Identification of Temperature Dependent Parameters in a Simplified Radiative Heat Transfer

Transcription

1 Australian Journal of Basic and Applied Sciences, 5(): 7-4, 0 ISSN Identification of Temperature Dependent Parameters in a Simplified Radiative Heat Transfer Oliver Tse, Renè Pinnau, Norbert Siedow Technomathematics University of Kaiserslautern Kaiserslautern, Germany Transport Processes Fraunhofer ITWM Kaiserslautern, Germany Abstract: CLaser-induced thermotherapy (LITT) is an established minimally invasive technique of tumor ablation. Therefore, there is a need to predict the effect of laser applications and optimizing irradiation planning in LITT. The radiative transfer model is approximated by simplified approximations of spherical harmonics (SP n ), which is coupled with a bioheat transfer equation. Changes in optical attributes (absorption, scattering) occur due to thermal denaturation. The work presents the possibility to identify these temperature dependent parameters from given temperature measurements via an optimal control problem/inverse problem. The solvability of the optimal control problem is analyzed and results of successful implementations are shown. Key words: Component; radiative heat transfer; optimal control; inverse problem; SP n -approximation. INTRODUCTION For the mathematical modeling of radiative heat transfer in biological tissue, the heat transfer equation has to be coupled with the radiative transfer equation. Due to the high dimensionality of the latter problem, the simpler SP -approximation is used instead of the full radiative transfer equation. A justification to this simplification for radiative transfer in biological tissues can be found in (Klose, A.D. and E.W. Larsen, 006). We refer the reader to (Klose, A.D. and E.W. Larsen, 006) for an in-depth analysis of the optimal control problem in this paper. We also refer the reader to (Adams,) for the complete definition of Lebesgue and Sobolev spaces used here. 3 Let be a bounded Lipschitz domain and I be an open bounded interval. We further denote the space-time cylinder by Q: I and : I. For convenience we denote the parameter space, state space and image space by U u H u ( ) 0, X Vp, K, p3, Z V V L ( ),,, respectively, where Vpr, Lr(; I Wp( )) and K as defined in Theorem. We also denote the space X V W, where W V, C(; I L( )). pr, pr, Problem Statement: We introduce the PDE-constrained optimal control problem consisting of the cost functional (CMP) and the radiative heat transfer equations (RHT). The optimal control problem reads: Find a damage function u U and its corresponding state y ( y, y ) X i.e. incident radiation and temperature, solving y y Corresponding Author: Oliver Tse, Renè PinnauTechnomathematics, University of Kaiserslautern Kaiserslautern, Germany tse@mathematik.uni-kl.de 7

2 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 the problem Minimize J ( uy, ) J( y) J( u) (CMP) subject to the constraints (RHT), where J and J are defined respectively by J ( y) y ( x, ), i m, i L ( I) i J( u) uuc, H ( ) for finitely many given temperature measurements ( ) mi, L I at points xi, common parameter U and Tikhonov regularization parameter 0 ; with constraints given by uc Euy (, ) 0 in Z*, (RHT) where the nonlinear operator system E: X Z is induced by the weak formulation of the radiative-bioheat y a ( u, y) y 0 3 ( uy, ) t y y b ( y ) ( u, y ) y 0 b a in Q and y (0) y0 0 in L ( ), with boundary conditions n y ( y ) 0 n y ( y ) 0 on 3. The functions b,, and y0 represent the blood temperature, the irradiation on the boundary, the boundary temperature and the initial temperature of the body respectively. as Here, we model the thermal denaturation of the absorption parameter ( g), ( uy, ) i i,c ( i,c i,n) e, i a, s, t a s where a and scattering parameter s ( u, y) u( y)( ) d 0 with constants i,n, i,c 0, i a, s for the natural and coagulated optical parameters respectively, and anisotropy factor g. For (RHT) we have the following result on existence and uniqueness of states. 8

3 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 Theorem : Suppose the function, L ( ), and are given. Then, for any b L ( Q) y0 ( ) given parameter uu, there exists a unique state y X that solves (RHT), where X V K with K=W C(Q\ ) p, Note that the assumptions on the given functions b,, and y0 may be relaxed. Details, along with the proof of the theorem, may be found in (Tse, O.,). Optimal Control Problem: For the solution of the optimal control problem, we adopt theories from adjoint calculus, which may be found, for example, in (Hinze, M., 009). The corresponding linearized adjoint problem is to find, for given ( uy, ) UX and functions h( h, h ) [ X ], an adjoint state (,, ) Z fulfilling, X A( u, y)[ ] h in, (arht) with terminal condition ( T ) 0 in L ( ) 0, where the linear operator A: Z X, is induced by the weak formulation of the adjoint to the linearized system given by a ( u, y )( ) h 3 t ( u, y ) a ( u, y) ( y y) ( uy, ) ( y) h t 3 t ( uy, ) in Q, with boundary conditions L n 0 n 0 on 3. Note that the adjoint operator A is, by definition, none other than DEuy (, ) y. For (arht) we have the following result on existence, uniqueness and regularity of adjoint states. h[ X ] Theorem : Let ( uy, ) UX, and. Then, there exists a unique adjoint state Z that, solves (arht). If h[ V, ], we further have that (, ) X, with (0) 0 in L ( ). Since the main focus of this paper is the optimal control problem, we refer the reader to (Tse, O.,) for details on the existence, uniqueness, and regularity results for (RHT) and (arht). Under the assumptions of Theorem and Theorem, we state the following theorem, which guarantees the existence of a solution to (CMP). Theorem 3: There exists ( u, y ) U X solving (CMP). For the first order optimality condition, we have the following statement. 9

4 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 Theorem 4: Let ( u, y ) U Xbe a solution to (CMP). Then there exists a unique Lagrange multiplier X,, which satisfies the first-order optimality system Eu (, y) 0 in Z, Au (, y)[ ] DJ( u, y) 0 in X, y c, D Eu (, y) [ ] ( u u), uu 0, uu u The reduced cost functional J ˆ : J(, y( )): H ( ) H ( ) is continuously Fréchet differentiable and its derivative may be given explicitly by ˆ J( u) ˆ ( u) ( y ) 3 ( u) t ˆ t ˆ ( u) ( y y ) ( uu ), a c where ˆ (, ( )) and ˆ (, ( )) are mappings from U to L ( Q). a a y t t y Numerical Results: Consider an Ansatz for uu given by the Arrhenius law, a u( y ) Ae E ( Ry ) with frequency factor A, activation energy E a and universal gas constant R. Notice that the problem reduces to finding parameters ( AE, a ) V, where V is a compact subset, which solves (CMP). Fig. : Initial temperature state. 0

5 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 Fig. shows, for example, the initial temperature state for a given starting point ( A, E ) V. 0 a,0 Due to the complexity of the equality constraints given by a system of PDEs for the reduced cost functional Ĵ, it is extremely difficult to compute the exact Hessian. Thus, the optimization was performed using a projected BFGS-Armijo algorithm (c.f. (Kelly, C.T., 999)), where the regularization parameter λ was set to 0-5. The convergence analysis for the projected BFGS-Armijo algorithm can be found in (Kelly, C.T., 999).We give an outline to the computation of the reduced cost functional derivative 4 along with the algorithm:. We start by choosing an initial point u o and positive definite matrix B o.. Solve for y k the forward system (RHT). Ĵ following Theorem 3. Solve for k the adjoint problem (arht). Jˆ ( u ) 4. Compute k and update u : [ ] k PV uk sk k and Bk according to the projected BFGS-Armijo algorithm, where k is the decent direction, sk is the step size and k k P V : V and go to., while is the projection operator. u u () r k k a r 0 where uk() PV[ uk ] and r u u () k The constants r and a are given relative and absolute tolerances respectively. Fig. and Fig. 3 show results of the optimization procedure under noiseless and noisy measurement data respectively. Note that the results of their respective gradient norm and cost functional show fast convergence of the projected BFGS-Armijo algorithm in obtaining stationary solutions ( A, E ) a, V to the optimal control problem. In conclusion, the parameter identification problem, formulated, as an optimal control problem is solvable, both theoretically and numerically. Numerical results show fast convergence to an optimal solution for both exact and noisy data.

6 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 Fig. : Top: Optimized temperature state, Center: Gradient norm, Bottom: Cost functional.

7 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 Fig. 3: Top: Optimized temperature state with 5% noise, Center: Gradient norm, Bottom: Cost functional. 3

8 Aust. J. Basic & Appl. Sci., 5(): 7-4, 00 REFERENCES Adams, Sobolev Spaces. Hinze, M., R. Pinnau, M. Ulbrich and S. Ulbrich, 009. Optimization with PDE constraints, Springer Science + Business Medie B.V. Kelly, C.T., 999. Iterative Methods for Optimization, Society for Industrial and Applied Mathematics. Klose, A.D. and E.W. Larsen, 006. ALight Transport in Biological Tissue based on the Simplified Spherical Harmonics Equations,@ Journal of Computational Physics. Tse, O., R. Pinnau and N. Siedow, AIdentification of Temperature Dependent Parameters in Radiative Heat Transfer,@ unpublished. 4