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We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal Ministerio de Ciencia, Innovación y Universidades PID2019-109175GB-C22 Junta de Andalucía US-1380977

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal.

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms., 38 pages, 10 figures

We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations., 24 pages, 16 figures

PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. For doing that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modelling parts, and the worst-case analysis is performed numerically via a standard solver., Reference work for the PEPit package (available at https://github.com/bgoujaud/PEPit)

In this thesis, we propose, analyse and implement numerical methods for time-dependent non-linear parabolic and Schrödinger-type equations with uncertain parameters. The discretisation of the parameter space which incorporates the uncertainty of the problem is performed via single- and multi-level collocation strategies. To deal with the possibly large dimension of the parameter space, sparse grid collocation techniques are used to alleviate the curse of dimensionality to a certain extent. We prove that the multi-level method is capable of reducing the overall computational costs significantly. In the parabolic case, the time discretisation is performed via an implicit-explicit splitting strategy of order two which consists shortly speaking of a combination of an implicit trapezoidal rule for the stiff linear part and Heun's method for the non-linear part. In the Schrödinger case, time is discretised via the famous second-order Strang splitting method. For both problem classes we review known error bounds for both discretizations and prove new error bounds for the time discretisations which take the regularity in the parameter space into account. In the parabolic case, a new error bound for the "implicit-explicit trapezoidal method" (IMEXT) method is proved. To our knowledge, this error bound stating second-order convergence of the IMEXT method closes a current gap in the literature. Utilising the aforementioned new error bounds for both problem classes, we can rigorously prove convergence of the single- and multi-level methods. Additionally, cost savings of the multi-level methods compared to the single-level approach are predicted and verifed by numerical examples. The results mentioned above are novel contributions in two areas of mathematics. The first one is (analysis of) numerical methods for uncertainty quantification and the second one is numerical analysis of time-integration schemes for PDEs.

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms. Linz Center of Mechatronics COMET-K2 Fonds zur Förderung der Wissenschaftlichen Forschung (DE-588)2054142-9 W1214-N15 Version of record

Splitting-Verfahren bieten eine Klasse von expliziten Integratoren für Systeme von Differentialgleichungen, welche in einzelne Teile aufgesplittet werden können. Diese Verfahren sind in der Praxis leicht zu implementieren, brauchen nur geringe Rechenkosten und bieten interessante Eigenschaften wie die Erhaltung der Systemenergie. Das Hauptziel dieser Arbeit ist die Entwicklung von verschiedene Ansätze für das Splitting von Systemen von stochastische Differentialgleichungen. Diese Splitting Ansätze werden dann verwendet für einen numerischen Vergleich von Lie-Trotter- und Strang Splitting-Verfahren mit stochastische nicht-Splitting Verfahren. Insbesondere der Mittelwert der Systemenergie ist der Hauptaugenmerk der angewandten numerischen Tests. Zuallerst gibt es eine Einführung in die Theorie von stochastische Differentialgleichungen. Dann werden mehrere lineare und nichtlineare stochastische Systeme von Oszillatoren mit additiven Rauschen sowie multiplikativem Rauschen vorgestellt und besprochen. Insgesamt werden vier verschiedene Splitting Ansätze für die stochastische Systeme von Oszillatoren mit additiven Rauschen getestet. Diese Splitting Ansätze werden angewandt auf lineare Modelle als auch auf ein nichtlineare Modell. Es wird gezeigt, dass die für die linearen stochastische Modelle entwickelten expliziten Splitting Schemen ohne viel Anstrengung auf ähnliche nichtlineare stochastische Modellen erweitert werden können. Der zweite Modell Typ den wir betrachten werden, ist das Modell eines echten mechanischen invertierten Pendels. Für dieses Modell werden wir drei verschiedene Splitting Ansätze präsentieren und analysieren. Diese Ansätze werden für ein deterministisches sowie ein stochastisches Modell des invertierten Pendels getestet. Splitting and composition methods provide a class of explicit integrators for systems of differen- tial equations that can be dismantled in multiple parts. These methods are in practice simple to implement, have low computational cost and provide interesting features like system energy preserving properties. The main goal of this thesis is to construct various splitting approaches for different stochastic differential equation systems. These splitting approaches are then used for a numerical comparison of the Lie-Trotter and Strang splitting methods with stochastic non-splitting methods. Especially the mean system energy behavior is the point of focus in the applied numeric tests. At first an introduction into stochastic differential equation theory is provided. Then several linear and nonlinear stochastic oscillatory systems with additive as with multiplicative noise are proposed and discussed. Overall, four different splitting approaches for the stochastic oscillatory system with additive noise will be tested. These splitting schemes will be applied on the linear models as well as on the nonlinear model. It will be shown that the explicit splitting schemes designed for the linear stochastic models can be extended to similar nonlinear stochastic models without much effort. The second considered model type is the model of a real mechanical inverted pendulum. For this model we will present and analyze three different splitting approaches. These splitting approaches will be tested for a deterministic and for a stochastic model of the inverted pendulum. submitted by Philipp Wagner Masterarbeit Universität Linz 2022

Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy and efficiency. Simulations were performed for a benchmark configuration with the Beeler–Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this benchmark configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time.

Efficient simulation of contact is of interest for numerous physics-based animation applications. For instance, virtual reality training, video games, rapid digital prototyping, and robotics simulation are all examples of applications that involve contact modeling and simulation. However, despite its extensive use in modern computer graphics, contact simulation remains one of the most challenging problems in physics-based animation. This course covers fundamental topics on the nature of contact modeling and simulation for computer graphics. Specifically, we provide mathematical details about formulating contact as a complementarity problem in rigid body and soft body animations. We briefly cover several approaches for contact generation using discrete collision detection. Then, we present a range of numerical techniques for solving the associated LCPs and NCPs. The advantages and disadvantages of each technique are further discussed in a practical manner, and best practices for implementation are discussed. Finally, we conclude the course with several advanced topics such as methods for soft body contact problems, barrier functions, and anisotropic friction modeling. Programming examples are provided in our appendix as well as on the course website to accompany the course notes.

The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

International audience; We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker-Planck equations.

In (Bernier in Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators. arXiv:1912.13219 , (2019)), some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, Fokker–Planck equations, and Schrodinger type equations with an angular momentum rotation term. In this work, these exact splittings are used combined with pseudo-spectral methods in space. High accuracy and efficiency of exact splitting methods are illustrated and comparison are performed with the numerical methods in literature. We show that our methods can be used to improve significantly some classical splitting methods for some nonlinear or non-quadratic equations.

We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group $\mathrm{SU}(2)$. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space., Comment: 18 pages, 7 figures. To be published in Journal of Computational Dynamics

The initial boundary-value problem associated to a semilinear wave equation with time dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases., This research was funded by the Ministerio de Ciencia y Educación, grant number PGC2018- 101443-B-I00, and the first author by Consejería de Educación, Junta de Castilla y León y Leónand Feder funds, grant number VA193P20.

Das menschliche Gehirn ist ein hochkomplexes und stark vernetztes dynamisches System. Zu verstehen, wie Informationen in diesem System generiert, verarbeitet und übermittelt werden, ist eines der wesentlichen Ziele auf dem Gebiet der Neurowissenschaften. Um neuronale Daten erklären und reproduzieren zu können, werden interpretierbare mathematische Modelle benötigt. Stochastische Differentialgleichungen (SDEs) berücksichtigen nicht prognostizierbare, zufällige Einflüsse und bilden daher ein etabliertes Modellierungshandwerk auf diesem Gebiet. Diese kumulative Dissertation beinhaltet mehrere Beiträge zur numerischen und statistischen Analyse von SDEs, inklusive einer Reihe von biologisch relevanten Modellen zur Beschreibung neuronaler Dynamiken. Sie besteht aus einem einleitenden Kapitel gefolgt von vier wissenschaftlichen Artikeln. Ein spezieller Fokus dieser Arbeit liegt auf der Konstruktion und Analyse von strukturerhaltenden numerischen Methoden für SDEs, diese reproduzieren wichtige strukturelle Eigenschaften ihrer Lösungen. Wir entwickeln numerische Verfahren, basierend auf der Splitting-Methode, deren Stärke in der Zusammensetzung von explizit gelösten Teilen der SDE, welche die Erhaltung zugrundeliegender Eigenschaften garantieren, besteht. Im Besonderen beweisen wir die Erhaltung von Hypoelliptizität, Ergodizität, Momenten, Schranken von Momenten und Grenzeigenschaften. Ein besonderer Schwerpunkt dieser Dissertation liegt darüber hinaus auf der Untersuchung des Zusammenspiels von numerischen und statistischen Methoden. Dabei schaffen wir Bewusstsein für die Bedeutung von Strukturerhaltung, wenn das numerische Verfahren als essentieller Baustein bei der Konstruktion von statistischen Inferenzmethoden dient. Im Speziellen entwickeln wir eine strukturerhaltende Approximate Bayesian Computation Methode für das Schätzen der Parameter von SDEs. Die Stärke dieser Methode liegt in der Kombination von zusammenfassenden Statistiken, welche auf der Struktureigenschaft eines zugrundeliegenden invarianten Maßes beruhen, und strukturerhaltenden Splitting-Methoden zur Simulation synthetischer Daten. Aufgrund der Verwendung von strukturerhaltenden numerischen Verfahren, ist der vorgeschlagene Inferenzalgorithmus zuverlässig, rechnerisch realisierbar und liefert zufriedenstellende Schätzergebnisse, sowohl basierend auf simulierten Daten als auch unter Verwendung von Elektroenzephalographie (EEG) Daten. Im Gegensatz zu den entwickelten Splitting-Verfahren, sind häufig verwendete numerische Methoden, beispielsweise das Euler-Maruyama Verfahren, im Allgemeinen nicht strukturerhaltend. Folglich liefern sie schlecht konditionierte oder rechnerisch nicht realisierbare Inferenzmethoden. The human brain is a highly complex interconnected dynamical system. Understanding how information is generated, processed and transmitted in this system is one of the major goals in the field of neuroscience. The aim to explain and reproduce neuronal data is reflected in an increasing demand for developing and analysing interpretable mathematical models. Taking into account unpredictable forces and the effect of noise, stochastic differential equations (SDEs) have become an established and powerful modelling tool in this field. This cumulative thesis presents several contributions to the numerical and statistical analysis of SDEs, including various biologically relevant models describing neuronal dynamics. It consists of an introductory essay followed by four scientific articles. A particular focus of this thesis lies on the construction and analysis of structure preserving numerical methods for SDEs which are capable to reproduce important structural properties of their solutions. We develop several numerical methods based on the splitting approach. The strength of the proposed methods lies in the composition of explicitly solved subparts of the SDE, through which the preservation of underlying properties can be guaranteed. In particular, we prove the preservation of hypoellipticity, ergodicity, moments, moment bounds and boundary properties. In addition, a special emphasis of this thesis is laid on the investigation of the interplay of numerical and statistical methods. This means that we draw attention to the importance of structure preservation when using numerical methods as essential building blocks in the construction of statistical inference tools. In particular, we develop a structure preserving approximate Bayesian computation method to estimate the parameters of SDEs. The strength of this method lies in the combination of summary statistics, which are based on the structural property of an underlying invariant distribution, and structure preserving splitting methods for the simulation of huge amounts of required synthetic datasets. The use of structure preserving numerical methods makes the proposed inference algorithm reliable, computationally feasible and enables successful parameter estimation from both simulated and real electroencephalography (EEG) data. In contrast to the proposed splitting methods, frequently applied numerical methods, such as the Euler-Maruyama discretisation, fail to preserve structural properties of solutions of SDEs in general. Consequently, they may yield ill-conditioned or computationally infeasible inference methods. eingereicht von DI Irene Tubikanec Universität Linz, Dissertation, 2021 (VLID)5811749

The research community has been making significant progress in hardware implementation, numerical computing and algorithm development for optimization-based control. However, there are two key challenges that still have to be overcome for optimization-based control to be a viable option in the context of advanced industrial applications. First, the large existing gap between algorithm development and its deployment on platforms used by practitioners in industry. Second, from a more theoretical viewpoint, the lack of robustness of certain approaches, which are based on the unreasonable assumption that the model at hand perfectly represents the object under investigation. This thesis addresses the aforementioned challenges by establishing software toolboxes for automatic code generation, and proposing a data-driven methodology to enhance the performance of real-time optimization strategies during operation. The first part of this thesis focuses on the efficient implementation of Model Predictive Control (MPC) based on first-order operator splitting methods. Because of the cheap numerical operations associated with them, splitting methods are favorable candidates for applications with limited computing power. We first identify the computational bottlenecks and, subsequently, discuss their efficient deployment on processors, Field Programmable Gate Arrays (FPGA), and heterogeneous platforms. For rapid prototyping and deployment, two code generation toolboxes are developed: SPLIT and LAFF. These possess a high-level parsing interface for MATLAB and yield optimized C code that can be directly used in a variety of FPGA platforms. Features such as pipelining, memory partitioning, and parallelization are automatically incorporated, not requiring users to have in-depth knowledge about computer architecture and low-level programming. We then propose a framework to a priori solve the co-design problem arising in splitting method-based MPC to provide trade-offs between resources and latency. We provide analytical expressions that can avoid the daunting and time-consuming task of exploring the design space manually, thus reducing the final application development time. The second part of the thesis deals with learning plant-model mismatch using Gaussian processes (GPs) in Real Time Optimization (RTO) schemes. Inaccurate models, the presence of disturbances, and time-varying conditions typically lead to the suboptimal operation of many plants. We use data-driven global surrogate models in the form of GPs to cope with such problems and show better numerical convergence and handling of noise effectively when compared to standard RTO techniques. We moreover prove that GPs can be certified as probabilistic and deterministic fully linear models, a key property to guarantee global convergence of derivative-free trust region (DFT) methods. We then propose a novel DFT methodology to incorporate noise, which requires less plant evaluations than other alternatives. Finally, we conclude this work by performing experiments on a Solid-Oxide Fuel Cell system.

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature.

Most of the existing numerical schemes developed to solve Burgers' equation cannot exhibit its correct physical behavior for very small values of viscosity. This difficulty can be overcome by using splitting methods derived for near-integrable system. This class of methods has positive real coefficients and can be used for non-reversible systems such as Burgers' equation. It also has the advantage of being able to account small viscosity in the accuracy. The algorithm is based on the combination of implicit-explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability. The resulting algorithm is accurate, efficient and easy to implement. The new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity. (C) 2018 Elsevier B.V. All rights reserved.

Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'skiĭ-Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward-backward and a Douglas-Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal-dual algorithms of forward-backward and Douglas-Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems. The theoretical results are illustrated by numerical experiments on the split feasibility problem in infinite dimensional spaces.

We study the solution of a time-varying optimization problem which is observed, i.e. , it is known, only intermittently. We propose three approaches based on the prediction-correction scheme for solving this problem by exploiting splitting methods. We present convergence results in mean to a bounded asymptotical error, and showcase them in a numerical example featuring a regression problem.

This study focuses on symplectic integrators for numerical evaluation of the asymptotic solutions of the nonlinear Airy-type equations obtained by reducing the nonlinear dispersive equations. Since the nature of Airy-type equations has both highly oscillatory slow decay and exponential fast decay, most of classical integrators are not able to correctly exhibit challenging physical behaviour. We use specially designed symplectic integrators combining splitting methods with Magnus integrators to catch asymptotic behaviour of nonlinear Airy-type equations efficiently, even for large step sizes. Efficiency of the proposed methods for given problems is discussed. Moreover, numerical results obtained by the proposed methods are compared with the existing results in the literature. (C) 2020 Elsevier B.V. All rights reserved. C1 [Seydaoglu, Muaz] Mus Alparslan Univ, Fac Art & Sci, Dept Math, TR-49100 Mus, Turkey. [Kocak, Huseyin] Pamukkale Univ, Quantitat Methods Div, TR-20160 Denizli, Turkey. [Erdogan, Utku] Eskisehir Tech Univ, Dept Math, TR-26555 Eskisehir, Turkey.

In this paper, two structure-preserving nonstandard finite difference (NSFD) operator splitting schemes are designed for the solution of reaction diffusion epidemic models. The proposed schemes preserve all the essential properties possessed by the continuous systems. These schemes are applied on a diffusive SEIQV epidemic model with a saturated incidence rate to validate the results. Furthermore, the stability of the continuous system is proved, and the bifurcation value is evaluated. A comparison is also made with the existing operator splitting numerical scheme. Simulations are also performed for numerical experiments.

We compare adaptive time integrators for the numerical solution of linear Schrödinger equations where the Hamiltonian explicitly depends on time. The approximation methods considered are splitting methods, where the time variable is split off and advanced separately, and commutator-free Magnus-type methods. The time-steps are chosen adaptively based on asymptotically correct estimators of the local error in both cases. It is found that splitting methods are more efficient when the Hamiltonian naturally suggests a separation into kinetic and potential part, whereas Magnus-type integrators excel when the structure of the problem only allows to advance the time variable separately.

In this paper, we discuss the different splitting approaches to numerically solvethe Gross–Pitaevskii equation (GPE). The models are motivated from spinor Bose–Einsteincondensate (BEC). This system is formed of coupled mean-field equations, which are based oncoupled Gross–Pitaevskii equations. We consider conservative finite-difference schemes and spectralmethods for the spatial discretisation. Furthermore, we apply implicit or explicit time-integrators andcombine these schemes with different splitting approaches. The numerical solutions are comparedbased on the conservation of the L2-norm with the analytical solutions. The advantages of the novelsplitting methods for large time-domains are based on the asymptotic conservation of the solution ofthe soliton’s applications. Furthermore, we have the benefit of larger local time-steps and thereforeobtain faster numerical schemes.

[ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial. La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo. El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis. El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo. La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético. En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock. El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente. El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura., [CA] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest. L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps. El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes fonamentals utilitzades al llarg de la tesi. El disseny dels integradors proposats es basa en els mètodes de composició i escissió i en el desenvolupament de Magnus. En el Capítol 3, es descriu el primer. La seua idea principal consta d'una recombinació d'uns integradors senzills per a obtenir la solució del problema. El concepte important de les condicions d'orde es descriu en eixe capítol. El Capítol 4 fa un resum de les àlgebres de Lie i del desenvolupament de Magnus que són les ferramentes algebraiques que permeten expressar la solució d'equacions diferencials dependents del temps. L'equació lineal de Schrödinger amb potencial dependent del temps està examinada en el Capítol 5. Donat la seua estructura particular, nous mètodes quasi sense commutadors, basats en el desenvolupament de Magnus, són construïts. La seua eficiència és demostrada en uns experiments numèrics amb el model de Walker-Preston d'una molècula dins d'un camp electromagnètic. En el Capítol 6 es dissenyen els mètodes de Magnus-escissió per a les equacions d'onda i de Hill. El seu rendiment està demostrat en els experiments numèrics amb diversos sistemes oscil·latoris: amb l'equació de Mathieu, l'ec. de Hill matricial, les equacions d'onda i de Klein-Gordon-Fock. El Capítol 7 explica com l'enfocament algebraic i el desenvolupament de Magnus poden generalitzar-se als problemes no lineals. L'exemple utilitzat és el problema de Kepler amb massa decreixent. El Capítol 8 conclou la tesi, ressenya els resultats i traça les possibles direccions de la investigació futura., [EN] The present thesis addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics because they come from quantum, classical and celestial mechanics. The goal of the thesis is to construct integrators for several import ant non-autonomous problems: the Schrödinger equation, which is the cornerstone of quantum mechanics; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass. Chapter 1 describes the motivation and the aims of the work in the historical context of numerical integration. In Chapter 2 essential concepts and some fundamental tools used throughout the thesis are introduced. The design of the proposed integrators is based on the composition and splitting methods and the Magnus expansion. In Chapter 3, the former is described. Their main idea is to recombine some simpler integrators to obtain the solution. The salient concept of order conditions is described in that chapter. Chapter 4 summarises Lie algebras and the Magnus expansion ¿ algebraic tools that help to express the solution of time-dependent differential equations. The linear Schrödinger equation with time-dependent potential is considered in Chapter 5. Given its particular structure, new, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker-Preston model of a molecule in an electromagnetic field. In Chapter 6, Magnus-splitting methods for the wave and the Hill equations are designed. Their performance is demonstrated in numerical experiments with various oscillatory systems: the Mathieu equation, the matrix Hill eq., the wave and the Klein-Gordon-Fock eq. Chapter 7 shows how the algebraic approach and the Magnus expansion can be generalised to non-linear problems. The example used is the Kepler problem with decreasing mass. The thesis is concluded by Chapter 8, in which the results are reviewed and possible directions of future work are outlined.

SEYDAOGLU, MUAZ/0000-0003-3211-0864; Kocak, Huseyin/0000-0001-9683-6096; Erdogan, Utku/0000-0003-0089-0114 This study focuses on symplectic integrators for numerical evaluation of the asymptotic solutions of the nonlinear Airy-type equations obtained by reducing the nonlinear dispersive equations. Since the nature of Airy-type equations has both highly oscillatory slow decay and exponential fast decay, most of classical integrators are not able to correctly exhibit challenging physical behaviour. We use specially designed symplectic integrators combining splitting methods with Magnus integrators to catch asymptotic behaviour of nonlinear Airy-type equations efficiently, even for large step sizes. Efficiency of the proposed methods for given problems is discussed. Moreover, numerical results obtained by the proposed methods are compared with the existing results in the literature. (C) 2020 Elsevier B.V. All rights reserved.

An ordinary differential equation (ODE) can be split into simpler sub equations and each of the sub equations is solved subsequently by a numerical method. Such a procedure involves splitting error and numerical error caused by the time stepping methods applied to sub equations. The aim of the paper is to present an integral formula for the global error expansion of a splitting procedure combined with any numerical ODE solver.

We address the solution of time-varying optimization problems characterized by the sum of a time-varying strongly convex function and a time-invariant nonsmooth convex function. We design an online algorithmic framework based on prediction-correction, which employs splitting methods to solve the sampled instances of the time-varying problem. We describe the prediction-correction scheme and two splitting methods, the forward-backward and the Douglas-Rachford. Then by using a result for generalized equations, we prove convergence of the generated sequence of approximate optimizers to a neighborhood of the optimal solution trajectory. Simulation results for a leader following formation in robotics assess the performance of the proposed algorithm., To be presented at the European Control Conference 2019

[EN] We propose new local error estimators for splitting and composition methods. They are based on the construction of lower order schemes obtained at each step as a linear combination of the intermediate stages of the integrator, so that the additional computational cost required for their evaluation is almost insignificant. These estimators can be subsequently used to adapt the step size along the integration. Numerical examples show the efficiency of the procedure., Part of this work was developed during a research stay at the Wolfgang Pauli Institute Vienna; the authors are grateful to the director Norbert Mauser and the staff members for their support and hospitality. The first two authors acknowledge funding by the Ministerio de Economia y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).

In this paper we analyze the use of time splitting techniques for solving shallow water equations. We discuss some properties that these schemes should satisfy so that interactions between the source term and the shock waves are controlled. This work shows that these schemes must be well balanced in the meaning expressed by Greenberg and Leroux [7]. More specifically, we analyze in what cases it is enough to verify an Approximate C-property and in which cases it is required to verify an Exact C-property (see [1, 2]). We also discuss this technique in two dimensions and include some numerical tests in order to justify our argument.

In this work, high order splitting methods have been used for calculating the numerical solutions of Burgers' equation in one space dimension with periodic, Dirichlet, Neumann and Robin boundary conditions. However, splitting methods with real coefficients of order higher than two necessarily have negative coefficients and cannot be used for time-irreversible systems, such as Burgers' equations, due to the time-irreversibility of the Laplacian operator. Therefore, the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed. If we consider the system as the perturbation of an exactly solvable problem (or one that can be easily approximated numerically), it is possible to employ highly efficient methods to approximate Burgers' equation. The numerical results show that both the methods with complex time steps having one set of coefficients real and positive, say ai ? ?+ and bi ? ?+, and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of Burgers' equation. © 2015 Published by Elsevier B.V.

We apply first- and second-order splitting schemes to the differential Riccati equation. Such equations are very important in, e.g., linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g., arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.

We propose a numerical method for time-dependent, semilinear partial differential equations (PDEs) with random parameters and random initial data. The method is based on an operator splitting approach. The linear part of the right-hand side is discretized by a stochastic Galerkin method in the stochastic variables and a pseudospectral method in the physical space, whereas the nonlinear part is approximated by a stochastic collocation method in the stochastic variables. In this setting both parts of the random PDE can be propagated very efficiently. The Galerkin method and the collocation method are combined with sparse grids in order to reduce the computational costs. This approach is discussed in detail for the Lugiato-Lefever equation, which serves as a motivating example throughout, but also applies to a much larger class of random PDEs. For such problems our method is computationally much cheaper than standard stochastic Galerkin methods, and numerical tests show that it outperforms standard stochastic collocation methods, too.

Nonlinear Dirac equations describe the motion of relativistic spin-$\frac{1}{2}$ particles in presence of external electromagnetic felds, modelled by an electric and magnetic potential, and taking into account a nonlinear particle self-interaction. In recent years, the construction of numerical splitting schemes for the solution of these systems in the nonrelativistic limit regime, i.e., the speed of light c formally tending to infnity, has gained a lot of attention. In this paper, we consider a nonlinear Dirac equation with Thirring type interaction, where in contrast to the case of the Soler type nonlinearity a classical twoterm splitting scheme cannot be straightforwardly applied. Thus, we propose and analyze a three-term Strang splitting scheme which relies on splitting the full problem into the free Dirac subproblem, a potential subproblem, and a nonlinear subproblem, where each subproblem can be solved exactly in time. Moreover, our analysis shows that the error of our scheme improves from $\mathcal{O} $ ($r$$^{2}$$c$$^{4}$) to $\mathcal{O} $ ($r$$^{2}$$c$$^{3}$) if the magnetic potential in the system vanishes. Furthermore, we propose an effcient limit approximation scheme for solving nonlinear Dirac systems in the nonrelativistic limit regime $c$ $\gg$ 1 which allows errors of $\mathcal{O}$ ($c$$^{-1}$) without any $c$-dependent time step restriction.

In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMM-like algorithm that we refer to as relaxed ADMM (R-ADMM) via a generalized Peaceman-Rachford Splitting operator on the Lagrange dual formulation of the original optimization problem. This specific algorithm depends on two parameters, namely the averaging coefficient $\alpha$ and the augmented Lagrangian coefficient $\rho$. We show that by setting $\alpha=1/2$ we recover the standard ADMM algorithm as a special case of our algorithm. Moreover, by properly manipulating the proposed R-ADMM, we are able to provide two alternative ADMM-like algorithms that present easier implementation and reduced complexity in terms of memory, communication and computational requirements. Most importantly the latter of these two algorithms provides the first ADMM-like algorithm which has guaranteed convergence even in the presence of lossy communication under the same assumption of standard ADMM with lossless communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over cycle graphs and random geometric graphs subject to i.i.d. random packet losses., Comment: Full version of the paper presented at European Control Conference (ECC) 2018

Diese Arbeit motiviert und fasst die Konstruktion von Splittingschemata f��r Evolutionsgleichungen zusammen. Weiters werden asymptotisch korrekte lokale Fehlersch��tzer vorgestellt, um die Schrittweite in der Zeitevolution anzupassen um die gesamte Rechenzeit zu reduzieren. Kommutatorfreie Magnus-Typ-Integratoren werden mit dem Splitting in der Zeitvariablen verglichen. Die Methoden werden an drei Anwendungen getestet: Modelle f��r Solarzellen, Reaktions-Diffusions-System von Gray-Scott und Schr��dinger-Wellengleichungen., This thesis motivates and summarizes the construction of splitting schemes for evolution equations. Further asymptotic correct local error estimators are proposed, to adapt the step-size in time propagation and reduce total computation time. Commutator free Magnus-type integrators are compared to splitting off the time variable. The Methods are tested by three applications: Models for solar cells, reaction-diffusion system by Gray-Scott and Schr��dinger wave equations.

This paper presents SPLIT, a C code generation tool for Model Predictive Control (MPC) based on operator splitting methods. In contrast to existing code generation packages, SPLIT is capable of generating both software and hardware-oriented C code to allow quick prototyping of optimization algorithms on conventional CPUs and field-programmable gate arrays (FPGAs). A Matlab interface is provided for compatibility with existing commercial and open-source software packages. A numerical study compares software, hardware and heterogeneous implementations of splitting methods and investigates MPC design trade-offs. For the considered testcases the reported speedup of hardware implementations over software realizations is 3x to 11x.

Quantitative Structure-Activity Relationship not only provides guidelines regarding structural features responsible for biological activity but it can be used also for prediction of desired activity prior to synthesis of untested chemicals. Therefore, an appropriate validation of any QSAR is of utmost importance to judge its external predictive ability. Generally, internal and external validations (preferred by many) are used in the absence of a true external dataset. The model developed using external method may not be reliable as it may not capture all essential features required for the particular SAR due to omission of some compounds, especially for small datasets. In external validation, the splitting is done either rationally or in random manner before descriptor selection. In the present study, rational splitting of dataset was performed using a novel method and its effect on statistical parameters was analyzed. The analysis reveals that the predictive ability of a QSAR model is sensitive toward (1) the method of splitting and (2) distribution of the training and the prediction sets. In addition, purposeful selection can be used to influence the statistical parameters ; therefore, external validation based on single split is insufficient to guarantee the true predictive ability of a QSAR model. Besides, it appears that the selection of descriptors prior to splitting (information leakage) has little role to play in deciding external predictivity of the model. The present study reveals that as many as possible statistical parameters should be examined along with boot-strapping instead of single external validation.

High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in the Stratonovich form. in the setting of weighted spaces, the necessary analyticity of the split semigroups can easily be proved. A numerical example from interest rate theory, the CIR2 model, is considered. The numerical results are robust for drift-dominated problems and confirm our theoretical results. (C) 2013 Elsevier B.V. All rights reserved.

In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh&ndash, Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature.

This thesis concerns the long time behavior of certain Vlasov equations, mainly the Vlasov- HMF model. We are in particular interested in the celebrated phenomenon of Landau damp- ing, proved mathematically in various frameworks, foar several Vlasov equations, such as the Vlasov-Poisson equation or the Vlasov-HMF model, and exhibiting certain analogies with the inviscid damping phenomenon for the 2D Euler equation. The results described in the document are the following.The first one is a Landau damping theorem for numerical solutions of the Vlasov-HMF model, constructed by means of time-discretizations by splitting methods. We prove more- over the convergence of the schemes. The second result is a Landau damping theorem for solutions of the Vlasov-HMF model linearized around inhomogeneous stationary states. We provide moreover a quite large amount of numerical simulations, which are designed to study numerically the nonlinear case, and which seem to show new phenomenons. The last result is the convergence of a scheme that discretizes in time the 2D Euler equation by means of a symplectic Crouch-Grossmann integrator., Cette thèse porte sur le comportement en temps long de solutions d’équations de type Vlasov, principalement le modèle Vlasov-HMF. On s’intéresse en particulier au phénomène d’amortissement Landau, prouvé mathématiquement dans divers cadres, pour plusieurs équations de type Vlasov, comme l’équation de Vlasov-Poisson ou le modèle Vlasov-HMF, et présentant certaines analogies avec le phénomène d’amortissement non visqueux pour l’équation d’Euler 2D. Les résultats qui y sont décrits sont les suivants. Le premier est un théorème d’amortissement Landau pour des solutions numériques du modèle Vlasov-HMF, obtenues par discrétisation en temps de ce dernier via des méthodes de splitting. Nous prouvons en outre la convergence des schémas numériques. Le second est un théorème d’amortissment Landau pour des solutions du modéle Vlasov-HMF linéarisé autour d’états stationnaires inhomogènes. Ce théorème est accompagné de nombreuses simulations numériques destinées à étudier numériquement le cas non-linéaire, et semblant mettre en lumière de nouveaux phénomènes. Enfin, le dernier résultat porte sur la discrétisation en temps de l’équation d’Euler 2D par un intégrateur de Crouch-Grossman symplectique. Nous prouvons la convergence du schéma.

[EN] Several symplectic splitting methods of orders four and six are presented for the step-by-step time numerical integration of the Schrodinger equation when the Hamiltonian is a general explicitly time-dependent real operator. They involve linear combinations of the Hamiltonian evaluated at some intermediate points. We provide the algorithm and the coefficients of the methods, as well as some numerical examples showing their superior performance with respect to other available schemes. Published by AIP Publishing., The authors acknowledge Ministerio de Economia y Competitividad (Spain) for financial support through Project Nos. MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE). Additionally, A.M. has been partially supported by the Basque Government (Consolidated Research Group No. IT649-13).

In this paper an efficient and accurate method for simulating the propagation of a localized solution of the Schrodinger equation with a smooth potential near the semiclassical limit is presented. We are interested computing arbitrarily accurate solutions when the non dimensional Planck's constant, @e, is small, but not negligible. The method is based on a time dependent transformation of the independent variables, closely related to Gaussian wave packets. A rescaled wave function, w, satisfies a new Schrodinger equation with a time dependent potential which is a perturbation of the harmonic oscillator, the perturbation being O(@e), so that all stiffness (in space and time) is greatly reduced. In fact, for integration in a fixed time interval, the number of modes required to fully resolve the problem decreases when @e is decreased. The original wave function may be reconstructed by Fourier interpolation, although expectation values of the observables can be computed directly from the function w itself. If the initial condition is a Gaussian wave packet, very few modes are necessary to fully resolve the w variable, so for short time very accurate solutions can be obtained at low computational cost. Initial conditions other than Gaussians wave packets can also be used. In this paper, the Gaussian wave packet transform is carefully outlined and applied to the Schrodinger equation in one dimension. Detailed numerical tests show the efficiency and accuracy of the approach. In the sequel of this paper, this approach has been extended to the multidimensional case.

This thesis address the issue of extreme event simulation. From a original understanding of the Splitting methods, a new theoretical framework is proposed, regardless of any algorithm. This framework is based on a point process associated with any real-valued random variable and lets defined probability, quantile and moment estimators without any hypothesis on this random variable. The artificial selection of threshold in Splitting vanishes and the estimator of the probability of exceeding a threshold is indeed an estimator of the whole cumulative distribution function until the given threshold. These estimators are based on the simulation of iid. replicas of the point process. So they allow for the use of massively parallel computer cluster. Suitable practical algorithms are thus proposed.Finally it can happen that these advanced statistics still require too much samples. In this context the computer code is considered as a random process with known distribution. The point process framework lets handle this additional source of uncertainty and estimate easily the conditional expectation and variance of the resulting random variable. It also defines new SUR enrichment criteria designed for extreme event probability estimation., Cette thèse est une contribution à la problématique de la simulation d’événements rares. A partir de l’étude des méthodes de Splitting, un nouveau cadre théorique est développé, indépendant de tout algorithme. Ce cadre, basé sur la définition d’un processus ponctuel associé à toute variable aléatoire réelle, permet de définir des estimateurs de probabilités, quantiles et moments sans aucune hypothèse sur la variable aléatoire. Le caractère artificiel du Splitting (sélection de seuils) disparaît et l’estimateur de la probabilité de dépasser un seuil est en fait un estimateur de la fonction de répartition jusqu’au seuil considéré. De plus, les estimateurs sont basés sur des processus ponctuels iid. et permettent donc l’utilisation de machine de calcul massivement parallèle. Des algorithmes pratiques sont ainsi également proposés.Enfin l’utilisation de métamodèles est parfois nécessaire à cause d’un temps de calcul toujours trop important. Le cas de la modélisation par processus aléatoire est abordé. L’approche par processus ponctuel permet une estimation simplifiée de l’espérance et de la variance conditionnelles de la variable aléaoire résultante et définit un nouveau critère d’enrichissement SUR adapté aux événements rares.

In this work, high-order splitting methods of integration without negative steps are shown which can be used in irreversible problems, like reaction–diffusion or complex Ginzburg–Landau equations. These methods consist of suitable affine combinations of Lie–Tortter schemes with different positive steps. The number of basic steps for these methods grows quadratically with the order, while for symplectic methods, the growth is exponential. Furthermore, the calculations can be performed in parallel, so that the computation time can be significantly reduced using multiple processors. Convergence results of these methods are proved for a large range of semilinear problems, which includes reaction–diffusion systems and dissipative perturbation of Hamiltonian systems. Fil: de Leo, Mariano Fernando. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Sanchez Fernandez de la Vega, Constanza Mariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina

The works presented in this thesis are devoted to the study of partial differential equations systems (PDE). In particular, we are interested in constrained systems coming from the fluid mechanics field which allow to described, in time and space, physical quantities such as density or speed. In this context, we build models for biology which are then numerically tested. We also present an original numerical method for a road traffic model.In the first part, using the theory of mixtures, we model the development of a photosynthetic micro-algae biofilm. The growth of micro-algae is precisely described by taking into account their composition and their access to nutrients dissolved in the surrounding liquid and light. Then, using numerical simulations, we estimate the biofilm productivity.In the second part, using the mixture theory we propose a model describing the rheology of the large intestine and its mucus layer. Thanks to this model we can give an accurate description of the velocity field induced by intestinal flow. This velocity field will then be used to build a modeldescribing precisely interactions between the intestinal microbiota, the gastric broth and the host. For these two models numerical schemes are proposed and allow a first validation of the models.The last part is devoted to developing an asymptotic preserving scheme for the constraint Aw-Rascle system for road traffic. We present an explicit-implicit method based on a splitting technique in order to approximate the solutions of Aw-Rascle system with constraint, while relaxing the stability condition (CFL).; Les travaux présentés dans cette thèse sont consacrés à l’étude de systèmes d’équations aux dérivées partielles (EDP). En particulier, nous nous intéressons à des systèmes issus de la mécanique des fluides avec contraintes, qui permettent de décrire de manière continue, en temps et en espace, des quantités physiques telles que la densité ou la vitesse. Dans ce cadre, nous construisons des modèles pour la biologie, qu’ensuite nous testons numériquement. Nous proposons également avec des méthodes similaires une approche numérique originale pour un système de trafic routier.Dans une première partie, à l’aide de la théorie des mélanges, nous modélisons le développement d’un biofilm de micro-algues photosynthétiques. La croissance des micro-algues y est précisément décrite, en tenant compte de leur composition et de l’accès aux nutriments dissouts, contenus dans le liquide environnant ainsi que de la lumière. Puis, à l’aide de simulations numériques, nous estimons la productivité du biofilm.Dans la seconde partie, en utilisant la théorie des mélanges, nous proposons un modèle permettant de décrire la rhéologie du gros intestin et de sa couche de mucus. Grâce à ce modèle, nous donnerons une description précise du champ de vitesse, induit par le flux intestinal. Puis, ce champ de vitesse sera utilisé pour construire un modèle décrivant les interactions entre le microbiote intestinal, le bouillon gastrique et l’hôte. Pour ces deux modèles, un schéma numérique est proposé et permet une première validation.Enfin, la dernière partie est consacrée à l’élaboration d’un schéma asymptotic preserving pour le système de trafic routier d’Aw-Rascle avec contrainte. Nous y présentons une méthode explicite-implicite basée sur une technique de splitting permettant d’approcher les solutions du systèmed’Aw-Rascle avec contrainte, tout en réduisant la contrainte de stabilité (CFL).

The works presented in this thesis are devoted to the study of partial differential equations systems (PDE). In particular, we are interested in constrained systems coming from the fluid mechanics field which allow to described, in time and space, physical quantities such as density or speed. In this context, we build models for biology which are then numerically tested. We also present an original numerical method for a road traffic model.In the first part, using the theory of mixtures, we model the development of a photosynthetic micro-algae biofilm. The growth of micro-algae is precisely described by taking into account their composition and their access to nutrients dissolved in the surrounding liquid and light. Then, using numerical simulations, we estimate the biofilm productivity.In the second part, using the mixture theory we propose a model describing the rheology of the large intestine and its mucus layer. Thanks to this model we can give an accurate description of the velocity field induced by intestinal flow. This velocity field will then be used to build a modeldescribing precisely interactions between the intestinal microbiota, the gastric broth and the host. For these two models numerical schemes are proposed and allow a first validation of the models.The last part is devoted to developing an asymptotic preserving scheme for the constraint Aw-Rascle system for road traffic. We present an explicit-implicit method based on a splitting technique in order to approximate the solutions of Aw-Rascle system with constraint, while relaxing the stability condition (CFL).; Les travaux présentés dans cette thèse sont consacrés à l’étude de systèmes d’équations aux dérivées partielles (EDP). En particulier, nous nous intéressons à des systèmes issus de la mécanique des fluides avec contraintes, qui permettent de décrire de manière continue, en temps et en espace, des quantités physiques telles que la densité ou la vitesse. Dans ce cadre, nous construisons des modèles pour la biologie, qu’ensuite nous testons numériquement. Nous proposons également avec des méthodes similaires une approche numérique originale pour un système de trafic routier.Dans une première partie, à l’aide de la théorie des mélanges, nous modélisons le développement d’un biofilm de micro-algues photosynthétiques. La croissance des micro-algues y est précisément décrite, en tenant compte de leur composition et de l’accès aux nutriments dissouts, contenus dans le liquide environnant ainsi que de la lumière. Puis, à l’aide de simulations numériques, nous estimons la productivité du biofilm.Dans la seconde partie, en utilisant la théorie des mélanges, nous proposons un modèle permettant de décrire la rhéologie du gros intestin et de sa couche de mucus. Grâce à ce modèle, nous donnerons une description précise du champ de vitesse, induit par le flux intestinal. Puis, ce champ de vitesse sera utilisé pour construire un modèle décrivant les interactions entre le microbiote intestinal, le bouillon gastrique et l’hôte. Pour ces deux modèles, un schéma numérique est proposé et permet une première validation.Enfin, la dernière partie est consacrée à l’élaboration d’un schéma asymptotic preserving pour le système de trafic routier d’Aw-Rascle avec contrainte. Nous y présentons une méthode explicite-implicite basée sur une technique de splitting permettant d’approcher les solutions du systèmed’Aw-Rascle avec contrainte, tout en réduisant la contrainte de stabilité (CFL).

For operator splitting methods, an approach based on Taylor expansion and the particular structure of its leading term as an element of a free Lie algebra is used for the setup of a system of order conditions. Along with a brief review of the underlying theoretical background, we discuss the implementation of the resulting algorithm in computer algebra, in particular using Maple 18 (Maple is a trademark of MapleSoft\(^\mathrm{TM}\).). A parallel version of such a code is described, and its performance on a computational node with 16 threads is documented.