Mathematik und Statistikhttps://kops.uni-konstanz.de:443/handle/123456789/82021-03-09T02:20:57Z2021-03-09T02:20:57ZModulation Equation and SPDEs on Unbounded DomainsBianchi, Luigi Amedeopop516179Blömker, DirkSchneider, Guido123456789/530272021-02-27T04:00:22Z2019Modulation Equation and SPDEs on Unbounded Domains
Bianchi, Luigi Amedeo; Blömker, Dirk; Schneider, Guido
We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space-time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space–time white noise on the dynamics close to this bifurcation. As a first example we study the stochastic Swift–Hohenberg equation on the whole real line. Here, due to the weak regularity of solutions, the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation. As an additional result, we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift–Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. We use energy estimates to show that solutions of the Ginzburg–Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.
2019Bianchi, Luigi AmedeoBlömker, DirkSchneider, Guido510We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space-time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space–time white noise on the dynamics close to this bifurcation. As a first example we study the stochastic Swift–Hohenberg equation on the whole real line. Here, due to the weak regularity of solutions, the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation. As an additional result, we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift–Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. We use energy estimates to show that solutions of the Ginzburg–Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.SpringerJOURNAL_ARTICLEeng10.1007/s00220-019-03573-70010-36161432-091619543711Communications in Mathematical Physics2021-02-26T14:21:05+01:00123456789/39Communications in Mathematical Physics ; 371 (2019), 1. - S. 19-54. - Springer. - ISSN 0010-3616. - eISSN 1432-0916true2021-02-26T13:21:05ZtrueOptiDose : Computing the Individualized Optimal Drug Dosing Regimen Using Optimal ControlBachmann, Freyapop259379Koch, Gilbertpop164441Pfister, MarcSzinnai, GaborSchropp, Johannespop03713123456789/50993.32021-03-05T12:57:36Z2021OptiDose : Computing the Individualized Optimal Drug Dosing Regimen Using Optimal Control
Bachmann, Freya; Koch, Gilbert; Pfister, Marc; Szinnai, Gabor; Schropp, Johannes
Providing the optimal dosing strategy of a drug for an individual patient is an important task in pharmaceutical sciences and daily clinical application. We developed and validated an optimal dosing algorithm (OptiDose) that computes the optimal individualized dosing regimen for pharmacokinetic–pharmacodynamic models in substantially different scenarios with various routes of administration by solving an optimal control problem. The aim is to compute a control that brings the underlying system as closely as possible to a desired reference function by minimizing a cost functional. In pharmacokinetic–pharmacodynamic modeling, the controls are the administered doses and the reference function can be the disease progression. Drug administration at certain time points provides a finite number of discrete controls, the drug doses, determining the drug concentration and its effect on the disease progression. Consequently, rewriting the cost functional gives a finite-dimensional optimal control problem depending only on the doses. Adjoint techniques allow to compute the gradient of the cost functional efficiently. This admits to solve the optimal control problem with robust algorithms such as quasi-Newton methods from finite-dimensional optimization. OptiDose is applied to three relevant but substantially different pharmacokinetic–pharmacodynamic examples.
2021Bachmann, FreyaKoch, GilbertPfister, MarcSzinnai, GaborSchropp, Johannes51049K15Providing the optimal dosing strategy of a drug for an individual patient is an important task in pharmaceutical sciences and daily clinical application. We developed and validated an optimal dosing algorithm (OptiDose) that computes the optimal individualized dosing regimen for pharmacokinetic–pharmacodynamic models in substantially different scenarios with various routes of administration by solving an optimal control problem. The aim is to compute a control that brings the underlying system as closely as possible to a desired reference function by minimizing a cost functional. In pharmacokinetic–pharmacodynamic modeling, the controls are the administered doses and the reference function can be the disease progression. Drug administration at certain time points provides a finite number of discrete controls, the drug doses, determining the drug concentration and its effect on the disease progression. Consequently, rewriting the cost functional gives a finite-dimensional optimal control problem depending only on the doses. Adjoint techniques allow to compute the gradient of the cost functional efficiently. This admits to solve the optimal control problem with robust algorithms such as quasi-Newton methods from finite-dimensional optimization. OptiDose is applied to three relevant but substantially different pharmacokinetic–pharmacodynamic examples.SpringerJOURNAL_ARTICLEeng10.1007/s10957-021-01819-w0022-32391573-2878Journal of Optimization Theory and Applications2021-02-26T09:37:32+01:00123456789/39Journal of Optimization Theory and Applications ; 2021. - Springer. - ISSN 0022-3239. - eISSN 1573-2878true2021-02-26T08:37:32ZtrueROM-based inexact subdivision methods for PDE-constrained multiobjective optimizationBanholzer, Stefanpop216783Gebken, BennetReichle, Lenapop256988Volkwein, Stefanpop214121123456789/529952021-03-01T07:59:37Z2021ROM-based inexact subdivision methods for PDE-constrained multiobjective optimization
Banholzer, Stefan; Gebken, Bennet; Reichle, Lena; Volkwein, Stefan
The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter dependent semilinear elliptic partial differential equation (PDE). To solve it, we use a gradient based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach.
2021Banholzer, StefanGebken, BennetReichle, LenaVolkwein, Stefan510The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter dependent semilinear elliptic partial differential equation (PDE). To solve it, we use a gradient based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach.MDPIJOURNAL_ARTICLEeng2297-87471300-686XMathematical and Computational Applications2021-02-25T10:39:00+01:00123456789/39Mathematical and Computational Applications ; 2021. - MDPI. - ISSN 2297-8747. - eISSN 1300-686Xunknown2021-02-25T09:39:00ZReduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDEBernreuther, Marcopop253089Müller, Georgpop518805Volkwein, Stefanpop214121123456789/49208.32021-02-24T02:02:47Z2021Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE
Bernreuther, Marco; Müller, Georg; Volkwein, Stefan
In this paper, an optimization problem governed by a nonsmooth semilinear elliptic partial differential equation is considered. A reduced order approach is applied in order to obtain a computationally fast and certified numerical solution approach. Using the reduced basis method and efficient a-posteriori error estimation for the primal and dual equations, an adaptive algorithm is developed and tested successfully for several numerical examples.
2021Bernreuther, MarcoMüller, GeorgVolkwein, Stefan510In this paper, an optimization problem governed by a nonsmooth semilinear elliptic partial differential equation is considered. A reduced order approach is applied in order to obtain a computationally fast and certified numerical solution approach. Using the reduced basis method and efficient a-posteriori error estimation for the primal and dual equations, an adaptive algorithm is developed and tested successfully for several numerical examples.De GruyterINCOLLECTIONurn:nbn:de:bsz:352-2-14ynfk4rb5c8r3engRadon Series on Computational and Applied Mathematics2021-02-23T13:57:31+01:00123456789/39Radon Series on Computational and Applied Mathematics. - Berlin : De Gruyter, 2021Berlin2021-02-23T12:57:31Z