Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating the fractional problem either by discretizing the fractional term or expanding the fractional derivatives as a series involving integer order derivatives. The former method, as a subclass of direct methods in the theory of calculus of variations, uses finite differences, Grunwald-Letnikov definition in this case, to discretize the fractional term. Any quadrature rule for integration, regarding the desired accuracy, is then used to discretize the whole problem including constraints. The final task in this method is to solve a static optimization problem to reach approximated values of the unknown functions on some mesh points. The latter method, however, approximates fractional problems by classical ones in which only derivatives of integer order are present. Precisely, two continuous approximations for fractional derivatives by series involving ordinary derivatives are introduced. Local upper bounds for truncation errors are provided and, through some test functions, the accuracy of the approximations are justified. Then we substitute the fractional term in the original problem with these series and transform the fractional problem to an ordinary one. Hereafter, we use indirect methods of classical theory, e.g. Euler-Lagrange equations, to solve the approximated problem. The methods are mainly developed through some concrete examples which either have obvious solutions or the solution is computed using the fractional Euler-Lagrange equation.Comment: This is a preprint of a paper whose final and definite form appeared in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control (Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova Science Publishers, New York, 2014. (See http://www.novapublishers.com/catalog/product_info.php?products_id=46851). Consists of 39 page

Métodos computacionais no cálculo das variações e controlo óptimo fraccionais

Doutoramento em MatemáticaThe fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.O cálculo das variações e controlo óptimo fraccionais são generalizações das correspondentes teorias clássicas, que permitem formulações e modelar problemas com derivadas e integrais de ordem arbitrária. Devido à carência de métodos analíticos para resolver tais problemas fraccionais, técnicas numéricas são desenvolvidas. Nesta tese, investigamos a aproximação de operadores fraccionais recorrendo a séries de derivadas de ordem inteira e diferenças finitas generalizadas. Obtemos majorantes para o erro das aproximações propostas e estudamos a sua eficiência. Métodos directos e indirectos para a resolução de problemas variacionais fraccionais são estudados em detalhe. Discutimos também condições de optimalidade para diferentes tipos de problemas variacionais, sem e com restrições, e para problemas de controlo óptimo fraccionais. As técnicas numéricas introduzidas são ilustradas recorrendo a exemplos

Fractional variational problems depending on indefinite integrals

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered.Comment: Submitted 29-Dec-2010; revised 14-Feb-2011; accepted 16-Feb-2011; for publication in Nonlinear Analysis Series A: Theory, Methods & Application

Fractional derivatives in Dengue epidemics

We introduce the use of fractional calculus, i.e., the use of integrals and derivatives of non-integer (arbitrary) order, in epidemiology. The proposed approach is illustrated with an outbreak of dengue disease, which is motivated by the first dengue epidemic ever recorded in the Cape Verde islands off the coast of west Africa, in 2009. Numerical simulations show that in some cases the fractional models fit better the reality when compared with the standard differential models. The classical results are obtained as particular cases by considering the order of the derivatives to take an integer value.Comment: This is a preprint of a paper accepted for presentation at ICNAAM 2011, Numerical Optimization and Applications Symposium, whose final and definite form will appear in AIP Conference Proceeding

Arbitration between model-free and model-based control is not affected by transient changes in tonic serotonin levels

Background: Serotonin has been suggested to modulate decision-making by influencing the arbitration between model-based and model-free control. Disruptions in these control mechanisms are involved in mental disorders such as drug dependence or obsessive-compulsive disorder. While previous reports indicate that lower brain serotonin levels reduce model-based control, it remains unknown whether increases in serotonergic availability might thus increase model-based control. Moreover, the mediating neural mechanisms have not been studied yet. Aim: The first aim of this study was to investigate whether increased/decreased tonic serotonin levels affect the arbitration between model-free and model-based control. Second, we aimed to identify the underlying neural processes. Methods: We employed a sequential two-stage Markov decision-task and measured brain responses during functional magnetic resonance imaging in 98 participants in a randomized, double-blind cross-over within-subject design. To investigate the influence of serotonin on the balance between model-free and model-based control, we used a tryptophan intervention with three intervention levels (loading, balanced, depletion). We hypothesized that model-based behaviour would increase with higher serotonin levels. Results: We found evidence that neither model-free nor model-based control were affected by changes in tonic serotonin levels. Furthermore, our tryptophan intervention did not elicit relevant changes in Blood-Oxygenation-Level Dependent activity

Discrete Direct Methods in the Fractional Calculus of Variations

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.Comment: This work was partially presented 16-May-2012 by Shakoor Pooseh, at FDA'2012, who received a 'Best Oral Presentation Award'. Submitted 26-Aug-2012; revised 25-Jan-2013; accepted 29-Jan-2013; for publication in Computers and Mathematics with Application

Dynamic Modelling of Mental Resilience in Young Adults: Protocol for a Longitudinal Observational Study (DynaM-OBS)

Background Stress-related mental disorders are highly prevalent and pose a substantial burden on individuals and society. Improving strategies for the prevention and treatment of mental disorders requires a better understanding of their risk and resilience factors. This multicenter study aims to contribute to this endeavor by investigating psychological resilience in healthy but susceptible young adults over 9 months. Resilience is conceptualized in this study as the maintenance of mental health or quick recovery from mental health perturbations upon exposure to stressors, assessed longitudinally via frequent monitoring of stressors and mental health. Objective This study aims to investigate the factors predicting mental resilience and adaptive processes and mechanisms contributing to mental resilience and to provide a methodological and evidence-based framework for later intervention studies. Methods In a multicenter setting, across 5 research sites, a sample with a total target size of 250 young male and female adults was assessed longitudinally over 9 months. Participants were included if they reported at least 3 past stressful life events and an elevated level of (internalizing) mental health problems but were not presently affected by any mental disorder other than mild depression. At baseline, sociodemographic, psychological, neuropsychological, structural, and functional brain imaging; salivary cortisol and α-amylase levels; and cardiovascular data were acquired. In a 6-month longitudinal phase 1, stressor exposure, mental health problems, and perceived positive appraisal were monitored biweekly in a web-based environment, while ecological momentary assessments and ecological physiological assessments took place once per month for 1 week, using mobile phones and wristbands. In a subsequent 3-month longitudinal phase 2, web-based monitoring was reduced to once a month, and psychological resilience and risk factors were assessed again at the end of the 9-month period. In addition, samples for genetic, epigenetic, and microbiome analyses were collected at baseline and at months 3 and 6. As an approximation of resilience, an individual stressor reactivity score will be calculated. Using regularized regression methods, network modeling, ordinary differential equations, landmarking methods, and neural net–based methods for imputation and dimension reduction, we will identify the predictors and mechanisms of stressor reactivity and thus be able to identify resilience factors and mechanisms that facilitate adaptation to stressors. Results Participant inclusion began in October 2020, and data acquisition was completed in June 2022. A total of 249 participants were assessed at baseline, 209 finished longitudinal phase 1, and 153 finished longitudinal phase 2. Conclusions The Dynamic Modelling of Resilience–Observational Study provides a methodological framework and data set to identify predictors and mechanisms of mental resilience, which are intended to serve as an empirical foundation for future intervention studies. International Registered Report Identifier (IRRID) DERR1-10.2196/3981

Temporal discounting and smoking cessation: choice consistency predicts nicotine abstinence in treatment-seeking smokers

Introduction: Smokers discount delayed rewards steeper than non-smokers or ex-smokers, possibly due to neuropharmacological effects of tobacco on brain circuitry, or lower abstinence rates in smokers with steep discounting. To delineate both theories from each other, we tested if temporal discounting, choice inconsistency, and related brain activity in treatment-seeking smokers (1) are higher compared to non-smokers, (2) decrease after smoking cessation, and (3) predict relapse. Methods: At T1, 44 dependent smokers, 29 non-smokers, and 30 occasional smokers underwent fMRI while performing an intertemporal choice task. Smokers were measured before and 21 days after cessation if abstinent from nicotine. In total, 27 smokers, 28 non-smokers, and 29 occasional smokers were scanned again at T2. Discounting rate k and inconsistency var(k) were estimated with Bayesian analysis. Results: First, k and var(k) in smokers in treatment were not higher than in non-smokers or occasional smokers. Second, neither k nor var(k) changed after smoking cessation. Third, k did not predict relapse, but high var(k) was associated with relapse during treatment and over 6 months. Brain activity in valuation and decision networks did not significantly differ between groups and conditions. Conclusion: Our data from treatment-seeking smokers do not support the pharmacological hypothesis of pronounced reversible changes in discounting behavior and brain activity, possibly due to limited power. Behavioral data rather suggest that differences between current and ex-smokers might be due to selection. The association of choice consistency and treatment outcome possibly links consistent intertemporal decisions to remaining abstinent