A Milstein scheme for SPDEs

This article studies an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite dimensional commutativity condition. In particular, a suitable infinite dimensional analog of Milstein's algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation and stochastic reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some numerical simulations are remove

Monte Carlo Methods for Backward Equations in Nonlinear Filtering

We consider Monte Carlo methods for the classical nonlinear filtering problem. The first method is based on a backward pathwise filtering equation and the second method is related to a backward linear stochastic partial differential equation. We study convergence of the proposed numerical algorithms.The considered methods have such advantages as a capability in principle to solve filtering problems of large dimensionality, reliable error control, and recurrency. Their efficiency is achieved due to the numerical procedures which use effective numerical schemes and variance reduction techniques. The results obtained are supported by numerical experiments. © Applied Probability Trust 2009.Engineering and Physical Sciences Research Council, EPSRC: EP/D049792/1

Identification and control of deposition processes

The electrochemical deposition process is defined as the production of a coating on a surface from an aqueous solution composed of several substances. Electrochemical deposition processes are characterized by strong nonlinearity, large complexity and disturbances. Therefore, improving production quality requires the identification of a reasonably accurate model which should be found from data in a reasonable amount of time and with a reasonable computational effort. This identification makes it possible to predict the behavior of unmeasured signals and design a control algorithm to meet the demands of consumers. This thesis addresses the identification and control of the deposition processes. A model for an electrochemical cell that takes into account both electrode interfaces and the activity of ions participating in the deposition process is developed and a method for taking into account uncompensated resistance is proposed. Identifiability of two models, the conventional model and the developed model, is investigated under step and sweep form of applied voltage. It is proven that conventional electrochemical cell model can be identified uniquely using a series of step voltage experiments or in a single linear sweep voltammetry experiment on the basis of the measurements of cell current. The Zakai filtering and pathwise filtering methods are applied to a nonlinear in the parameters electrochemical cell model to estimate the electrode kinetics and mass-transfer parameters of the copper electrodeposition process. In the case of known parameters the feedforward controllers that force the concentration at the boundary to follow the desired reference concentration are designed for the deposition processes. The adaptive boundary concentration control problem for the electrochemical cell with simultaneous parameter identification is solved using the Zakai filtering method. Using such a control, depletion in industrial applications, such as copper deposition baths, can be avoided. An identification method for identifying kinetic parameters and a time-varying mixed potential process of the nonlinear electroless nickel plating model is proposed. The method converts the original nonlinear time-varying identification problem into a time-invariant quadratic optimization problem solvable by conventional least squares

Simulation of Stochastic Partial Differential Equations and Stochastic Active Contours

This thesis discusses several aspects of the simulation of stochastic partial differential equations. First, two fast algorithms for the approximation of infinite dimensional Gaussian random fields with given covariance are introduced. Later Hilbert space-valued Wiener processes are constructed out of these random fields. A short introduction to infinite-dimensional stochastic analysis and stochastic differential equations is given. Furthermore different definitions of numerical stability for the discretization of stochastic partial differential equations are presented and the numerical stability of the heat equation with additive and multiplicative noise is explicitely computed using semigroup theory. Finally stochastic active contours are used for segmentation. This thesis generalizes work done by Juan et al. and does the simulation of different stochastic partial differential equations. The results are compared to equations without stochastics