Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

Kunisch, Karl; Wagner, Marcus

doi:10.1051/m2an/2012058

Cited by 17 publications

(13 citation statements)

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“…This is true not only for DA but also for the identification of the optimal realization of a therapy or, more specifically, of a surgical intervention. For instance, in [49,50,56] the identification of the optimal placement of leads for optimizing pacemaking action in the heart is addressed; the computation of a personalized patient-specific peritoneal dialysis is addressed in [63,Chap. 7].…”

Section: Discussionmentioning

confidence: 99%

Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction

Bertagna

D’Elia

Perego

et al. 2014

Fluid-Structure Interaction and Biomedical Applications

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Numerical methods for incompressible fluid dynamics have recently received a strong impulse from the applications to the cardiovascular system. In particular, fluid-structure interaction methods have been extensively investigated in view of an accurate and possibly fast simulation of blood flow in arteries and veins. This has been strongly motivated by the progressive interest in using numerical tools not only for understanding the general physiology and pathology of the vascular system. The opportunity offered by medical images properly preprocessed and elaborated to simulate blood flow in real patients highlighted the potential impact of scientific computing on the clinical practice. Therefore, in silico experiments are currently extensively used in bioengineering for completing (and sometimes driving) more traditional in vivo and in vitro investigations. Parallel to the development of numerical models, the need for quantitative analysis for diagnostic purposes has strongly stimulated the design of new methods and instruments for measurements and imaging. Thanks to these developments, a huge amount of data is nowadays available. Data Assimilation is the accurate merging of measures (including images) and numerical simulations for a mathematically sound integration of different sources of information. The outcome of this process includes both the patient-specific measures and the general principles underlying the development of mathematical models. In this way, simulations are adapted to the availability of individual data and are therefore supposed to be more reliable; measures are correspondingly filtered by the mathematical models assumed to describe the underlying phenomena, resulting in a (hopefully) significant reduction of the noise.This chapter provides an introduction to methods for data assimilation, mostly developed in fields like meteorology, applied to computational hemodynamics. We focus mainly on two of them: methods based on stochastic arguments (Kalman filtering) and variational methods. We also address some examples that have been approached with different techniques, in particular the estimation of vascular compliance from displacement measures.

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Section: Discussionmentioning

confidence: 99%

Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction

Bertagna

D’Elia

Perego

et al. 2014

Fluid-Structure Interaction and Biomedical Applications

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“…For the case of distributed control action the details are provided in [12]. For computational purposes the first order necessary conditions are of paramount importance.…”

Section: The Optimal Control Problemmentioning

confidence: 99%

“…We give a formal derivation of these conditions here. They can be made rigorous by modifications of the proofs given in [12].…”

Section: The Optimal Control Problemmentioning

confidence: 99%

On boundary stimulation and optimal boundary control of the bidomain equations

Chamakuri

Kunisch

Plank

2013

Mathematical Biosciences

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The bidomain equations with Neumann boundary stimulation and optimal control of these stimuli are investigated. First an analytical framework for boundary control is provided. Then a parallel finite element based algorithm is devised and its efficiency is demonstrated not only for the direct problem but also for the optimal control problem. The computations realize a model configuration corresponding to optimal boundary defibrillation of a reentry phenomenon by applying current density stimuli.

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“…2. It should be mentioned that there exists a large literature concerning the optimal control problems governed by the deterministic FitzHugh-Nagumo equation, see, e.g., [10,19], while to the best of our knowledge, the stochastic case that we are interested in, lacks of such results. The motivation is that existence of an optimal control for the stochastic problem we consider here is quite a delicate problem which cannot be solved with standard optimization arguments which require the weak lower semicontinuity of cost functional in the control basic space and a more subtle argument based on Eckelands's variational principle was used.…”

Section: Introductionmentioning

confidence: 98%