Transcript SPDE-constrained optimization with stochastic collocation

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
03.06.2009
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Outline
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Motivation
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Stochastic Processes
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How to solve SPDEs
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Numerical tests

Optimization with SPDEs

Numerical examples
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Motivation
Motivation- Planung
- Planung Motivation
lesion
local vessels
RF-Ablation
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Uncertainty in Material Properties
1,00E+00
Material properties
9,00E-01
8,00E-01
–
7,00E-01
are different for each patient
6,00E-01
5,00E-01
–
change with vaporisation of water
4,00E-01
3,00E-01
–
2,00E-01
change with coagulation of the cells
1,00E-01
0,00E+00
0
10
20
30
40
50
60
70
80
90
100
Experimental Data: K. Lehmann, B. Frericks, U. Zurbuchen, Charite, Berlin
Output depends on uncertain parameters
P( )
x
x
x
x
x
x
x
Random process
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PDF
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Stochastic Process

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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
space , Hilbert space
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-dimensional
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Stochastic Collocation Method

Combine stochastic Galerkin method and Monte Carlo Method
Random
sample
points
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
Sparse grid,
generated
with
Smolyak‘s
algorithm
use polynomial approximation in random spaces and sample at
discrete points
orthogonal Lagrange interpolation polynomials
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Stochastic Galerkin method
stochastic elliptic PDE
is weak solution of the SPDE if
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Numerical Tests
Variance of the solution of the
SPDE for different coefficients
Different realizations for
with

Stochastic solution for
deterministic solution with
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converges for
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to the
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Numerical Tests for the SPDEs
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Cauchy Criterion
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Norm in tensor product space
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Ratio Criterion
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Objective Functionals
Simple data measurements:
Several moments for the measurements:
Cumulative distribution function:
Zabaras, Ganapathysubramanian
With
and
with the spanning variable
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is the inverse CDF of the random variable
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Optimization Problem with SPDE
Constraints
subject to
with
such that
and
and the measurements
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Optimality System
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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

Calculate stepwidth

Update optimization variables
for penalty function
such that
and Hessian matrix.
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Computational Aspects
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Second derivative of objective functional
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Expectation value is omnipresent
convenient to be solve with collocation method
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Stochastic Model for RFA
Electric potential:
Steady State Heat-equation:
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First Applications for the Probe Position*
* I. Altrogge, CeVis, University of Bremen
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Expectation of the maximal volume on destroyed tissue
Highest probability
for successful
Therapy
Confidence
interval
optimal probe position
for the deterministic
model
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Probe positon for the
expected maximal volume
of destroyed tissue
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Conclusion and Outlook
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Derivation of optimality system for SPDE-constrained problems
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Gradient descent method and SQP method
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First applications for RFA
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Apply for more problems/objective functionals
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Confidence interval
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Hierarchical basis functions
Thank You!
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