$L^1$ penalization of volumetric dose objectives in optimal control of PDEs

Barnard, Richard C.; Clason, Christian

doi:10.1007/s10589-017-9897-6

$L^1$ penalization of volumetric dose objectives in optimal control of PDEs

Journal Article · Fri Feb 10 23:00:00 EST 2017 · Computational Optimization and Applications

DOI:https://doi.org/10.1007/s10589-017-9897-6· OSTI ID:1344291

Barnard, Richard C. ^[1]; ^[2]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Univ. of Duisburg-Essen, Essen (Germany)

This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on L¹ penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. Finally, the performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.

View Accepted Manuscript (DOE)

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1344291

Journal Information:: Computational Optimization and Applications, Journal Name: Computational Optimization and Applications Journal Issue: 2 Vol. 67; ISSN 0926-6003

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (21)

Perturbations of nonlinear maximal monotone sets in banach space Brezis, H.; Crandall, M. G.; Pazy, A. Communications on Pure and Applied Mathematics, Vol. 23, Issue 1 https://doi.org/10.1002/cpa.3160230107	journal	January 1970
Instantaneous closed loop control of the radiative transfer equations with applications in radiotherapy Frank, M.; Herty, M.; Hinze, M. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 92, Issue 1 https://doi.org/10.1002/zamm.201000191	journal	August 2011
Convex Analysis and Monotone Operator Theory in Hilbert Spaces Bauschke, Heinz H.; Combettes, Patrick L. CMS Books in Mathematics https://doi.org/10.1007/978-1-4419-9467-7	book	December 2013
Fundamentals of Convex Analysis Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude Springer Berlin, Heidelberg https://doi.org/10.1007/978-3-642-56468-0	book	January 2001
Elektronenfelder in der klinischen Anwendung: Ein Vergleich von Pencil-Beam- und Monte-Carlo-Algorithmus Treutwein, Marius; Bogner, Ludwig Strahlentherapie und Onkologie, Vol. 183, Issue 8 https://doi.org/10.1007/s00066-007-1687-0	journal	August 2007
Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices Stadler, Georg Computational Optimization and Applications, Vol. 44, Issue 2 https://doi.org/10.1007/s10589-007-9150-9	journal	November 2007
Semi-smooth Newton methods for state-constrained optimal control problems Ito, Kazufumi; Kunisch, Karl Systems & Control Letters, Vol. 50, Issue 3 https://doi.org/10.1016/S0167-6911(03)00156-7	journal	October 2003
Optimal radiotherapy treatment planning using minimum entropy models Barnard, Richard; Frank, Martin; Herty, Michael Applied Mathematics and Computation, Vol. 219, Issue 5 https://doi.org/10.1016/j.amc.2012.08.099	journal	November 2012
Multi-bang control of elliptic systems Clason, Christian; Kunisch, Karl Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 31, Issue 6 https://doi.org/10.1016/j.anihpc.2013.08.005	journal	November 2014
A convex penalty for switching control of partial differential equations Clason, Christian; Rund, Armin; Kunisch, Karl Systems & Control Letters, Vol. 89 https://doi.org/10.1016/j.sysconle.2015.12.013	journal	March 2016
A convex analysis approach to optimal controls with switching structure for partial differential equations Clason, Christian; Ito, Kazufumi; Kunisch, Karl ESAIM: Control, Optimisation and Calculus of Variations, Vol. 22, Issue 2 https://doi.org/10.1051/cocv/2015017	journal	April 2016
The smoothed-penalty algorithm for state constrained optimal control problems for partial differential equations Gugat, Martin; Herty, Michael Optimization Methods and Software, Vol. 25, Issue 4 https://doi.org/10.1080/10556780903002750	journal	August 2010
A moment-based approach for DVH-guided radiotherapy treatment plan optimization Zarepisheh, M.; Shakourifar, M.; Trigila, G. Physics in Medicine and Biology, Vol. 58, Issue 6 https://doi.org/10.1088/0031-9155/58/6/1869	journal	February 2013
Report 50 Landberg, T.; Chavaudra, J.; Dobbs, J. Journal of the International Commission on Radiation Units and Measurements, Vol. os26, Issue 1 https://doi.org/10.1093/jicru/os26.1.Report50	journal	September 1993
Penalty Techniques for State Constrained Optimal Control Problems with the Wave Equation Gugat, Martin SIAM Journal on Control and Optimization, Vol. 48, Issue 5 https://doi.org/10.1137/080725921	journal	January 2010
Lagrange Multiplier Approach to Variational Problems and Applications Ito, Kazufumi; Kunisch, Karl https://doi.org/10.1137/1.9780898718614	book	January 2008
Directional Sparsity in Optimal Control of Partial Differential Equations Herzog, Roland; Stadler, Georg; Wachsmuth, Gerd SIAM Journal on Control and Optimization, Vol. 50, Issue 2 https://doi.org/10.1137/100815037	journal	January 2012
A Semismooth Newton Method for Nonlinear Parameter Identification Problems with Impulsive Noise Clason, Christian; Jin, Bangti SIAM Journal on Imaging Sciences, Vol. 5, Issue 2 https://doi.org/10.1137/110826187	journal	January 2012
Erratum: Directional Sparsity In Optimal Control of Partial Differential Equations Herzog, Roland; Stadler, Georg; Wachsmuth, Gerd SIAM Journal on Control and Optimization, Vol. 53, Issue 4 https://doi.org/10.1137/15M102544X	journal	January 2015
Optimizing the Delivery of Radiation Therapy to Cancer Patients Shepard, David M.; Ferris, Michael C.; Olivera, Gustavo H. SIAM Review, Vol. 41, Issue 4 https://doi.org/10.1137/S0036144598342032	journal	January 1999
Optimal Radiotherapy Treatment Planning Governed by Kinetic Equations Frank, Martin; Herty, Michael; Sandjo, Albert N. Mathematical Models and Methods in Applied Sciences, Vol. 20, Issue 04 https://doi.org/10.1142/S0218202510004386	journal	April 2010

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Related Subjects

97 MATHEMATICS AND COMPUTING
L1 penalization
dose volume constraints
optimal control
semi-smooth Newton method

$L^1$ penalization of volumetric dose objectives in optimal control of PDEs

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