We present a new reduced basis approach for the efficient and reliable solution of parametrized PDE-constrained optimization problems. This kind of problems arise when a system modeled by PDEs is optimized in order to achieve a desired behavior. Typically, the numerical solution of these optimization problems is challenging in terms of required computational cost, in particular if one is interested in multiple solutions for different parameter values. The reduced basis method is a technique for parametric model order reduction, which provides both efficient and computationally stable approximations. It further provides rigorous error bounds and is well established for a large class of parametrized PDEs of elliptic and parabolic nature. In this thesis, we extend the reduced basis methodology to parametrized PDE-constrained optimization problems. These optimization problems may contain parameters in the PDE constraint itself, as well as in the objective function (such as the desired state or regularization parameter). We consider both elliptic and parabolic problems, as well as finite-dimensional and distributed controls. Our approach is completely online-efficient: The online computational cost for the reduced basis approximation and the associated error bounds depends only on the dimension of the reduced basis space. It is independent of the original high-dimensional finite element approximation. Our approximation is based on a reduced basis space for the state and the adjoint variables. By integrating state and adjoint snapshots into a single reduced basis space, we ensure both a consistent approximation and the stability of the reduced model. In addition, we propose a reduction of the control space for distributed controls. The focus of this thesis is to derive efficiently computable and rigorous a posteriori error bounds for various quantities of interest. To this end, we present two approaches: The first approach is based on a perturbation argument, whereas the second bound is directly derived from the error residual equations of the optimality system. Both approaches yield bounds for the errors in the optimal control, in the optimal value of the cost functional, and in the optimal state and adjoint variables. More generally, by introducing an additional dual problem, it is possible to bound the error of arbitrary linear output functionals depending on the state, adjoint, and control variables. All of the proposed bounds are completely offline-online separable and hence can be evaluated efficiently. This makes our approach relevant in the many-query or real-time context. In particular, the bounds involve only constants (or their upper/lower bounds), which can be computed with low effort. We present numerical results for various parametrized PDE-constrained optimization problems to demonstrate the effectiveness of the proposed method.

Certified reduced basis methods for parametrized PDE-constrained optimization problems

Preface [Special issue on cancer modeling (II)]

Abstract This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> - or $$\mathcal {H}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples. 

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On Robustly Convergent and Efficient Iterative Methods for Anisotropic Radiative Transfer

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that $\mathcal{H}$- or $\mathcal{H}^2$-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.

On robustly convergent and efficient iterative methods for anisotropic radiative transfer.

We study numerical methods for inverse problems arising in cancer therapy treatment under uncertainty. The interest is on efficient and reliable numerical methods that allow to determine the influence of possible unknown parameters on the treatment plan for cancer therapy. A kinetic transport equation is used to model the evolution of charged particles in tissue. A mixed variational framework is presented and existence and uniqueness of a weak solution is established. The optimality system is approximated using a low-dimensional reduced basis formulation based on a PN-FE discretization. We derive a posteriori bounds for the error in the reduced basis solution of the optimal control problem with respect to the solution of the PN-FE discretization. Numerical results in slab geometry are presented to confirm the validity of our approach.

/pdf/certified-reduced-order-methods-for-optimal-treatment-e2vp2lng6w.pdf

Certified reduced-order methods for optimal treatment planning

Radiation therapy becomes an important and common tool for cancer treatment with increasing the number of cancer deceases. Most common tools in clinics and medical institutions are based on statistical and experimental methods, e.g. a Pencil Beam, a Collapsed Cone or a Monte Carlo method. Nevertheless, in the present work we consider deterministic models for radiation therapy which are based on transport theory of the particles. The Boltzmann transport equation is used to describe evolution of charged particles in tissue. The radiation transport problem is approximated in terms of moments of the transfer equation. We introduce two different methods to solve spherical harmonics moments systems. We study properties of the numerical methods and show that the presented numerical method preserves asymptotic limit while the transport theory transmits into diffusion theory. The process of radiotherapy is an inverse problem. The main aim of the radiation therapy is to find position and dose of external beam such that particles will be distributed in a domain in desired way: much more coupled in cancer region and almost no particles in other regions. Since deterministic methods have explicit mathematical structures it allows us to formulate the radiation treatment process as optimal control problem constrained by the Boltzmann equation. We show existence, uniqueness and regularity of the optimal control and present the first–order optimality conditions. Geometrical or physical parameters of the problem may change during the treatment process which requires quick and efficient re-computation of the treatment plan. We propose low–dimensional reduced basis formulation which allows us to solve optimal control problem efficiently. We derive a posteriori error estimators for reduced basis solution with respect to the solution of high dimensional system.

Approximation methods and reduced models for optimal radiation treatment

We consider the problem of cancer treatment using radiation therapy. A deterministic Boltzmann equation is considered as the particle transport model. The PN system of moment equations is derived using angular discretization of the transport equation. This PN system is solved using StaRMAP [3]. An optimization problem to get the desired particle distribution is formulated and solved using adjoint equations. The algorithm is validated with several simulations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Bahodir Ahmedov

Papers

Certified reduced-order methods for optimal treatment planning

Approximation methods and reduced models for optimal radiation treatment

Optimization of Approximation Models for Solving Kinetic Equations