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Some numerical techniques for approximating semilinear parabolic (stochastic) partial differential equationsMukam, Jean Daniel 11 October 2021 (has links)
Partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) are powerful tools in modeling real-world phenomena in many fields
such as geo-engineering. For instance processes such as oil or gas recovery from hydrocarbon reservoirs and mining heat from geothermal reservoirs can be modeled
by PDEs or SPDEs. An important task is to understand the behavior of such phenomena. This can be achieved through explicit solutions of equations. Since
explicit solutions of many PDEs and SPDEs are rarely known, developing numerical schemes is a good alternative to provide approximations of these explicit solutions.
The study of numerical solutions of PDEs and SPDEs is therefore an active research area and has attracted a lot of attentions since at least two decades. The aims of this dissertation is to develop numerical schemes to approximate semilinear parabolic PDEs and SPDEs in space and in time. The approximation in space is done via the standard Galerkin finite element method and the approximation in time, which is the core of our work is done via various numerical integrators. This dissertation consists of two general parts.
The first part of this thesis deals with autonomous PDEs and SPDEs. Here, our main interest is on semilinear PDEs and SPDEs where the nonlinear part is stronger
than the linear part also called (stochastic) reactive dominated transport equations, or stiff problems. For such problems, many numerical techniques in the current
scientific literature lose their good stability properties. We develop a new explicit exponential integrator (called exponential Rosenbrock-type method) and a new
semi-implicit method (called linear implicit Rosenbrock-type method), appropriate for such PDEs and SPDEs. We analyze the strong convergence of our novel fully
discrete schemes towards the mild solution of the (S)PDE and obtain convergence orders similar to existing ones in the literature.
The second part of this thesis focuses on numerical approximations of semilinear non-autonomous parabolic PDEs and SPDEs. Such equations are more realistic than the autonomous ones and find applications in many fields such as fluid mechanics, quantum field theory, electromagnetism, etc. Numerics of non-autonomous semilinear parabolic PDEs and SPDEs are far from being well understood in the literature. We fill that gap in this thesis by developing a new exponential integrator (called Magnus-type method) and the semi-implicit method for such problems and provide their strong convergence towards the mild solution. Moreover, for both autonomous and non-autonomous SPDEs driven by additive noise, we achieve optimal convergence order in time 1 or approximately 1. Numerical simulations are provided to illustrate our theoretical findings in both autonomous and non-autonomous cases.
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Geometric Integrators for Schrödinger EquationsBader, Philipp Karl-Heinz 11 July 2014 (has links)
The celebrated Schrödinger equation is the key to understanding the dynamics of
quantum mechanical particles and comes in a variety of forms. Its numerical solution
poses numerous challenges, some of which are addressed in this work.
Arguably the most important problem in quantum mechanics is the so-called harmonic
oscillator due to its good approximation properties for trapping potentials. In
Chapter 2, an algebraic correspondence-technique is introduced and applied to construct
efficient splitting algorithms, based solely on fast Fourier transforms, which
solve quadratic potentials in any number of dimensions exactly - including the important
case of rotating particles and non-autonomous trappings after averaging by Magnus
expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii
equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is
introduced and it is shown how to efficiently compute them using Fourier transforms.
It is shown how to apply complex coefficient splittings to this nonlinear equation and
numerical results corroborate the findings.
In the semiclassical limit, the evolution operator becomes highly oscillatory and standard
splitting methods suffer from exponentially increasing complexity when raising
the order of the method. Algorithms with only quadratic order-dependence of the
computational cost are found using the Zassenhaus algorithm. In contrast to classical
splittings, special commutators are allowed to appear in the exponents. By construction,
they are rapidly decreasing in size with the semiclassical parameter and can be
exponentiated using only a few Lanczos iterations. For completeness, an alternative
technique based on Hagedorn wavepackets is revisited and interpreted in the light of
Magnus expansions and minor improvements are suggested. In the presence of explicit
time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm
requires a special initiation step. Distinguishing the case of smooth and fast frequencies,
it is shown how to adapt the mechanism to obtain an efficiently computable
decomposition of an effective Hamiltonian that has been obtained after Magnus expansion,
without having to resolve the oscillations by taking a prohibitively small
time-step.
Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as
an initial value problem after a Wick-rotating the Schrödinger equation to imaginary
time. The elliptic nature of the evolution operator restricts standard splittings to
low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps
that correspond to the ill-posed integration backwards in time. The inclusion
of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be
circumvented using complex fractional time-steps with positive real part and sixthorder
methods optimized for near-integrable Hamiltonians are presented.
Conclusions and pointers to further research are detailed in Chapter 6, with a special
focus on optimal quantum control. / Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716 / TESIS / Premios Extraordinarios de tesis doctorales
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From single decisions to sequential choice patterns: Extending the dynamics of value-based decision-makingScherbaum, Stefan, Lade, Steven J., Siegmund, Stefan, Goschke, Thomas, Dshemuchadse, Maja 04 June 2024 (has links)
Every day, we make many value-based decisions where we weigh the value of options with other properties, e.g. their time of delivery. In the laboratory, such value-based decision-making is usually studied on a trial by trial basis and each decision is assumed to represent an isolated choice process. Real-life decisions however are usually embedded in a rich context of previous choices at different time scales. A fundamental question is therefore how the dynamics of value-based decision processes unfold on a time scale across several decisions. Indeed, findings from perceptual decision making suggest that sequential decisions patterns might also be present for vale-based decision making. Here, we use a neural-inspired attractor model as an instance of dynamic models from perceptual decision making, as such models incorporate inherent activation dynamics across decisions. We use the model to predict sequential patterns, namely oscillatory switching, perseveration and dependence of perseveration on the delay between decisions. Furthermore, we predict RT effects for specific sequences of trials. We validate the predictions in two new studies and a reanalysis of existing data from a novel decision game in which participants have to perform delay discounting decisions. Applying the validated reasoning to a well-established choice questionnaire, we illustrate and discuss that taking sequential choice patterns into account may be necessary to accurately analyse and model value-based decision processes, especially when considering differences between individuals.
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Intégrateurs temporels basés sur la resommation des séries divergentes : applications en mécanique / Time integrators based on divergent series resummation : applications in mechanicsDeeb, Ahmad 17 December 2015 (has links)
Les systèmes dynamiques qui évoluent sur un grand intervalle de temps (dynamique moléculaire, prédiction astronomique, turbulence...) occupent une place importante dans le domaine de la science de l'ingénieur. Leur résolution numérique constitue, jusqu'à l'heure actuelle, un défi. En effet, la simulation de la solution nécessite un solveur non seulement rapide mais aussi qui respecte les propriétés physiques du problème, pour garantir la stabilité. Dans cette thèse, on se propose d'étudier, vis-à-vis de cette problématique, un schéma d'intégration temporelle basée sur la décomposition de la solution en série temporelle, suivie de la technique de resommation de Borel des séries divergentes. On analyse alors la rapidité du schéma sur des problèmes modèles. Ensuite, on montre sa capacité à préserver la structure des équations (symplecticité, iso-spectralité, conservation de l'énergie...) à un ordre arbitrairement élevé. Par la suite, on applique le schéma à la résolution d'équations aux dérivées partielles issues de la mécanique, dont les équations de la chaleur, de Burgers et de Navier-Stokes bidimensionnelles. Pour cela, on associe le schéma à une méthode de discrétisation par éléments finis en espace. Enfin, dans le but de rendre l'algorithme plus robuste, on s'intéresse à la représentation de la somme de Borel par une série de factorielle généralisée. / Dynamical systems which evolve in a large time interval (molecular dynamic, astronomical prediction, turbulence…) take an important place in engineering science. Their numerical resolution has so far constituted a challenge. Indeed, the simulation of the solution requires a solver which is not only fast but also respects the physical properties of the problem, to ensure the stability. In this thesis, we propose to study, regarding this issue, a time integration scheme based on the decomposition of the solution into time series, followed by Borel's resummation technique of divergent series. We analyse the speed of scheme on model problems. Next, we show its capability to preserve the structure of the equation (symplecticity, iso-spectrality, conservation of energy…) up to an arbitrary high order. Thereafter, we use the scheme to resolve partial differential equations coming from mechanics, including the two-dimensional heat equation, Burger’s equation and the Navier-Stokes equation. To this aim, we choose a finite element method for space discretisation. Finally, and in order to make the algorithm more robust, we are interested in the representation of the Borel sum by a generalized factorials series.
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Contributions au calcul des variations et au principe du maximum de Pontryagin en calculs time scale et fractionnaire / Contributions to calculus of variations and to Pontryagin maximum principle in time scale calculus and fractional calculusBourdin, Loïc 18 June 2013 (has links)
Cette thèse est une contribution au calcul des variations et à la théorie du contrôle optimal dans les cadres discret, plus généralement time scale, et fractionnaire. Ces deux domaines ont récemment connu un développement considérable dû pour l’un à son application en informatique et pour l’autre à son essor dans des problèmes physiques de diffusion anormale. Que ce soit dans le cadre time scale ou dans le cadre fractionnaire, nos objectifs sont de : a) développer un calcul des variations et étendre quelques résultats classiques (voir plus bas); b) établir un principe du maximum de Pontryagin (PMP en abrégé) pour des problèmes de contrôle optimal. Dans ce but, nous généralisons plusieurs méthodes variationnelles usuelles, allant du simple calcul des variations au principe variationnel d’Ekeland (couplé avec la technique des variations-aiguilles), en passant par l’étude d’invariances variationnelles par des groupes de transformations. Les démonstrations des PMPs nous amènent également à employer des théorèmes de point fixe et à prendre en considération la technique des multiplicateurs de Lagrange ou encore une méthode basée sur un théorème d’inversion locale conique. Ce manuscrit est donc composé de deux parties : la Partie 1 traite de problèmes variationnels posés sur time scale et la Partie 2 est consacrée à leurs pendants fractionnaires. Dans chacune de ces deux parties, nous suivons l’organisation suivante : 1. détermination de l’équation d’Euler-Lagrange caractérisant les points critiques d’une fonctionnelle Lagrangienne ; 2. énoncé d’un théorème de type Noether assurant l’existence d’une constante de mouvement pour les équations d’Euler-Lagrange admettant une symétrie ; 3. énoncé d’un théorème de type Tonelli assurant l’existence d’un minimiseur pour une fonctionnelle Lagrangienne et donc, par la même occasion, d’une solution pour l’équation d’Euler-Lagrange associée (uniquement en Partie 2) ; 4. énoncé d’un PMP (version forte en Partie 1, version faible en Partie 2) donnant une condition nécessaire pour les trajectoires qui sont solutions de problèmes de contrôle optimal généraux non-linéaires ; 5. détermination d’une condition de type Helmholtz caractérisant les équations provenant d’un calcul des variations (uniquement en Partie 1 et uniquement dans les cas purement continu et purement discret). Des théorèmes de type Cauchy-Lipschitz nécessaires à l’étude de problèmes de contrôle optimal sont démontrés en Annexe. / This dissertation deals with the mathematical fields called calculus of variations and optimal control theory. More precisely, we develop some aspects of these two domains in discrete, more generally time scale, and fractional frameworks. Indeed, these two settings have recently experience a significant development due to its applications in computing for the first one and to its emergence in physical contexts of anomalous diffusion for the second one. In both frameworks, our goals are: a) to develop a calculus of variations and extend some classical results (see below); b) to state a Pontryagin maximum principle (denoted in short PMP) for optimal control problems. Towards these purposes, we generalize several classical variational methods, including the Ekeland’s variational principle (combined with needle-like variations) as well as variational invariances via the action of groups of transformations. Furthermore, the investigations for PMPs lead us to use fixed point theorems and to consider the Lagrange multiplier technique and a method based on a conic implicit function theorem. This manuscript is made up of two parts : Part A deals with variational problems on time scale and Part B is devoted to their fractional analogues. In each of these parts, we follow (with minor differences) the following organization: 1. obtaining of an Euler-Lagrange equation characterizing the critical points of a Lagrangian functional; 2. statement of a Noether-type theorem ensuring the existence of a constant of motion for Euler-Lagrange equations admitting a symmetry;3. statement of a Tonelli-type theorem ensuring the existence of a minimizer for a Lagrangian functional and, consequently, of a solution for the corresponding Euler-Lagrange equation (only in Part B); 4. statement of a PMP (strong version in Part A and weak version in Part B) giving a necessary condition for the solutions of general nonlinear optimal control problems; 5. obtaining of a Helmholtz condition characterizing the equations deriving from a calculus of variations (only in Part A and only in the purely continuous and purely discrete cases). Some Picard-Lindelöf type theorems necessary for the analysis of optimal control problems are obtained in Appendices.
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