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Chapter 3 SPDEs and Polynomial Chaos Expansions Figure 3.3: PDFs of initial uniformly distributed input intervals (gray) and the PDFs of the results of the polynomial chaos computation (black) for squaring an interval (left) and dividing an interval by itself (right). sin([30 ◦ ,150 ◦ ]) = [0.5,0.5] in the above arithmetic. Other problems are that there are no simple methods to link realizations of intervals, e.g. the naive multiplication yields [−2,2] 2 = [−4,4], because there is no information in the interval arithmetic calculus that the realizations of both intervals must be the same. This is also problematic for division, because dividing an interval by itself should result into an interval with zero width, but, e.g. 2x/x for x = [2,4] yields the interval [1,4], not the desired interval [2,2]. Furthermore, the result is forced to be uniformly distributed inside the resulting interval, which is not the case for nonlinear operations and the resulting intervals can become arbitrarily large. Nevertheless, interval arithmetic is used in applications [70]. The polynomial chaos expansion can be thought as an extension of the interval arithmetic calculus. The results of polynomial chaos calculations are not forced to be uniformly distributed. Instead, they can have every distribution that can be represented in the chosen polynomial chaos basis. Results that can not be represented in this basis are projected onto the basis using the Galerkin projection introduced earlier. Fig. 3.3 shows the polynomial chaos result of the problematic operations squaring an interval and dividing by an interval. In both cases the polynomial chaos expansions yields the exact result, up to the machine precision. Furthermore, the problem of the huge resulting intervals of interval arithmetic operations is solved by using polynomial chaos expansions, because events at the tails of the intervals have a very low probability. Conclusion This chapter provided us with a possibility for the finite-dimensional approximation of arbitrary random variables, the polynomial chaos expansion. Even if the representation is possible for second order random variables only, this is sufficient for random variables arising in the image processing context. The finite-dimensional approximations and the associated closed computations provide a powerful toolbox for the discretization of SPDEs and a stochastic modeling of image processing problems. With the presented theoretical background, it is also possible to prove existence and uniqueness of solutions for the stochastic image processing models in the preceding chapters. It remains to show the few, easy to verify, assumptions. 36
Chapter 4 Discretization of SPDEs The discretization of SPDEs is an active research field [47, 159]. Besides the discretization based on sampling approaches like Monte Carlo simulation or stochastic collocation, there are methods for the intrusive computation of stochastic solutions in the literature. Intrusive means that we do not generate the solutions based on sampling strategies and we cannot reuse deterministic algorithms for the solution at the sampling points. This makes it necessary to develop new algorithms, but has the advantage that these algorithms are more efficient than the classical sampling approaches. This thesis focuses on intrusive methods. We present the sampling based approaches as well, but use them to verify the correctness of the intrusive algorithms and implementations only. The intrusive methods presented in this thesis range from the stochastic finite difference method based on polynomial chaos expansions to the generalized spectral decomposition [113–115, 117, 118], a method allowing to speed up the solution process of the stochastic finite element method (SFEM) [54]. 4.1 Sampling Based Discretization of SPDEs Since the mid of the 20 th century [100, 101] authors developed sampling based algorithms for the simulation of stochastic processes, starting with the development of the Monte Carlo method. Later, advanced sampling-based methods like the stochastic collocation method and improvements of these methods e.g. the combination of stochastic collocation and polynomial chaos expansions or the use of sparse grids based on a Smolyak construction [140] have been developed. 4.1.1 Monte Carlo Simulation Monte Carlo simulation is the simplest technique for the discretization of random variables and SPDEs. A set of samples is generated randomly from the known distribution of the random variables via a pseudo random number generator like [97]. We can use the well-known deterministic algorithms on these samples and compute from the results approximations to stochastic quantities like expected value, variance, etc. using well-known formulas. For example, when we computed the solution of R samples, approximate expected value and variance are E(x) ≈ ¯x = 1 R ∑R i=1 x i and Var(x) ≈ 1 R − 1 ∑R i=1 (x i − ¯x) 2 . (4.1) The main drawback of the Monte Carlo method is the slow convergence. In fact, Kendall [79] showed for the Monte Carlo method that the convergence of the samples mean towards the expected value is of order O(σ/ √ R). Despite the slow convergence rate, Monte Carlo methods are widely used (see e.g. [88, 94, 133]) due to the simple implementation and the possibility to reuse deterministic code. 4.1.2 Stochastic Collocation During the last years, a variety of stochastic collocation (SC) techniques was developed. These techniques range from simple collocation techniques over sparse grid techniques to SC techniques allowing to obtain polynomial chaos coefficient (see [159] for a more detailed review). 37
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
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Chapter 9 Summary, Discussion, and
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Appendix A Publications Written Dur
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Bibliography [14] L. Ambrosio and M
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Bibliography [46] B. Echebarria, R.
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Bibliography [81] B. N. Khoromskij
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Bibliography [147] J. Tryoen, O. Le
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