Segmentation of Stochastic Images using ... - Jacobs University

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).
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