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SPDE-Constrained Optimization With Stochastic Collocation. Hanne Tiesler CeVis/ZeTeM @ University of Bremen DFG SPP 1253 Mike Kirby, University of Utah Tobias Preusser, Jacobs University Bremen/Fraunhofer MEVIS. Outline. Motivation Stochastic Processes How to solve SPDEs Numerical tests - PowerPoint PPT Presentation
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SPDE-Constrained Optimization With
Stochastic Collocation
Hanne TieslerCeVis/ZeTeM @ University of Bremen
DFG SPP 1253
Mike Kirby, University of Utah
Tobias Preusser, Jacobs University Bremen/Fraunhofer MEVIS
103.06.2009
Motivation
Stochastic Processes
How to solve SPDEs
Numerical tests
Optimization with SPDEs
Numerical examples
Hanne Tiesler 2
Outline
03.06.2009
Hanne Tiesler 3
MotivationMotivation - Planung
5
lesion
RF-Ablation
Motivation - Planung
5
03.06.2009
Hanne Tiesler
Uncertainty in Material Properties
Material properties
– are different for each patient
– change with vaporisation of water
– change with coagulation of the cells0 10 20 30 40 50 60 70 80 90 100
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
6.00E-01
7.00E-01
8.00E-01
9.00E-01
1.00E+00
Experimental Data: K. Lehmann, B. Frericks, U. Zurbuchen, Charite,
Berlin
4
P( )
x
x
xxx
xx
Random process
Output depends on uncertain parameters
03.06.2009
Hanne Tiesler 5
Stochastic Process
Let be a probability space
Stochastic process decomposed into finite set of independent
random variables
Joint probability density function of
reduce infinite dimensional probability space to -dimensional
space , Hilbert space
03.06.2009
Stochastic Collocation Method Combine stochastic Galerkin method and Monte Carlo Method
use polynomial approximation in random spaces and sample at
discrete points
orthogonal Lagrange interpolation polynomials
Hanne Tiesler 6
Random
sample
points
Sparse grid,
generated
with
Smolyak‘s
algorithm
03.06.2009
Stochastic Galerkin method
Hanne Tiesler 7
stochastic elliptic PDE
is weak solution of the SPDE if
03.06.2009
Hanne Tiesler 8
Numerical Tests
VVariance of the solution of the SPDE for different coefficients
Different realizations for
with
Stochastic solution for converges for to the
deterministic solution with
03.06.2009
Hanne Tiesler 9
Cauchy Criterion Ratio Criterion
Norm in tensor product space
Numerical Tests for the SPDEs
03.06.2009
Hanne Tiesler 10
Objective Functionals
With and is the inverse CDF of the random variable
with the spanning variable
Simple data measurements:
Several moments for the measurements:
Cumulative distribution function:
Zabaras, Ganapathysubramanian
03.06.2009
Hanne Tiesler 11
Optimization Problem with SPDE
Constraints
subject to
with
such that and
and the measurements
03.06.2009
Numerical Solution
Sequential quadratic programming (SQP)
Determine search direction by solving the quadratic problem
Define weighting factor for penalty function
Calculate stepwidth such that
Update optimization variables
and Hessian matrix.
Hanne Tiesler 1303.06.2009
Computational Aspects
Second derivative of objective functional
Expectation value is omnipresent
convenient to be solve with collocation method
Hanne Tiesler 1403.06.2009
First Applications for the Probe Position*
Expectation of the maximal volume on destroyed tissue
Highest probability
for successful
Therapy
Confidence
interval
Hanne Tiesler
optimal probe position for the deterministic
model
Probe positon for the expected maximal volume
of destroyed tissue
1603.06.2009
* I. Altrogge, CeVis, University of Bremen