SPDE-Constrained Optimization With Stochastic Collocation

1. 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

2. Hanne Tiesler Outline • Motivation • Stochastic Processes • How to solve SPDEs • Numerical tests • Optimization with SPDEs • Numerical examples

3. Hanne Tiesler Motivation RF-Ablation lesion local vessels Motivation - Planung Motivation - Planung 5 5

4. Hanne Tiesler Uncertainty in Material Properties x x x • Material properties • are different for each patient • change with vaporisation of water • change with coagulation of the cells x x x x P( ) Experimental Data: K. Lehmann, B. Frericks, U. Zurbuchen, Charite, Berlin Output depends on uncertain parameters PDF Random process

5. Hanne Tiesler 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

6. Stochastic Collocation Method Hanne Tiesler • Combine stochastic Galerkin method and Monte Carlo Method • use polynomial approximation in random spaces and sample at discrete points • orthogonal Lagrange interpolation polynomials Sparse grid, generated with Smolyak‘s algorithm Random sample points

7. Stochastic Galerkin method Hanne Tiesler stochastic elliptic PDE is weak solution of the SPDE if

8. Hanne Tiesler VVariance of the solution of the SPDE for different coefficients Different realizations for with Numerical Tests • Stochastic solution for converges for to the deterministic solution with

9. Hanne Tiesler Numerical Tests for the SPDEs • Cauchy Criterion Ratio Criterion • Norm in tensor product space

10. Hanne Tiesler Objective Functionals Simple data measurements: Several moments for the measurements: Cumulative distribution function: Zabaras, Ganapathysubramanian With and is the inverse CDF of the random variable with the spanning variable

11. Hanne Tiesler Optimization Problem with SPDE Constraints subject to with such that and and the measurements

12. Hanne Tiesler Optimality System • Adjoint equation • Derivative with respect to

13. Numerical Solution Hanne Tiesler 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.

14. Computational Aspects Hanne Tiesler • Second derivative of objective functional • Expectation value is omnipresent convenient to be solve with collocation method

15. Stochastic Model for RFA Hanne Tiesler Electric potential: Steady State Heat-equation:

16. First Applications for the Probe Position* Hanne Tiesler Probe positon for the expected maximal volume of destroyed tissue * I. Altrogge, CeVis, University of Bremen • Expectation of the maximal volume on destroyed tissue Highest probability for successful Therapy Confidence interval optimal probe position for the deterministic model

17. Conclusion and Outlook Hanne Tiesler • 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 Thank You!