Relevant bibliographies by topics / Infinite-Dimensional statistics

Academic literature on the topic 'Infinite-Dimensional statistics'

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Published: 1 June 2024

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Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Journal articles on the topic "Infinite-Dimensional statistics":

1

Kleijn, B. J. K., and A. W. van der Vaart. "Misspecification in infinite-dimensional Bayesian statistics." Annals of Statistics 34, no. 2 (April 2006): 837–77. http://dx.doi.org/10.1214/009053606000000029.

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Cuchiero, Christa, and Sara Svaluto-Ferro. "Infinite-dimensional polynomial processes." Finance and Stochastics 25, no. 2 (March 4, 2021): 383–426. http://dx.doi.org/10.1007/s00780-021-00450-x.

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Song, Yanglei, Xiaohui Chen, and Kengo Kato. "Approximating high-dimensional infinite-order $U$-statistics: Statistical and computational guarantees." Electronic Journal of Statistics 13, no. 2 (2019): 4794–848. http://dx.doi.org/10.1214/19-ejs1643.

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Palev, Tchavdar D. "Lie superalgebras, infinite-dimensional algebras and quantum statistics." Reports on Mathematical Physics 31, no. 3 (June 1992): 241–62. http://dx.doi.org/10.1016/0034-4877(92)90017-u.

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Bojdecki, Tomasz, and Luis G. Gorostiza. "Inhomogenous infinite dimensional langevin equations." Stochastic Analysis and Applications 6, no. 1 (January 1988): 1–9. http://dx.doi.org/10.1080/07362998808809133.

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Dalecky, Yu L., and V. R. Steblovskaya. "On infinite-dimensional variational problems." Stochastic Analysis and Applications 14, no. 1 (January 1996): 47–71. http://dx.doi.org/10.1080/07362999608809425.

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Nishikawa, Naoki, Taiji Suzuki, Atsushi Nitanda, and Denny Wu. "Two-layer neural network on infinite-dimensional data: global optimization guarantee in the mean-field regime *." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 11 (November 1, 2023): 114007. http://dx.doi.org/10.1088/1742-5468/ad01b2.

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Abstract The analysis of neural network optimization in the mean-field regime is important as the setting allows for feature learning. The existing theory has been developed mainly for neural networks in finite dimensions, i.e. each neuron has a finite-dimensional parameter. However, the setting of infinite-dimensional input naturally arises in machine learning problems such as nonparametric functional data analysis and graph classification. In this paper, we develop a new mean-field analysis of a two-layer neural network in an infinite-dimensional parameter space. We first give a generalization error bound, which shows that the regularized empirical risk minimizer properly generalizes when the data size is sufficiently large, despite the neurons being infinite-dimensional. Next, we present two gradient-based optimization algorithms for infinite-dimensional mean-field networks, by extending the recently developed particle optimization framework to the infinite-dimensional setting. We show that the proposed algorithms converge to the (regularized) global optimal solution, and moreover, their rates of convergence are of polynomial order in the online setting and exponential order in the finite sample setting, respectively. To the best of our knowledge, this is the first quantitative global optimization guarantee of a neural network on infinite-dimensional input and in the presence of feature learning.
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Schmuland, Byron. "Dirichlet forms: Some infinite-dimensional examples." Canadian Journal of Statistics 27, no. 4 (December 1999): 683–700. http://dx.doi.org/10.2307/3316125.

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Vaart, A. W. "Efficiency. of infinite dimensional M- estimators." Statistica Neerlandica 49, no. 1 (March 1995): 9–30. http://dx.doi.org/10.1111/j.1467-9574.1995.tb01452.x.

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Heintze, Ernst, and Xiaobo Liu. "Homogeneity of Infinite Dimensional Isoparametric Submanifolds." Annals of Mathematics 149, no. 1 (January 1999): 149. http://dx.doi.org/10.2307/121022.

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Journal articles Books

Dissertations / Theses on the topic "Infinite-Dimensional statistics":

1

Romon, Gabriel. "Contributions to high-dimensional, infinite-dimensional and nonlinear statistics." Electronic Thesis or Diss., Institut polytechnique de Paris, 2023. http://www.theses.fr/2023IPPAG013.

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Trois problèmes sont abordés dans cette thèse: l'inférence en régression multi-tâche de grande dimension, les quantiles géométriques dans les espaces normés de dimension infinie, et les moyennes de Fréchet généralisées dans les arbres métriques. Premièrement, nous considérons un modèle de régression multi-tâche avec une hypothèse de sparsité sur les lignes de la matrice paramètre. L'estimation est faite en haute dimension avec l'estimateur Lasso multi-tâche. Afin de corriger le biais induit par la pénalité, nous introduisons un nouvel objet dépendant uniquement des données que nous appelons matrice d'interaction. Cet outil nous permet d'établir des résultats asymptotiques avec des lois limites normales ou chi². Il en découle des intervalles de confiance et des ellipsoïdes de confiance, qui sont valides dans des régimes de sparsité qui ne sont pas couverts par la littérature existante. Deuxièmement, nous étudions le quantile géométrique, qui généralise le quantile classique au cadre des espaces normés. Nous commençons par fournir de nouveaux résultats sur l'existence et l'unicité des quantiles géométriques. L'estimation est effectuée avec un M-estimateur approché et nous examinons ses propriétés asymptotiques en dimension infinie. Quand le quantile théorique n'est pas unique, nous utilisons la théorie de la convergence variationnelle pour obtenir des résultats asymptotiques sur les sous-suites dans la topologie faible. Quand le quantile théorique est unique, nous montrons que l'estimateur est consistant pour la topologie de la norme dans une large classe d'espaces de Banach, en particulier dans les espaces séparables et uniformément convexes. Dans les Hilbert séparables nous démontrons des représentations de Bahadur-Kiefer de l'estimateur, dont découle immédiatement la normalité asymptotique à la vitesse paramétrique. Finalement, nous considérons des mesures de tendance centrale pour des données vivant sur un réseau, qui est modélisé par un arbre métrique. Les paramètres de localisation que nous étudions sont appelés moyennes de Fréchet généralisées: elles sont obtenues en remplaçant le carré dans la définition de la moyenne de Fréchet par une fonction de perte convexe et croissante. Nous élaborons une notion de dérivée directionnelle dans l'arbre, ce qui nous aide à localiser et caractériser les minimiseurs. Nous examinons les propriétés statistiques du M-estimateur correspondant: nous étendons le concept de moyenne collante au contexte des arbres métriques, puis nous obtenons un théorème collant non-asymptotique et une loi des grands nombres collante. Pour la médiane de Fréchet, nous établissons des bornes de concentration non-asymptotiques et des théorèmes central limite collants
Three topics are explored in this thesis: inference in high-dimensional multi-task regression, geometric quantiles in infinite-dimensional Banach spaces and generalized Fréchet means in metric trees. First, we consider a multi-task regression model with a sparsity assumption on the rows of the unknown parameter matrix. Estimation is performed in the high-dimensional regime using the multi-task Lasso estimator. To correct for the bias induced by the penalty, we introduce a new data-driven object that we call the interaction matrix. This tool lets us develop normal and chi-square asymptotic distribution results, from which we obtain confidence intervals and confidence ellipsoids in sparsity regimes that are not covered by the existing literature. Second, we study the geometric quantile, which generalizes the classical univariate quantile to normed spaces. We begin by providing new results on the existence and uniqueness of geometric quantiles. Estimation is then conducted with an approximate M-estimator and we investigate its large-sample properties in infinite dimension. When the population quantile is not uniquely defined, we leverage the theory of variational convergence to obtain asymptotic statements on subsequences in the weak topology. When there is a unique population quantile, we show that the estimator is consistent in the norm topology for a wide range of Banach spaces including every separable uniformly convex space. In separable Hilbert spaces, we establish novel Bahadur-Kiefer representations of the estimator, from which asymptotic normality at the parametric rate follows. Lastly, we consider measures of central tendency for data that lives on a network, which is modeled by a metric tree. The location parameters that we study are called generalized Fréchet means: they obtained by relaxing the square in the definition of the Fréchet mean to an arbitrary convex nondecreasing loss. We develop a notion of directional derivative in the tree, which helps us locate and characterize the minimizers. We examine the statistical properties of the corresponding M-estimator: we extend the notion of stickiness to the setting of metrics trees, and we state a non-asymptotic sticky theorem, as well as a sticky law of large numbers. For the Fréchet median, we develop non-asymptotic concentration bounds and sticky central limit theorems
2

Blacque-Florentin, Pierre. "Some infinite dimensional topics in probability and statistics." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43537.

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This thesis comprises two independent parts. In the first part, we develop a pathwise calculus for functionals of integer-valued measures and extend the framework of Functional Itô Calculus to functionals of integer-valued random measures by constructing a ’stochastic derivative’ operator with respect to such integer-valued random measures. This allows us to obtain weak martingale representation formulae holding beyond the class of Poisson random measures, and allowing for random and time-dependent compensators. We study the behaviour of this operator and compare it with other previous approaches in the literature, providing in passing a review of the various Malliavin approaches for jump processes. Finally, some examples of computations are provided. The second part is oriented towards nonparametric statistics, with a financial application as our main goal: we aim at recovering a surface of FX call options on a pegged currency such as the Hong Kong dollar against the U.S. dollar, based on a small number of noisy measurements (the market bid-ask quotes). Inspiring ourselves from the Compressed Sensing literature, we develop a methodology that aims at recovering an arbitrage-free call surface. We first apply this methodology, based on tensor polynomial decomposition of the surface, to a sparse set of call-option prices on the S&P500, recovering the call option prices within desired tolerance, as well as a smooth local-volatility surface. On a pegged currency such as the HKD/USD, it appears that tensor polynomials may not be an adequate way to model the smiles across maturities. Modifying the methodology in favour of structure-preserving functions, we apply the new methodology to our HKD/USD dataset, recovering the smiles, and the corresponding state-price density.
3

Horta, Eduardo de Oliveira. "Essays in nonparametric econometrics and infinite dimensional mathematical statistics." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2015. http://hdl.handle.net/10183/133007.

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A presente Tese de Doutorado é composta de quatro artigos científicos em duas áreas distintas. Em Horta, Guerre e Fernandes (2015), o qual constitui o Capítulo 2 desta Tese, é proposto um estimador suavizado no contexto de modelos de regressão quantílica linear (Koenker e Basset, 1978). Uma representação de Bahadur-Kiefer uniforme é obtida, a qual apresenta uma ordem assintótica que domina aquela correspondente ao estimador clássico. Em seguida, prova-se que o viés associado à suavização é negligenciável, no sentido de que o termo de viés é equivalente, em primeira ordem, ao verdadeiro parâmetro. A taxa precisa de convergência é dada, a qual pode ser controlada uniformemente pela escolha do parâmetro de suavização. Em seguida, são estudadas propriedades de segunda ordem do estimador proposto, em termos do seu erro quadrático médio assintótico, e mostra-se que o estimador suavizado apresenta uma melhoria em relação ao usual. Como corolário, tem-se que o estimador é assintoticamente normal e consistente à ordem p n. Em seguida, é proposto um estimador consistente para a matriz de covariância assintótica, o qual não depende de estimação de parâmetros auxiliares e a partir do qual pode-se obter diretamente intervalos de confiança assintóticos. A qualidade do método proposto é por fim ilustrada em um estudo de simulação. Os artigos Horta e Ziegelmann (2015a, 2015b, 2015c) se originam de um ímpeto inicial destinado a generalizar os resultados de Bathia et al. (2010). Em Horta e Ziegelmann (2015a), Capítulo 3 da presente Tese, é investigada a questão de existência de certos processos estocásticos, ditos processos conjugados, os quais são conduzidos por um segundo processo cujo espaço de estados tem como elementos medidas de probabilidade. Através dos conceitos de coerência e compatibilidade, obtémse uma resposta afirmativa à questão anterior. Baseado nas noções de medida aleatória (Kallenberg, 1973) e desintegração (Chang e Pollard, 1997; Pollard, 2002), é proposto um método geral para construção de processos conjugados. A teoria permite um rico conjunto de exemplos, e inclui uma classe de modelos de mudança de regime. Em Horta e Ziegelmann (2015b), Capítulo 4 desta Tese, é proposto – em relação com a construção obtida em Horta e Ziegelmann (2015a) – o conceito de processo fracamente conjugado: um processo estocástico real a tempo contínuo, conduzido por uma sequência de funções de distribuição aleatórias, ambos conectados por uma condição de compatibilidade a qual impõe que aspectos da distribuição do primeiro processo são divisíveis em uma quantidade enumerável de ciclos, dentro dos quais este tem como marginais, precisamente, o segundo processo. Em seguida, mostra-se que a metodologia de Bathia et al. (2010) pode ser aplicada para se estudar a estrutura de dependência de processos fracamente conjugados, e com isso obtém-se resultados de consistência à ordem p n para os estimadores que surgem naturalmente na teoria. Adicionalmente, a metodologia é ilustrada através de uma implementação a dados financeiros. Especificamente, o método proposto permite que características da dinâmica das distribuições de processos de retornos sejam traduzidas em termos de um processo escalar latente, a partir do qual podem ser obtidas previsões de quantidades associadas a essas distribuições. Em Horta e Ziegelmann (2015c), Capítulo 5 da presente Tese, são obtidos resultados de consistência à ordem p n em relação à estimação de representações espectrais de operadores de autocovariância de séries de tempo Hilbertianas estacionárias, em um contexto de medições imperfeitas. Os resultados são uma generalização do método desenvolvido em Bathia et al. (2010), e baseiam-se no importante fato de que elementos aleatórios em um espaço de Hilbert separável são quase certamente ortogonais ao núcleo de seu respectivo operador de covariância. É dada uma prova direta deste fato.
The present Thesis is composed of 4 research papers in two distinct areas. In Horta, Guerre, and Fernandes (2015), which constitutes Chapter 2 of this Thesis, we propose a smoothed estimator in the framework of the linear quantile regression model of Koenker and Bassett (1978). A uniform Bahadur-Kiefer representation is provided, with an asymptotic rate which dominates the standard quantile regression estimator. Next, we prove that the bias introduced by smoothing is negligible in the sense that the bias term is firstorder equivalent to the true parameter. A precise rate of convergence, which is controlled uniformly by choice of bandwidth, is provided. We then study second-order properties of the smoothed estimator, in terms of its asymptotic mean squared error, and show that it improves on the usual estimator when an optimal bandwidth is used. As corollaries to the above, one obtains that the proposed estimator is p n-consistent and asymptotically normal. Next, we provide a consistent estimator of the asymptotic covariance matrix which does not depend on ancillary estimation of nuisance parameters, and from which asymptotic confidence intervals are straightforwardly computable. The quality of the method is then illustrated through a simulation study. The research papers Horta and Ziegelmann (2015a;b;c) are all related in the sense that they stem from an initial impetus of generalizing the results in Bathia et al. (2010). In Horta and Ziegelmann (2015a), Chapter 3 of this Thesis, we address the question of existence of certain stochastic processes, which we call conjugate processes, driven by a second, measure-valued stochastic process. We investigate primitive conditions ensuring existence and, through the concepts of coherence and compatibility, obtain an affirmative answer to the former question. Relying on the notions of random measure (Kallenberg (1973)) and disintegration (Chang and Pollard (1997), Pollard (2002)), we provide a general approach for construction of conjugate processes. The theory allows for a rich set of examples, and includes a class of Regime Switching models. In Horta and Ziegelmann (2015b), Chapter 4 of the present Thesis, we introduce, in relation with the construction in Horta and Ziegelmann (2015a), the concept of a weakly conjugate process: a continuous time, real valued stochastic process driven by a sequence of random distribution functions, the connection between the two being given by a compatibility condition which says that distributional aspects of the former process are divisible into countably many cycles during which it has precisely the latter as marginal distributions. We then show that the methodology of Bathia et al. (2010) can be applied to study the dependence structure of weakly conjugate processes, and therewith provide p n-consistency results for the natural estimators appearing in the theory. Additionally, we illustrate the methodology through an implementation to financial data. Specifically, our method permits us to translate the dynamic character of the distribution of an asset returns process into the dynamics of a latent scalar process, which in turn allows us to generate forecasts of quantities associated to distributional aspects of the returns process. In Horta and Ziegelmann (2015c), Chapter 5 of this Thesis, we obtain p n-consistency results regarding estimation of the spectral representation of the zero-lag autocovariance operator of stationary Hilbertian time series, in a setting with imperfect measurements. This is a generalization of the method developed in Bathia et al. (2010). The generalization relies on the important property that centered random elements of strong second order in a separable Hilbert space lie almost surely in the closed linear span of the associated covariance operator. We provide a straightforward proof to this fact.
4

Karlsson, John. "A class of infinite dimensional stochastic processes with unbounded diffusion." Licentiate thesis, Linköpings universitet, Matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-96583.

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The aim of this work is to provide an introduction into the theory of infinite dimensional stochastic processes. The thesis contains the paper A class of infinite dimensional stochastic processes with unbounded diffusion written at Linköping University during 2012. The aim of that paper is to take results from the finite dimensional theory into the infinite dimensional case. This is done via the means of a coordinate representation. It is shown that for a certain kind of Dirichlet form with unbounded diffusion, we have properties such as closability, quasi-regularity, and existence of local first and second moment of the associated process. The starting chapters of this thesis contain the prerequisite theory for understanding the paper. It is my hope that any reader unfamiliar with the subject will find this thesis useful, as an introduction to the field of infinite dimensional processes.
5

Bassi, Mohamed. "Quantification d'incertitudes et objets en dimension infinie." Thesis, Normandie, 2019. http://www.theses.fr/2019NORMIR03.

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La théorie des polynômes de chaos, étant une alternative moins onéreuse et plus efficace de la simulation de Monte Carlo, reste limitée aux polynômes de variables gaussiennes. On présente une méthode de type hilbertien qui généralise cette théorie et on établit les conditions d’existence et de convergence d’une expansion en Série de Fourier Généralisée. Ensuite, on présente la Statistique des Objets qui permet d’étudier les caractéristiques statistiques d’un ensemble d’objets aléatoires en dimension infinie. En calculant les distances entre les hypervolumes, notamment la distance de Hausdorff, cette méthode permet de déterminer l’objet médian, les objets quantiles et un intervalle de confiance à un seuil donné pour un ensemble fini d’objets aléatoires. Une méthode pour simuler un échantillon de grande taille d’un objet aléatoire à coût computationnel très réduit, et calculer sa moyenne sans faire appel à la distance entre les hypervolumes, fait l’objet de la troisième partie
The Polynomial Chaos theory, being a less expensive and more efficient alternative of the Monte Carlo Simulation, remains limited to the polynomials of Gaussian variables. We present a Hilbertian method that generalizes this theory and we establish the conditions of existence and convergence of an expansion in Generalized Fourier Series. Then, we present the Statistics of Things that allows studying the statistical characteristics of a set of random infinite-dimensional objects. By computing the distances between the hypervolumes, namely the distance of Hausdorff, this method allows determining the median object, the quantile objects and a confidence interval at a given level for a finite set of random objects. In the third section, we address a method for simulating a large size sample of a random object at a much reduced computational cost, and calculating its mean without using the distance between the hypervolumes
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DE, VECCHI FRANCESCO CARLO. "LIE SYMMETRY ANALYSIS AND GEOMETRICAL METHODS FOR FINITE AND INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS." Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/565457.

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The main aim of the thesis is a systematic application (via suitable generalizations) of Lie symmetry analysis, or more generally, of the various geometric techniques for differential equations, to the study of finite and infinite dimensional stochastic differential equations (SDEs). The work can be divided in three main parts. In the first part a new geometric approach to finite dimensional SDEs driven by a multidimensional Brownian motion is proposed, which is based on a new notion of random transformations of a stochastic process called stochastic transformations. After having studied the probabilistic and geometric properties of stochastic transformations, we provide a useful generalization of the well-known results of reduction and reconstruction of symmetric ODEs to the stochastic setting. We give many applications of previous results to some interesting SDEs among which the two dimensional Brownian motion, the Kolmogorov-Pearson equation, a generalized Langevin equation and the SABR model. Finally, using the previous theorems, we propose a symmetry-adapted numerical scheme whose effectiveness is verified through both theoretical estimates and numerical simulations. The second part contains an extension of the results obtained in the first part to finite dimensional SDEs driven by a general semimartingale taking values in a Lie group. In order to provide such an extension we use the notion of geometrical SDEs introduced by Serge Choen, and we introduce some new notions of stochastic invariance for semimartingales called gauge and time symmetries of a semimartingale. Using these mathematical tools we generalize the notion of stochastic transformations in this setting and we propose the natural definition of symmetry based on this group of transformations. The formulated theory allows us to analyze in detail an important class of SDEs with possible relevant applications to iterated random maps theory. In the third part we take advantage of the geometry of the infinite jets bundle to develop a convenient algorithm for the explicit determination of finite dimensional solutions to stochastic partial differential equations (SPDEs). In this setting we are able to propose a generalization of Frobenius theorem in the infinite jet bundles setting, which, exploiting the classical notion of characteristics of a PDE, allows us to find some sufficient conditions for the existence of finite dimensional solutions to an SPDE and then to explicitly reduce the SPDE to a finite dimensional SDE. Our techniques permits to individuate new finite dimensional solutions to interesting SPDEs among which the proportional volatility equation in Heath-Jarrow-Morton framework, a stochastic perturbation of Hunter-Saxton equation and a filtering problem related to affine type processes.
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IBRAGIMOV, ANTON. "G - Expectations in infinite dimensional spaces and related PDES." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2013. http://hdl.handle.net/10281/44738.

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In this thesis, we extend the G-expectation theory to infinite dimensions. Such notions as a covariation set of G-normal distributed random variables, viscosity solution, a stochastic integral drive by G-Brownian motion are introduced and described in the given infinite dimensional case. We also give a probabilistic representation of the unique viscosity solution to the fully nonlinear parabolic PDE with unbounded first order term in Hilbert space in terms of G-expectation theory.
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Kidzinski, Lukasz. "Inference for stationary functional time series: dimension reduction and regression." Doctoral thesis, Universite Libre de Bruxelles, 2014. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209226.

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Les progrès continus dans les techniques du stockage et de la collection des données permettent d'observer et d'enregistrer des processus d’une façon presque continue. Des exemples incluent des données climatiques, des valeurs de transactions financières, des modèles des niveaux de pollution, etc. Pour analyser ces processus, nous avons besoin des outils statistiques appropriés. Une technique très connue est l'analyse de données fonctionnelles (ADF).

L'objectif principal de ce projet de doctorat est d'analyser la dépendance temporelle de l’ADF. Cette dépendance se produit, par exemple, si les données sont constituées à partir d'un processus en temps continu qui a été découpé en segments, les jours par exemple. Nous sommes alors dans le cadre des séries temporelles fonctionnelles.

La première partie de la thèse concerne la régression linéaire fonctionnelle, une extension de la régression multivariée. Nous avons découvert une méthode, basé sur les données, pour choisir la dimension de l’estimateur. Contrairement aux résultats existants, cette méthode n’exige pas d'assomptions invérifiables.

Dans la deuxième partie, on analyse les modèles linéaires fonctionnels dynamiques (MLFD), afin d'étendre les modèles linéaires, déjà reconnu, dans un cadre de la dépendance temporelle. Nous obtenons des estimateurs et des tests statistiques par des méthodes d’analyse harmonique. Nous nous inspirons par des idées de Brillinger qui a étudié ces models dans un contexte d’espaces vectoriels.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

9

Cangiotti, Nicolò. "Feynman path integral for Schrödinger equation with magnetic field." Doctoral thesis, Università degli studi di Trento, 2020. http://hdl.handle.net/11572/251697.

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Feynman path integrals introduced heuristically in the 1940s are a powerful tool used in many areas of physics, but also an intriguing mathematical challenge. In this work we used techniques of infinite dimensional integration (i.e. the infinite dimensional oscillatory integrals) in two different, but strictly connected, directions. On the one hand we construct a functional integral representation for solutions of a general high-order heat-type equations exploiting a recent generalization of infinite dimensional Fresnel integrals; in this framework we prove a a Girsanov-type formula, which is related, in the case of Schrödinger equation, to the Feynman path integral representation for the solution in presence of a magnetic field; eventually a new phase space path integral solution for higher-order heat-type equations is also presented. On the other hand for the three dimensional Schrödinger equation with magnetic field we provide a rigorous mathematical Feynman path integral formula still in the context of infinite dimensional oscillatory integrals; moreover, the requirement of independence of the integral on the approximation procedure forces the introduction of a counterterm, which has to be added to the classical action functional (this is done by the example of a linear vector potential). Thanks to that, it is possible to give a natural explanation for the appearance of the Stratonovich integral in the path integral formula for both the Schrödinger and the heat equation with magnetic field.
10

Cangiotti, Nicolò. "Feynman path integral for Schrödinger equation with magnetic field." Doctoral thesis, Università degli studi di Trento, 2020. http://hdl.handle.net/11572/251697.

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Feynman path integrals introduced heuristically in the 1940s are a powerful tool used in many areas of physics, but also an intriguing mathematical challenge. In this work we used techniques of infinite dimensional integration (i.e. the infinite dimensional oscillatory integrals) in two different, but strictly connected, directions. On the one hand we construct a functional integral representation for solutions of a general high-order heat-type equations exploiting a recent generalization of infinite dimensional Fresnel integrals; in this framework we prove a a Girsanov-type formula, which is related, in the case of Schrödinger equation, to the Feynman path integral representation for the solution in presence of a magnetic field; eventually a new phase space path integral solution for higher-order heat-type equations is also presented. On the other hand for the three dimensional Schrödinger equation with magnetic field we provide a rigorous mathematical Feynman path integral formula still in the context of infinite dimensional oscillatory integrals; moreover, the requirement of independence of the integral on the approximation procedure forces the introduction of a counterterm, which has to be added to the classical action functional (this is done by the example of a linear vector potential). Thanks to that, it is possible to give a natural explanation for the appearance of the Stratonovich integral in the path integral formula for both the Schrödinger and the heat equation with magnetic field.

Books on the topic "Infinite-Dimensional statistics":

1

Giné, Evarist. Mathematical foundations of infinite-dimensional statistical models. New York, NY: Cambridge University Press, 2016.

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2

Liu, Kai. Stability of infinite dimensional stochastic differential equations with applications. Boca Raton, FL: Chapman & Hall/CRC, 2006.

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Pantsulaia, Gogi. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces. Hauppauge, N.Y: Nova Science Publishers, 2007.

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Socolovsky, Eduardo A. A dissimilarity measure for clustering high- and infinite dimensional data that satisfies the triangle inequality. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2002.

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Osswald, Horst. Malliavin calculus for Lévy processes and infinite-dimensional Brownian motion: An introduction. Cambridge: Cambridge University Press, 2012.

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6

Accardi, Luigi. Recent Developments in Infinite-Dimensional Analysis and Quantum Probability: Papers in Honour of Takeyuki Hida's 70th Birthday. Dordrecht: Springer Netherlands, 2001.

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Conference on Quantum Probability and Infinite Dimensional Analysis (29th 2008 Ḥammāmāt, Tunisia). Quantum probability and infinite dimensional analysis: Proceedings of the 29th conference, Hammamet, Tunisia 13-18 October 2008. New Jersey: World Scientific, 2010.

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M, Berezanskiĭ I͡U. Spectral methods in infinite-dimensional analysis. Dordrecht: Kluwer Academic, 1994.

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Temam, Roger. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York, NY: Springer US, 1988.

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Arizona School of Analysis with Applications (2nd 2010 University of Arizona). Entropy and the quantum II: Arizona School of Analysis with Applications, March 15-19, 2010, University of Arizona. Edited by Sims Robert 1975- and Ueltschi Daniel 1969-. Providence, R.I: American Mathematical Society, 2011.

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Book chapters on the topic "Infinite-Dimensional statistics":

1

Pantsulaia, Gogi. "Infinite-Dimensional Monte Carlo Integration." In Applications of Measure Theory to Statistics, 19–46. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-45578-5_2.

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Eggermont, P. P. B., and V. N. LaRiccia. "Convex Optimization in Infinite-Dimensional Spaces." In Springer Series in Statistics, 377–403. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-1-0716-1244-6_10.

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Buldygin, V. V., and A. B. Kharazishvili. "Some infinite-dimensional vector spaces." In Geometric Aspects of Probability Theory and Mathematical Statistics, 57–70. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-017-1687-1_5.

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Sudakov, A. V., V. N. Sudakov, and H. v. Weizsäcker. "Typical Distributions: Infinite-Dimensional Approaches." In Asymptotic Methods in Probability and Statistics with Applications, 205–12. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0209-7_14.

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Nagy, Stanislav. "Depth in Infinite-dimensional Spaces." In Functional and High-Dimensional Statistics and Related Fields, 187–95. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-47756-1_25.

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Whittle, Peter. "Extension: Examples of the Infinite-Dimensional Case." In Springer Texts in Statistics, 317–28. New York, NY: Springer New York, 2000. http://dx.doi.org/10.1007/978-1-4612-0509-8_19.

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Whittle, Peter. "Extension: Examples of the Infinite-Dimensional Case." In Springer Texts in Statistics, 258–69. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-2892-9_15.

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Ferraty, Frédéric, and Philippe Vieu. "Nonparametric Statistics and High/Infinite Dimensional Data." In Springer Proceedings in Mathematics & Statistics, 357–67. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-0569-0_32.

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Nikulin, A. M. "Applications of Infinite-Dimensional Gaussian Integrals." In Asymptotic Methods in Probability and Statistics with Applications, 177–87. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0209-7_12.

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Draouil, Olfa, and Habib Ouerdiane. "Solutions of Infinite Dimensional Partial Differential Equations." In Springer Proceedings in Mathematics & Statistics, 239–50. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-06170-7_14.

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Conference papers on the topic "Infinite-Dimensional statistics":

1

Bongiorno, Enea G. "Contributions in infinite-dimensional statistics and related topics." In Contributions in infinite-dimensional statistics and related topics, edited by Ernesto Salinelli, Aldo Goia, and Philippe Vieu. Società Editrice Esculapio, 2014. http://dx.doi.org/10.15651/9788874887637.

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GOLDIN, GERALD A., UGO MOSCHELLA, and TAKAO SAKURABA. "MEASURES ON SPACES OF INFINITE-DIMENSIONAL CONFIGURATIONS, GROUP REPRESENTATIONS, AND STATISTICAL PHYSICS." In Proceedings of the Fifth International Workshop. WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812702562_0020.

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Van der Velden, Alex, Patrick Koch, Srikanth Devanathan, Jeff Haan, Dave Naehring, and David Fox. "Probabilistic Certificate of Correctness for Cyber Physical Systems." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70135.

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Abstract:
Cyber Physical Systems couple computational and physical elements, therefore the behavior of geometry (deformations, kinematics), physics and controls needs to be certified using many different tools over a very high dimensional space. Because of the near infinite number of ways such a system can fail meeting its requirements, we developed a Probabilistic Certificate of Correctness (PCC) metric which quantifies the probability of satisfying requirements with consistent statistical confidence. PCC can be implemented as a scalable engineering practice for certifying complex system behavior at every milestone in the product lifecycle. This is achieved by: creating virtual prototypes at different levels of model abstraction and fidelity; capturing and integrating these models into a simulation process flow; verifying requirements in parallel by deploying virtual prototypes across large organizations; reducing certification time proportional to additional computational resources and trading off sizing, modeling accuracy, technology and manufacturing tolerances against requirements and cost. This process is an improvement over the V-cycle because verification and validation happens at every stage of the system engineering process thus reducing rework in the more expensive implementation and physical certification phase. The PCC process is illustrated using the example of “Safe Range” certification for an UAV with active flutter control.
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Xu, Hongyi, Yang Li, Catherine Brinson, and Wei Chen. "Descriptor-Based Methodology for Designing Heterogeneous Microstructural Materials System." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12232.

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Abstract:
In designing a microstructural materials system, there are several key questions associated with design representation, design evaluation, and design synthesis: how to quantitatively represent the design space of a heterogeneous microstructure system using a small set of design variables, how to efficiently reconstruct statistically equivalent microstructures for design evaluation, and how to quickly search for the optimal microstructure design to achieve the desired material properties. This paper proposes a new descriptor-based methodology for designing microstructural materials systems. A descriptor-based characterization method is proposed to provide a quantitative representation of material morphology using a small set of microstructure descriptors covering features of material composition, dispersion status, and phase geometry at different levels of representation. A descriptor-based multi-phase microstructure reconstruction algorithm is developed which allows efficient stochastic reconstruction of microstructures for Finite Element Analysis (FEA) of material behavior. The choice of descriptors for polymer nanocomposites is verified by establishing a mapping between the finite set of descriptors and the infinite dimensional correlation function. Finally, the descriptor-based representation allows the use of parametric optimization approach to search the optimal microstructure design that meets the target material properties. To improve the search efficiency, this paper employs state-of-the-art computational design methods such as Design of Experiment (DOE), metamodeling, statistical sensitivity analysis, and multi-objective optimization. The proposed methodology is demonstrated using the design of a polymer nanocomposites system.
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Alhemdi, Aymen, and Ming Gu. "A Robust Workflow for Optimizing Drilling/Completion/Frac Design Using Machine Learning and Artificial Intelligence." In SPE Annual Technical Conference and Exhibition. SPE, 2022. http://dx.doi.org/10.2118/210160-ms.

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Abstract One of the biggest challenges in drilling/completion/hydraulic fracturing optimization is determining the optimal parameters in the infinite space of possible solutions. Applying a comprehensive parametric study with various geomechanical properties using both a frac simulator and a reservoir simulator is low efficient. This study proposes a workflow for optimizing unconventional reservoir development using machine learning and artificial intelligence (AI) in conjunction with advanced geomechanical modeling. The workflow consists of four steps: in Step1, appropriate acoustic interpretation models are used for geomechanical and in-situ stress characterization. In Step2, unsupervised machine learning optimizes completion designs based on formation anisotropy and heterogeneity along a well. In step3, a training database is built by generating multiple cases based on various simulations guided by a smart sampling algorithm. Proxy models are trained and validated by feeding the training datasets to supervised machine learning algorithms. Lastly, the tested proxy models are run for a multi-parameter sensitivity study for design optimization. The workflow was validated by a Marcellus field case. First, the newly proposed orthorhombic acoustic interpretation model yielded in-situ stress results more consistent with field measurements than the traditional acoustic models. Second, using the C-Means Fuzzy Clustering, the stage and cluster spacings were optimized to overcome the low cluster efficiency issue led by the current geometric completion design. Last, using the newly proposed smart sampling algorithm, a 200-critical-case database was built and fed into the Neural Network algorithm for training proxy models. After running the proxy models in a random-search algorithm, the optimal design parameter values were obtained statistically, leading to the Return-On-Frac-Investment (ROFI) improved by 22-40% from the current base case. The study introduces a robust four-step workflow combining unsupervised and supervised machine learning to examine high-dimensional multivariable drilling/completion/frac designs efficiently. The new workflow enables the evaluation of the statistical significance of the influencing parameters and, most importantly, their interactions, which have often been neglected in the current simulation-based optimization workflow. Moreover, the trained proxy models can be applied to optimize the design of the current wellbore as well as any other future wells drilled in the same basin in a convenient and time-efficient manner.

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