematica.The comparison criterion is CPU time
Books on the topic "Hessian matrices"
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Coleman, Thomas F. Structure and efficient Hessian calculation. Ithaca, N.Y: Cornell Theory Center, Cornell University, 1996.
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Arian, Eyal. Analysis of the Hessian for aeroelastic optimization. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1995.
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Arian, Eyal. Analysis of the Hessian for aerodynamic optimization: Inviscid flow. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1996.
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Mann, Peter. Matrices. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0031.
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This chapter looks at the calculus of a function of two or more variables, which is the subject of partial differentiation. The partial derivative of a function is the rate of change of the function with respect to the distance in the direction of a particular coordinate axis and is symbolised with the sign ∂. The chapter spends time on the implicit function theorem, since it is relied upon heavily elsewhere in the text. Lagrange multipliers are used to solve constrained optimisation problems. Topics include critical points, the product rule, the chain rule, directional derivatives, hypersurfaces and Taylor’s theorem. In addition, the chapter discusses Jacobian matrices, the inverse function theorem, gradients, level sets and Hessian matrices.
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A nonlinear programming perspective on sensitivity calculations for systems governed by state equations. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.
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Hui-Qian, Tan, Dong X, and United States. National Aeronautics and Space Administration., eds. Application of symbolic computations to the constitutive modeling of structural materials. [Washington, D.C.]: NASA, 1990.
Find full textBook chapters on the topic "Hessian matrices"
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Gill, Jeff, and Gary King. "Numerical Issues Involved in Inverting Hessian Matrices." In Wiley Series in Probability and Statistics, 143–76. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2004. http://dx.doi.org/10.1002/0471475769.ch6.
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Berchio, Elvise. "A Family of Hardy-Rellich Type Inequalities Involving the L 2-Norm of the Hessian Matrices." In Geometric Properties for Parabolic and Elliptic PDE's, 17–33. Milano: Springer Milan, 2013. http://dx.doi.org/10.1007/978-88-470-2841-8_2.
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Lin, Psang Dain. "Hessian Matrices of Ray R̄i with Respect to Incoming Ray R̄i-1 and Boundary Variable Vector X̄i." In Advanced Geometrical Optics, 405–23. Singapore: Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-2299-9_16.
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"Second-order differentials and Hessian matrices." In Matrix Differential Calculus with Applications in Statistics and Econometrics, 211–21. Chichester, UK: John Wiley & Sons, Ltd, 2019. http://dx.doi.org/10.1002/9781119541219.ch10.
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A., Georgi. "Symbolic Determination of Jacobian and Hessian Matrices and Sensitivities of Active Linear Networks by Using Chan-Mai Signal-Flow Graphs." In New Frontiers in Graph Theory. InTech, 2012. http://dx.doi.org/10.5772/34352.
Full textConference papers on the topic "Hessian matrices"
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Erleben, Kenny, and Sheldon Andrews. "Inverse kinematics problems with exact Hessian matrices." In MiG '17: Motion in Games. New York, NY, USA: ACM, 2017. http://dx.doi.org/10.1145/3136457.3136464.
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G. Pratt, R., and C. S. Shin. "Hessian matrices, diffraction tomography and multiple suppresion." In 58th EAEG Meeting. Netherlands: EAGE Publications BV, 1996. http://dx.doi.org/10.3997/2214-4609.201408702.
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Groenwold, Albert, and L. F. Pascal Etman. "Duality in Convex Nonlinear Multipoint Approximations with Diagonal Approximate Hessian Matrices Deriving from Incomplete Series Expansions." In 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2006. http://dx.doi.org/10.2514/6.2006-7090.
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Pigoski, T., M. Griffis, and J. Duffy. "Stiffness Mappings Employing Different Frames of Reference." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0411.
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Abstract The stiffness mapping matrix for a planar compliant mechanism is analyzed using two different reference frames. The first is rigidly attached to the fixed body of the coupling, whilst the second is attached to the moving body of the coupling. It was found that, in general, these matrices are asymmetric when the coupling is loaded, and that one is the transpose of the other. This is an important result and can be considered an extension of the work done by Dimentberg[1], who derived a symmetrical stiffness mapping for an unloaded coupling. These new mappings are essential for the control of the coupling as it moves away from its unloaded position. Additionally, a third frame of reference which produces a symmetric mapping is examined and found to be identical to the Hessian matrix obtained from the second differentials of the elastic potential energy of the system. However, this symmetric mapping is not useful for control purposes and is only included to show that such a frame can be realized. Finally, static force loci for each of the reference frames are drawn to support the notion of frame-of-reference dependency.
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Tari, Hafez, and Marcelo J. Dapino. "Fast and Globally Convergent Nonlinear System Model for 3D Magnetostrictive Systems." In ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/smasis2014-7714.
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A globally convergent and fully coupled magnetomechanical model for 3D magnetostrictive systems is presented. In magnetostrictive actuators, magnetic field and stress inputs generate magnetic flux density and strain. We refer to models that follow this scheme as direct models (no relation to the direct magnetomechanical effect). In certain design and control situations, inverse models are necessary in which the magnetic field and stress are found from specified magnetic flux density and strains. This inversion typically involves an iterative procedure, which may be prone to convergence issues. An inverse model approach is proposed for arbitrary magnetostrictive materials. The inversion requirement is a continuous and second order differentiable direct model for any chosen magnetostrictive material. The approach is globally convergent, which makes it ideal for use in finite element frameworks. The premise of the proposed iterative system model is to constitute a recursive correction formula based on second order approximations of a novel scalar error function which allows to achieve a faster convergence rate. A continuation approach is then used to achieve global convergence for arbitrary input parameters. To illustrate, Galfenol is chosen as the magnetostrictive material, and analytical derivations of the Jacobian and Hessian matrices are presented. Finally, the computational efficiency of the proposed approach is shown to compare favorably against existing models.
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Yang, Xuesen, Xiaofeng Guo, and Wei Dong. "On-Line Component Map Adaptive Procedure Based on Sensor Data." In ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/gt2020-16242.
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Abstract A key challenge in the gas turbine community is to adapt the engine model by matching measured data with simulation data. This study presents a procedure aiming to calibrate a certain type of gas turbine for power generation. To reproduce degradation, disturbance is injected into the healthy components maps at different time. Subsequently, six correction factors along with measured data and unmeasured parameters are coupled together using cooperative working equations and optimized based on primal-dual interior point method. When performing the adaptive procedure, Jacobian and hessian matrices are calculated using finite difference since the component maps have external, mapped, functions implemented as lookup-tables, and mode-switching statements. To improve the accuracy of first-order and second-order partial derivatives, the finite difference is enhanced by Richardson extrapolation method. The search scope of correction factors and unmeasured parameters are determined by the whole working conditions. Meanwhile, an adaptive update method of initial solution is proposed to make sure the convergence of the optimization procedure as quickly as possible. Finally, the proposed method is further applied to the on-line adaptation in case of performance degradation. The influence of measurement noise on optimization is also studied. It is demonstrated that the procedure is capable of refining the component maps progressively, which is significant for the model-based gas path diagnostics and prognostics.
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Hjorungnes, Are, and David Gesbert. "Hessians of scalar functions of complex-valued matrices: A systematic computational approach." In 2007 9th International Symposium on Signal Processing and Its Applications (ISSPA). IEEE, 2007. http://dx.doi.org/10.1109/isspa.2007.4555383.
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