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

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MotivationMotivation - Planung

5

lesion

RF-Ablation

Motivation - Planung

5

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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

PDF

P( )

x

x

xxx

xx

Random process

Output depends on uncertain parameters

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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

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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

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Random

sample

points

Sparse grid,

generated

with

Smolyak‘s

algorithm

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Stochastic Galerkin method

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stochastic elliptic PDE

is weak solution of the SPDE if

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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

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Cauchy Criterion Ratio Criterion

Norm in tensor product space

Numerical Tests for the SPDEs

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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

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Optimization Problem with SPDE

Constraints

subject to

with

such that and

and the measurements

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Optimality System

Adjoint equation

Derivative with respect to

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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.

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Computational Aspects

Second derivative of objective functional

Expectation value is omnipresent

convenient to be solve with collocation method

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Stochastic Model for RFA

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First Applications for the Probe Position*

Expectation of the maximal volume on destroyed tissue

Highest probability

for successful

Therapy

Confidence

interval

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optimal probe position for the deterministic

model

Probe positon for the expected maximal volume

of destroyed tissue

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* I. Altrogge, CeVis, University of Bremen

Conclusion and Outlook

Derivation of optimality system for SPDE-constrained problems

Gradient descent method and SQP method

First applications for RFA

Apply for more problems/objective functionals

Confidence interval

Hierarchical basis functions

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Thank You!

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