Distributed hydrological models have been used for decades to calculate and predict the movement of water and energy within watersheds. These models have evolved from relatively simple empirical applications into complex spatially distributed and physically-based programs. However, the evolution of distributed hydrological models has not involved the improvement of the numerical methods used to calculate the redistribution of water and energy in the watershed. Because of this, many models still use numerical methods that are potentially inaccurate. In order to simulate the transport of water and energy in a hydrological model, typical numerical methods employ an operator splitting approach. Operator splitting (OS) essentially breaks down the set of coupled ordinary differential equations (ODEs) that define a hydrological model into separate ODEs that can be solved individually. The dominant operator splitting method in surface water models is the ordered series approach. Because the ordered series approach treats parallel hydrological processes as if they happen in series, it is prone to errors that can significantly reduce the accuracy of model results. The impact that operator splitting errors have upon hydrologic model results is, to date, unknown. Using a new distributed hydrological model, Raven, the impact of operator splitting errors is investigated. Understanding these errors will lead to better numerical methods for reducing errors in models and to shed light on the shortcomings of hydrological models with respect to numerical method choice. Alternative numerical methods - the explicit Euler and the implicit iterative Heun methods - are implemented and assessed in their ability to minimize errors and produce more accurate distributed hydrological models.

Distributed hydrological models have been used for decades to calculate and predict the movement of water and energy within watersheds. These models have evolved from relatively simple empirical applications into complex spatially distributed and physically-based programs. However, the evolution of distributed hydrological models has not involved the improvement of the numerical methods used to calculate the redistribution of water and energy in the watershed. Because of this, many models still use numerical methods that are potentially inaccurate. In order to simulate the transport of water and energy in a hydrological model, typical numerical methods employ an operator splitting approach. Operator splitting (OS) essentially breaks down the set of coupled ordinary differential equations (ODEs) that define a hydrological model into separate ODEs that can be solved individually. The dominant operator splitting method in surface water models is the ordered series approach. Because the ordered series approach treats parallel hydrological processes as if they happen in series, it is prone to errors that can significantly reduce the accuracy of model results. The impact that operator splitting errors have upon hydrologic model results is, to date, unknown. Using a new distributed hydrological model, Raven, the impact of operator splitting errors is investigated. Understanding these errors will lead to better numerical methods for reducing errors in models and to shed light on the shortcomings of hydrological models with respect to numerical method choice. Alternative numerical methods - the explicit Euler and the implicit iterative Heun methods - are implemented and assessed in their ability to minimize errors and produce more accurate distributed hydrological models.

La modélisation du transport réactif dans les milieux poreux implique la simulation de plusieurs processus physico-chimiques : écoulement de phases fluides, transport de chaleur, réactions chimiques entre espèces en phases identiques ou différentes. La résolution du système d'équations qui décrit le problème peut être obtenue par une approche soit totalement couplée soit découplée. Les approches découplées simplifient le système d'équations en décomposant le problème sous-parties plus faciles à gérer. Chacune de ces sous-parties peut être résolue avec des techniques d'intégration appropriées. Les techniques de découplage peuvent être non‑itératives (operator splitting methods) ou itératives (fixed‑point iteration), chacunes ayant des avantages et des inconvénients. Les approches non‑iteratives génèrent une erreur associée à la séparation des sous­-parties couplées, et les approaches itératives peuvent présenter des problèmes de convergence. Dans cette thèse, nous développons un code sous licence libre en langage MATLAB (https://github.com/TReacLab/TReacLab) dédie à la modélisation du la problématique de la carbonatation atmosphérique du béton, dans le cadre du stockage de déchets de moyenne activité et longue vie en couche géologique profonde. Le code propose un ensemble d'approche découplée : classique, comme les approches de fractionnement séquentiel, alternatif ou Strang, et moins classique, comme les approches de fractionnement additif ou par répartition symétrique. En outre, deux approches itératives basées sur une formulation spécifique (SIA CC et SIA TC) ont également été implémentées. Le code été interfacé de manière générique avec différents solveurs de transport (COMSOL, pdepe MATLAB, FVTool, FD scripts) et géochimiques (iPhreeqc, PhreeqcRM). Afin de valider l'implémentations des différentes approches, plusieurs bancs d'essais classiques dans le domaine du transport réactif ont été utilises avec succès. L'erreur associée à la combinaison du fractionnement de l'opérateur et des techniques numériques étant complexe à évaluer, nous explorons les outils mathématiques existants permettant de l'estimer. Enfin, nous structurons le problème de la carbonatation atmosphérique et présentons des simulations préliminaires, en détaillant les problèmes pertinents et les étapes futures à suivre. / Reactive transport modeling in porous media involves the simulation of several physico‑chemical processes: flow of fluid phases, transport of species, heat transport, chemical reactions between species in the same phase or in different phases. The resolution of the system of equations that describes the problem can be obtained by a fully coupled approach or by a decoupled approach. Decoupled approaches can simplify the system of equations by breaking down the problem into smaller parts that are easier to handle. Each of the smaller parts can be solved with suitable integration techniques. The decoupling techniques might be non‑iterative (operator splitting methods) or iterative (fixed‑point iteration), having each its advantages and disadvantages. Non‑iterative approaches have an error associated with the separation of the coupled effects, and iterative approaches might have problems to converge. In this thesis, we develop an open‑source code written in MATLAB (https://github.com/TReacLab/TReacLab) in order to model the problematic of concrete atmospheric carbonation for an intermediate‑level long‑lived nuclear waste package in a deep geological repository. The code uses a decoupled approach. Classical operator splitting approaches, such as sequential, alternating or Strang splitting, and less classical splitting approaches, such as additive or symmetrically weighted splitting, have been implemented. Besides, two iterative approaches based on an specific formulation (SIA CC, and SIA TC) have also been implemented. The code has been interfaced in a generic way with different transport solvers (COMSOL, pdepe MATLAB, FVTool, FD scripts) and geochemical solvers (iPhreeqc, PhreeqcRM). In order to validate the implementation of the different approaches, a series of classical benchmarks in the field of reactive transport have been solved successfully and compared with analytical and external numerical solutions. Since the associated error due to the combination of operator splitting and numerical techniques may be complex to assess, we explore the existing mathematical tools used to evaluate it. Finally, we frame the atmospheric carbonation problem and run preliminary simulations, stating the relevant problems and future steps to follow.

Operator splitting methods have been applied to nonlinear partial differential equations that involve operators of different nature. The main idea of these methods is to decompose a complex equation into simpler sub-equations, which can be solved separately. The main advantage of the operator splitting methods is that they provide a great flexibility in choosing different numerical methods, depending on the feature of each sub-problem. In this dissertation, we have developed highly accurate and efficient numerical methods for several nonlinear partial differential equations, which involve both linear and nonlinear operators. We first propose a fast explicit operator splitting method for the modified Buckley-Leverett equations which include a third-order mixed derivatives term resulting from the dynamic effects in the pressure difference between the two phases. The method splits the original equation into two equations, one with a nonlinear convective term and the other one with high-order linear terms so that appropriate numerical methods can be applied to each of the split equations: The high-order linear equation is numerically solved using a pseudo-spectral method, while the nonlinear convective equation is integrated using the Godunov-type central-upwind scheme. The spatial order of the central-upwind scheme depends on the order of the piecewise polynomial reconstruction: We test both the second-order minmod-based reconstruction and fifth-order WENO5 one to demonstrate that using higher-order spatial reconstruction leads to more accurate approximation of solutions. We then propose fast and stable explicit operator splitting methods for two phase-field models (the molecular beam epitaxy equation with slope selection and the Cahn-Hilliard equation), numerical simulations of which require long time computations. The equations are split into nonlinear and linear parts. The nonlinear part is solved using a method of lines combined with an efficient large stability domain explicit ODE solver. The linear part is solved by a pseudo-spectral method, which is based on the exact solution and thus has no stability restriction on the time step size. We have verified the numerical accuracy of the proposed methods and demonstrated their performance on extensive one- and two-dimensional numerical examples, where different solution profiles can be clearly observed and are consistent with previous analytical studies. / Zhuolin Qu

In this paper we propose an algorithm for solving systems of coupled monotone inclusions in Hilbert spaces. The operators arising in each of the inclusions of the system are processed in each iteration separately, namely, the single-valued are evaluated explicitly (forward steps), while the set-valued ones via their resolvents (backward steps). In addition, most of the steps in the iterative scheme can be executed simultaneously, this making the method applicable to a variety of convex minimization problems. The numerical performances of the proposed splitting algorithm are emphasized through applications in average consensus on colored networks and image classification via support vector machines.

Convex optimization has developed a wide variety of useful tools critical to many applications in machine learning. However, unlike linear and quadratic programming, general convex solvers have not yet reached sufficient maturity to fully decouple the convex programming model from the numerical algorithms required for implementation. Especially as datasets grow in size, there is a significant gap in speed and scalability between general solvers and specialized algorithms. This thesis addresses this gap with a new model for convex programming based on an intermediate representation of convex problems as a sum of functions with efficient proximal operators. This representation serves two purposes: 1) many problems can be expressed in terms of functions with simple proximal operators, and 2) the proximal operator form serves as a general interface to any specialized algorithm that can incorporate additional `2-regularization. On a single CPU core, numerical results demonstrate that the prox-affine form results in significantly faster algorithms than existing general solvers based on conic forms. In addition, splitting problems into separable sums is attractive from the perspective of distributing solver work amongst multiple cores and machines. We apply large-scale convex programming to several problems arising from building the next-generation, information-enabled electrical grid. In these problems (as is common in many domains) large, high-dimensional datasets present opportunities for novel data-driven solutions. We present approaches based on convex models for several problems: probabilistic forecasting of electricity generation and demand, preventing failures in microgrids and source separation for whole-home energy disaggregation.

In this thesis we investigate Asian oating strike options. We particu-larly focus on options with early exercise - American options. This typeof options are very lucrative to the end-users of commodities or ener-gies who are tend to be exposed to the average prices over time. Asianoptions are also very popular with corporations, who have ongoing cur-rency exposures. The main idea of the pricing is to examine the freeboundary position on which the value of the option is depending. Wefocus on developing a ecient numerical algorithm for this boundary.In the rst Chapter we give an informative description of the nancialderivatives including Asian options. The second Chapter is devoted tothe analytical derivation of the corresponding partial dierential equa-tion coming from the original Black - Scholes equation. The problemis simplied using transformation methods and dimension reduction. Inthe third and fourth Chapter we describe important numerical methodsand discretize the problem. We use the rst order Lie splitting and thesecond order Strang splitting. Finally, in the fth Chapter we makenumerical experiments with the free boundary and compare the resultwith other known methods.

The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis.

Duisburg, Essen, Univ., Diss., 2005

Convex optimization problems are a class of mathematical problems which arise in numerous applications. Although interior-point methods can in principle solve these problems efficiently, they may become intractable for solving large-scale problems or be unsuitable for real-time embedded applications. Iterations of operator splitting methods are relatively simple and computationally inexpensive, which makes them suitable for these applications. However, some of their known limitations are slow asymptotic convergence, sensitivity to ill-conditioning, and inability to detect infeasible problems. The aim of this thesis is to better understand operator splitting methods and to develop reliable software tools for convex optimization. The main analytical tool in our investigation of these methods is their characterization as the fixed-point iteration of a nonexpansive operator. The fixed-point theory of nonexpansive operators has been studied for several decades. By exploiting the properties of such an operator, it is possible to show that the alternating direction method of multipliers (ADMM) can detect infeasible problems. Although ADMM iterates diverge when the problem at hand is unsolvable, the differences between subsequent iterates converge to a constant vector which is also a certificate of primal and/or dual infeasibility. Reliable termination criteria for detecting infeasibility are proposed based on this result. Similar ideas are used to derive necessary and sufficient conditions for linear (geometric) convergence of an operator splitting method and a bound on the achievable convergence rate. The new bound turns out to be tight for the class of averaged operators. Next, the OSQP solver is presented. OSQP is a novel general-purpose solver for quadratic programs (QPs) based on ADMM. The solver is very robust, is able to detect infeasible problems, and has been extensively tested on many problem instances from a wide variety of application areas. Finally, operator splitting methods can also be effective in nonconvex optimization. The developed algorithm significantly outperforms a common approach based on convex relaxation of the original nonconvex problem.