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\(L^1\) penalization of volumetric dose objectives in optimal control of PDEs

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Abstract

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 \(L^1\) 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. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.

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Authors and Affiliations

  1. Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

    Richard C. Barnard

  2. Faculty of Mathematics, University Duisburg-Essen, 45117, Essen, Germany

    Christian Clason

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  1. Richard C. Barnard
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  2. Christian Clason
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Correspondence to Richard C. Barnard.

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This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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Barnard, R.C., Clason, C. \(L^1\) penalization of volumetric dose objectives in optimal control of PDEs. Comput Optim Appl 67, 401–419 (2017). https://doi.org/10.1007/s10589-017-9897-6

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