How difficult is nonlinear optimization? A practical solver tuning approach, with illustrative results

Pintér, János D.

doi:10.1007/s10479-017-2518-z

Cited by 15 publications

(8 citation statements)

References 44 publications

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“…In [39], section 8 we showed some features (such as convexity and Lipschitz continuity properties) of the above object functions. The search of global minimum requires usually global optimization methods ( [32], [33]). Global optimization algorithms in turn require an initial point.…”

Section: Inverse Radiation Treatment Problemmentioning

confidence: 99%

“…Moreover, the applied optimization method should be reasonably fast. The initialization (that is, the determination of an initial solution for global optimization scheme) is necessary since the determination of a carefully chosen initial point for a large dimensional global optimization scheme is very essential for achieving time saving and satisfactory results ( [33]). We prove in section 5.3 that for a certain (related) convex object function, the optimal control exists, and we give formulas for it in a variational form.…”

Section: Introductionmentioning

confidence: 99%

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A system of transport equations and an inverse problem from radiation therapy

Tervo

2019

Preprint

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The paper considers existence results of solution for a linear coupled system of Boltzmann transport equations and related inverse problem. The system models the evolution of three species of particles, photons, electrons and positrons. Hyper-singularities of differential cross sections associated with charged particle transport cause that modelling contains the first order partial differential term with respect to energy and the second order partial differential term with respect to velocity angle. The overall system is a partial integro-differential equation. The model is intended especially for dose calculation in the forward problem of radiation therapy. Firstly we consider a single transport equation for charged particles. After that we verify under physically relevant assumptions that the coupled equation together with relevant initial and inflow boundary values has a unique solution in appropriate L 2 -based spaces. The existence of solutions for the adjoint problem is verified as well. Moreover, we deal with the related inverse problem, a so called inverse radiation treatment planning problem. It is formulated as an optimal boundary control problem. Variational equations for an optimal control related to an appropriate differentiable strictly convex object function are verified. Its solution can be used as an initial point for an actual optimization which needs global optimization methods.

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Section: Inverse Radiation Treatment Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A system of transport equations and an inverse problem from radiation therapy

Tervo

2019

Preprint

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“…for some constants (weights) c T , c C , c N , c > 0, where X := T 2 (Γ − ) × H 1 (I, T 2 (Γ ′ − )) 2 and it is equipped with the inner product g, h X := g 1 , h 1 T 2 (Γ − ) + dimensional global optimization scheme is very essential for achieving (time savings and) satisfactory results ( [73]). We also notice that if we contented ourselves with so-called mild solutions ([71], p. 146), then the existence of an optimal control g (the admissible set being a subset of T 2 (Γ − ) 3 ), together with the explicit formula g = 1 c (γ − (ψ * )) + for it, could be proven under quite weak assumptions.…”

Section: Outlook Towards Inverse Radiation Treatment Planningmentioning

confidence: 99%

On Existence of $L^2$-solutions of Coupled Boltzmann Continuous Slowing Down Transport Equation System

Tervo,

Kokkonen,

Frank

et al. 2016

Preprint

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The paper considers a coupled system of linear Boltzmann transport equations (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems in numerical methods. We shall find, after the preliminary treatments, that for some interactions CSDA-type modelling is actually necessary due to hyper-singularities in the differential cross-sections of certain interactions, that is, first or second order partial derivatives with respect to energy and angle must be included into the transport part of charged particles. The existence and uniqueness of (weak) solutions is shown, under sufficient criteria and in appropriate L 2 -based spaces, for a single (particle) CSDA-equation by using three techniques, the Lions-Lax-Milgram Theorem (variational approach), the theory of m-dissipative operators and the theory evolution operators (semigroup approach). The due a priori estimates are derived and the positivity of solutions are retrieved. In addition, we prove the corresponding results and estimates for the system of coupled transport equations. The related existence results are given for the adjoint problem as well. We also give some computational points (e.g. certain explicit formulas), and we outline a related inverse problem at the end of the paper.

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“…We emphasize that the linear models are a subclass of nonlinear models, nonlinear models can theoretically model both the linear and nonlinear data. However, most nonlinear approaches have well-known shortcomings as they are hard to optimize due to high computational complexity [12] and tend to overfit [13].…”

Section: Introductionmentioning

confidence: 99%