2019-07-04, Almeida, Luís, Bagnerini, Patrizia, Fabrini, Giulia, Hughes, Barry D., Lorenzi, Tommaso

We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the cell population density function. We develop a novel method for constructing exact solutions to the model equation, which allows for a systematic investigation of the way in which the size and the phenotypic composition of the cell population change in response to variations of the drug dose and other evolutionary parameters. Moreover, we address numerical optimal control for a calibrated version of the model based on biological data from the existing literature, in order to identify the drug delivery schedule that makes it possible to minimise either the population size at the end of the treatment or the average population size during the course of treatment. The results obtained challenge the notion that traditional high-dose therapy represents a “one-fits-all solution” in anticancer therapy by showing that the continuous administration of a relatively low dose of the cytotoxic drug performs more closely to i.e. the optimal dosing regimen to minimise the average size of the cancer cell population during the course of treatment.

2017-09-06, Alla, Alessandro, Fabrini, Giulia, Falcone, Maurizio

We consider an optimal control problem where the dynamics is given by the propagation of a one-dimensional graph controlled by its normal speed. A target corresponding to the final configuration of the front is given and we want to minimize the cost to reach the target. We want to solve this optimal control problem via the dynamic programming approach but it is well known that these methods suffer from the “curse of dimensionality” so that we can not apply the method to the semi-discrete version of the dynamical system. However, this is made possible by a reduced-order model for the level set equation which is based on Proper Orthogonal Decomposition. This results in a new low-dimensional dynamical system which is sufficient to track the dynamics. By the numerical solution of the Hamilton-Jacobi-Bellman equation related to the POD approximation we can compute the feedback law and the corresponding optimal trajectory for the nonlinear front propagation problem. We discuss some numerical issues of this approach and present a couple of numerical examples.

2018, Fabrini, Giulia, Iapichino, Laura, Volkwein, Stefan

Often a dynamical system is characterized by one or more parameters describing physical features of the problem or geometrical configurations of the computational domain. As a consequence, by assuming that the system is controllable, a range of optimal controls exists corresponding to different parameter values. The goal of the proposed approach is to avoid the computation of a control function for any instance of the parameters. The greedy controllability consists in the selection of the most representative values of the parameter set that allows a rapid approximation of the control function for any desired new parameter value, ensuring that the system is steered to the target within a certain accuracy. By proposing the reduced basis (RB) method in this framework, we are able to consider linear parametrized partial differential equations (PDEs) in our setting. The computational costs are drastically reduced and the efficiency of the greedy controllability approach is significantly improved. As a numerical example a heat equation with convection is studied to illustrate our proposed RB greedy controllability strategy.

2017, Fabrini, Giulia, Falcone, Maurizio, Volkwein, Stefan

In this paper we use a reference trajectory computed by a model predic- tive method to shrink the computational domain where we set the Hamilton-Jacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate the successful realization of the proposed strategy.

2018, Ciaramella, Gabriele, Fabrini, Giulia

The approximation of feedback control via the Dynamic Programming approach is a challenging problem. The computation of the feedback requires the knowledge of the value function, which can be characterized as the unique viscosity solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) equation. The major obstacle is that the numerical methods known in literature strongly suffer when the dimension of the discretized problem becomes large. This is a strong limitation to the application of classical numerical schemes for the solution of the HJB equation in real applications. To tackle this problem, a new multi-level numerical framework is proposed. Numerical evidences show that classical methods have good smoothing properties, which allow one to use them as smoothers in a multilevel strategy. Moreover, a new smoother iterative scheme based on the Anderson acceleration of the classical value function iteration is introduced. The effectiveness of our new framework is proved by several numerical experiments focusing on minimum-time control problems.