Kalman filtering on approximate state-space models

Several state-space models for estimating a second-order stochastic process are proposed in this paper on the basis of the approximate Karhunen-Loève expansion. Properties of these models are studied and then the Kalman filtering method is applied. The accuracy of the models on the basis of two different situations, deterministic or random inputs, is studied by means of a simulation of a Brownian motion.This work was supported in part by DGICYT, Project No. PS93-0201.     

Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs

In this paper we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations. The control objective is to minimize the expectation of a tracking cost functional, and the control is of the deterministic, distributed type. The main analytical tool is the Wiener-Itô chaos or the Karhunen-Loève expansion. Mathematically, we prove the existence of an optimal solution; we establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations; we represent the input data in their Wiener-Itô chaos expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method; we also derive error estimates in terms of both types of discretizations.     

A priori reduction method for solving the two-dimensional Burgers’ equations

The two-dimensional Burgers’ equations are solved here using the A Priori Reduction method. This method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved. The method is called a priori because it does not need any prior knowledge of the solution, which is not the case if the standard Karhunen-Loéve decomposition is used. The accuracy of the APR method is compared with the standard Newton-Raphson scheme and with results from the literature. The APR basis is also compared with the Karhunen-Loéve basis.     

Sliding mode control of the forced generalized Burgers equation

^** Email: smaoui{at}mcs.sci.kuniv.edu.kw This paper deals with the sliding mode control (SMC) of theforced generalized Burgers equation via the Karhunen-Loève(K-L) Galerkin method. The decomposition procedure of the K-Lmethod is presented to illustrate the use of this method inanalysing the numerical simulations data which represent thesolutions of the forced generalized Burgers equation for viscosityranging from 1 to 100. The K-L Galerkin projection is used asa model reduction technique for non-linear systems to derivea system of ordinary differential equations (ODEs) that mimicsthe dynamics of the forced generalized Burgers equation. Thedata coefficients derived from the ODE system are then usedto approximate the solutions of the forced Burgers equation.Finally, static and dynamic SMC schemes with the objective ofenhancing the stability of the forced generalized Burgers equationare proposed. Simulations of the controlled system are givento illustrate the developed theory.     

Solving the stochastic steady-state diffusion problem using multigrid

Darran Furnival ^{We study multigrid for solving the stochastic steady-state diffusionproblem. We operate under the mild assumption that the diffusioncoefficient takes the form of a finite Karhunen-Loèveexpansion. The problem is discretized using a finite-elementmethodology using the polynomial chaos method to discretizethe stochastic part of the problem. We apply a multigrid algorithmto the stochastic problem in which the spatial discretizationis varied from grid to grid while the stochastic discretizationis held constant. We then show, theoretically and experimentally,that the convergence rate is independent of the spatial discretization,as in the deterministic case, and the stochastic discretization.}     

On the characteristic functional of a doubly stochastic Poisson process: Application to a narrow-band process

The characteristic functional (c.fl.) of a doubly stochastic Poisson process (DSPP) is studied and it provides us the finite dimensional distributions of the process and so its moments. It is also studied the case of a DSPP which intensity is a narrow-band process. The Karhunen–Loève expansion of its intensity is used to obtain the probability distribution function and a decomposition of this Poisson process. The covariance derived from the general c.fl. is applied in this particular DSPP.     

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

Natural frequency analysis of functionally graded material beams with axially varying stochastic properties

The functionally graded material (FGM) has a potential to replace ordinary ones in engineering reality due to its superior thermal and dynamical characteristics. In this regard, the paper presents an effective approach for uncertain natural frequency analysis of composite beams with axially varying material properties. Rather than simply assuming the material model as a deterministic function, we further extend the FGM property as a random field, which is able to account for spatial variability in laboratory observations and in-field data. Due to the axially varying input uncertainty, natural frequencies of the stochastically FGM (S-FGM) beam become random variables. To this end, the Karhunen–Loève expansion is first introduced to represent the composite material random field as the summation of a finite number of random variables. Then, a generalized eigenvalue function is derived for stochastic natural frequency analysis of the composite beam. Once the mechanistic model is available, the brutal Monte-Carlo simulation (MCS) similar to the design of experiment can be used to estimate statistical characteristics of the uncertain natural frequency response. To alleviate the computational cost of the MCS method, a generalized polynomial chaos expansion model developed based on a rather small number of training samples is used to mimic the true natural frequency function. Case studies have demonstrated the effectiveness of the proposed approach for uncertain natural frequency analysis of functionally graded material beams with axially varying stochastic properties.     

Reconstructing phase space from PDE simulations

We propose the Karhunen-Loève (K-L) decomposition as a tool to analyze complex spatio-temporal structures in PDE simulations in terms of concepts from dynamical systems theory. Taking the Kuramoto-Sivashinsky equation as a model problem we discuss the K-L decomposition for 4 different values of its bifurcation parameter . We distinguish two modes of using the K-L decomposition: As an analytic and synthetic tool respectively. Using the analytic mode we find unstable fixed points and stable and unstable manifolds in a parameter regime with structurally stable homoclinic orbits (=17.75). Choosing the data for a K-L analysis carefully by restricting them to certain burst events, we can analyze a more complicated intermittent regime at =68. We establish that the spatially localized oscillations around a so called strange fixed point which are considered as fore-runners of spatially concentrated zones of turbulence are in fact created by a very specific limit cycle (=83.75) which, for =87, bifurcates into a modulated traveling wave. Using the K-L decomposition synthetically by determining an optimal Galerkin system, we present evidence that the K-L decomposition systematically destroys dissipation and leads to blow up solutions.We would like to dedicate this paper to Klaus Kirchgässner on the occasion of his 60th birthday     

Sur le comportement asymptotique des transformations linéaires des suites

Sans résuméSponcered by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No: DA-11-022-ORD-2059.Extrait de la thèse «Comportement asymptotique des transformations linéaires des suites», Genève, 1966 [8].     

High dimensional model representation for stochastic finite element analysis

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.     

Numerical study of the controlled Van der Pol oscillator in Chebyshev series

Summary The optimal control of the Van der Pol oscillator is investigated by a numerical method based on the expansion of the state function and the control strategy in Chebyshev series. The optimal control problem is reduced to a parameter optimization problem.

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Resumé On étudie le contrôle optimal de l'oscillateur de Van der Pol à l'aide d'une méthode numérique qui utilise le développement des fonctions de position et de contrôle en série de Chebyshev. Le problème de contrôle optimal est réduit à un problème d'optimisation de paramètres.

     

Solution of a stochastic Darcy equation by polynomial chaos expansion

This paper deals with solving a boundary value problem for the Darcy equation with a random hydraulic conductivity field.We use an approach based on polynomial chaos expansion in a probability space of input data.We use a probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. The computational complexity of this algorithm is determined by the order of the polynomial chaos expansion and the number of terms in the Karhunen–Loève expansion. We calculate various Eulerian and Lagrangian statistical characteristics of the flow by the conventional Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method over the directMonte Carlo algorithm.     

On a state-space modelling for functional data

The objective of this paper is to derive a state-space model for several continuous-time processes, by applying the Karhunen–Loève expansion, and then to apply the Kalman filter equations. The accuracy of the models on the basis of deterministic or random inputs is studied by means of simulation on two well-known processes.     

Stochastic modeling of fatigue crack propagation

This paper presents a stochastic model of fatigue-induced crack propagation in metallic materials. The crack growth rate predicted by the model is guaranteed to be non-negative. The model structure is built upon the underlying principle of Karhunen–Loève expansion and does not require solutions of stochastic differential equations in either Wiener integral or Itô integral setting. As such this crack propagation model can be readily adapted to damage monitoring and remaining life prediction of stressed structures. The model results have been verified by comparison with experimental data of time-dependent fatigue crack statistics for 2024-T3 and 7075-T6 aluminum alloys.     

Expansion of random field gradients using hierarchical matrices

We present two expansions for the gradient of a random field. In the first approach, we differentiate its truncated Karhunen-Loève expansion. In the second approach, the Karhunen-Loève expansion of the random field gradient is computed directly. Both strategies require the solution of dense, symmetric matrix eigenvalue problems which can be handled efficiently by combining hierachical matrix techniques with a thick-restart Lanczos method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Karhunen–Loève expansion for additive Brownian motions

For Gaussian random fields defined as additive processes based on standard Brownian motions and Brownian bridges, we find their Karhunen–Loève expansions and make connections with related mean centered processes in distribution. Moreover, Pythagorean type distribution identities are established for additive Brownian motions and Brownian bridges. As applications, the corresponding Laplace transform and small deviation estimates are given.     

Multivariate interpolation at arbitrary points made simple

The concrete method of surface spline interpolation is closely connected with the classical problem of minimizing a Sobolev seminorm under interpolatory constraints; the intrinsic structure of surface splines is accordingly that of a multivariate extension of natural splines. The proper abstract setting is a Hilbert function space whose reproducing kernel involves no functions more complicated than logarithms and is easily coded. Convenient representation formulas are given, as also a practical multivariate extension of the Peano kernel theorem. Owing to the numerical stability of Cholesky factorization of positive definite symmetric matrices, the whole construction process of a surface spline can be described as a recursive algorithm, the data relative to the various interpolation points being exploited in sequence.

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Résumé La méthode concrète d'interpolation par surfaces-spline est étroitement liée au problème classique de la minimisation d'une semi-norme de Soboleff sous des contraintes d'interpolation; la structure intrinsèque des surfaces-spline est dès lors celle d'une extension multivariée des fonctions-spline naturelles. Le cadre abstrait adéquat est un espace fonctionnel hilbertien dont le noyau reproduisant ne fait pas intervenir de fonctions plus compliquées que des logarithmes et est aisé à programmer. Des formules commodes de représentation sont données, ainsi qu'une extension multivariée d'intérêt pratique du théorème du noyau de Peano. Grâce à la stabilité numérique de la factorisation de Cholesky des matrices symétriques définies positives, la construction d'une surface-spline peut se faire en exploitant point après point les données d'interpolation.

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Dedicated to Professor E. Stiefel     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

About a generalized model of lymphoma

In this work we introduce and analyze a generalized model of precursor T-lymphoblastic lymphoma as a competition between two clonotypes of naïve T-cells, one “normal” and one tumorous. It is modeled as a continuous-time bivariate Markov process. Using an expansion of the master equation a deterministic approximation and the Fokker–Planck equation are derived. For a deterministic model we show existence and uniqueness of global solutions and positive invariance of the first quadrant of the phase space. Stability analysis of the model is performed, finding conditions guaranteeing existence of a unique, positive, steady state, which is proved to be globally stable. It is shown that expectations of fluctuations for both clonotypes tend to zero for large time. We also present numerical simulations in which two types of behavior of solutions are observed: either both clonotypes survive in the repertoire or the “normal” clonotype becomes extinct. Comparing this result with the rules of maintenance of naïve T-cell repertoire, which say that clonotypes with more specific set of receptors have longer life-span, it seems that “normal” clonotype follows them, whereas the tumorous one violates them and tends to the maximum possible expansion. The model supports the hypothesis of mutated precursor cells as an origin of cancer.