Méthodes sans factorisation pour l’optimisation non linéaire

RÉSUMÉ : Cette thèse a pour objectif de formuler mathématiquement, d'analyser et d'implémenter deux méthodes sans factorisation pour l'optimisation non linéaire. Dans les problèmes de grande taille, la jacobienne des contraintes n'est souvent pas disponible sous forme de matrice; seules son action et celle de sa transposée sur un vecteur le sont. L'optimisation sans factorisation consiste alors à utiliser des opérateurs linéaires abstraits représentant la jacobienne ou le hessien. De ce fait, seules les actions > sont autorisées et l'algèbre linéaire directe doit être remplacée par des méthodes itératives. Outre ces restrictions, une grande difficulté lors de l'introduction de méthodes sans factorisation dans des algorithmes d'optimisation concerne le contrôle de l'inexactitude de la résolution des systèmes linéaires. Il faut en effet s'assurer que la direction calculée est suffisamment précise pour garantir la convergence de l'algorithme concerné. En premier lieu, nous décrivons l'implémentation sans factorisation d'une méthode de lagrangien augmenté pouvant utiliser des approximations quasi-Newton des dérivées secondes. Nous montrons aussi que notre approche parvient à résoudre des problèmes d'optimisation de structure avec des milliers de variables et contraintes alors que les méthodes avec factorisation échouent. Afin d'obtenir une méthode possédant une convergence plus rapide, nous présentons ensuite un algorithme qui utilise un lagrangien augmenté proximal comme fonction de mérite et qui, asymptotiquement, se transforme en une méthode de programmation quadratique séquentielle stabilisée. L'utilisation d'approximations BFGS à mémoire limitée du hessien du lagrangien conduit à l'obtention de systèmes linéaires symétriques quasi-définis. Ceux-ci sont interprétés comme étant les conditions d'optimalité d'un problème aux moindres carrés linéaire, qui est résolu de manière inexacte par une méthode de Krylov. L'inexactitude de cette résolution est contrôlée par un critère d'arrêt facile à mettre en œuvre. Des tests numériques démontrent l'efficacité et la robustesse de notre méthode, qui se compare très favorablement à IPOPT, en particulier pour les problèmes dégénérés pour lesquels la LICQ n'est pas respectée à la solution ou lors de la minimisation. Finalement, l'écosystème de développement d'algorithmes d'optimisation en Python, baptisé NLP.py, est exposé. Cet environnement s'adresse aussi bien aux chercheurs en optimisation qu'aux étudiants désireux de découvrir ou d'approfondir l'optimisation. NLP.py donne accès à un ensemble de blocs constituant les éléments les plus importants des méthodes d'optimisation continue. Grâce à ceux-ci, le chercheur est en mesure d'implémenter son algorithme en se concentrant sur la logique de celui-ci plutôt que sur les subtilités techniques de son implémentation.----------ABSTRACT : This thesis focuses on the mathematical formulation, analysis and implementation of two factorization-free methods for nonlinear constrained optimization. In large-scale optimization, the Jacobian of the constraints may not be available in matrix form; only its action and that of its transpose on a vector are. Factorization-free optimization employs abstract linear operators representing the Jacobian or Hessian matrices. Therefore, only operator-vector products are allowed and direct linear algebra is replaced by iterative methods. Besides these implementation restrictions, a difficulty inherent to methods without factorization in optimization algorithms is the control of the inaccuracy in linear system solves. Indeed, we have to guarantee that the direction calculated is sufficiently accurate to ensure convergence. We first describe a factorization-free implementation of a classical augmented Lagrangian method that may use quasi-Newton second derivatives approximations. This method is applied to problems with thousands of variables and constraints coming from aircraft structural design optimization, for which methods based on factorizations fail. Results show that it is a viable approach for these problems. In order to obtain a method with a faster convergence rate, we present an algorithm that uses a proximal augmented Lagrangian as merit function and that asymptotically turns in a stabilized sequential quadratic programming method. The use of limited-memory BFGS approximations of the Hessian of the Lagrangian combined with regularization of the constraints leads to symmetric quasi-definite linear systems. Because such systems may be interpreted as the KKT conditions of linear least-squares problems, they can be efficiently solved using an appropriate Krylov method. Inaccuracy of their solutions is controlled by a stopping criterion which is easy to implement. Numerical tests demonstrate the effectiveness and robustness of our method, which compares very favorably with IPOPT, especially for degenerate problems for which LICQ is not satisfied at the optimal solution or during the minimization process. Finally, an ecosystem for optimization algorithm development in Python, code-named NLP.py, is exposed. This environment is aimed at researchers in optimization and students eager to discover or strengthen their knowledge in optimization. NLP.py provides access to a set of building blocks constituting the most important elements of continuous optimization methods. With these blocks, users are able to implement their own algorithm focusing on the logic of the algorithm rather than on the technicalities of its implementation

Non-Uniform Smoothness for Gradient Descent

The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given problem. In this work we introduce a local first-order smoothness oracle (LFSO) which generalizes the Lipschitz continuous gradients smoothness condition and is applicable to any twice-differentiable function. We show that this oracle can encode all relevant problem information for tuning stepsizes for a suitably modified gradient descent method and give global and local convergence results. We also show that LFSOs in this modified first-order method can yield global linear convergence rates for non-strongly convex problems with extremely flat minima, and thus improve over the lower bound on rates achievable by general (accelerated) first-order methods

Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM)

Development of robust and efficient solution strategies for coupled problems

Det er mange modeller i moderne vitenskap hvor sammenkoblingen mellom forskjellige fysiske prosesser er svært viktig. Disse finner man for eksempel i forbindelse med lagring av karbondioksid i undervannsreservoarer, flyt i kroppsvev, kreftsvulstvekst og geotermisk energiutvinning. Denne avhandlingen har to fokusområder som er knyttet til sammenkoblede modeller. Det første er å utvikle pålitelige og effektive tilnærmingsmetoder, og det andre er utviklingen av en ny modell som tar for seg flyt i et porøst medium som består av to forskjellige materialer. For tilnærmingsmetodene har det vært et spesielt fokus på splittemetoder. Dette er metoder hvor hver av de sammenkoblede modellene håndteres separat, og så itererer man mellom dem. Dette gjøres i hovedsak fordi man kan utnytte tilgjengelig teori og programvare for å løse hver undermodell svært effektivt. Ulempen er at man kan ende opp med løsningsalgoritmer for den sammenkoblede modellen som er trege, eller ikke kommer frem til noen løsning i det hele tatt. I denne avhandlingen har tre forskjellige metoder for å forbedre splittemetoder blitt utviklet for tre forskjellige sammenkoblede modeller. Den første modellen beskriver flyt gjennom deformerbart porøst medium og er kjent som Biot ligningene. For å anvende en splittemetode på denne modellen har et stabiliseringsledd blitt tilført. Dette sikrer at metoden konvergerer (kommer frem til en løsning), men dersom man ikke skalerer stabiliseringsleddet riktig kan det ta veldig lang tid. Derfor har et intervall hvor den optimale skaleringen av stabiliseringsleddet befinner seg blitt identifisert, og utfra dette presenteres det en måte å praktisk velge den riktige skaleringen på. Den andre modellen er en fasefeltmodell for sprekkpropagering. Denne modellen løses vanligvis med en splittemetode som er veldig treg, men konvergent. For å forbedre dette har en ny akselerasjonsmetode har blitt utviklet. Denne anvendes som et postprosesseringssteg til den klassiske splittemetoden, og utnytter både overrelaksering og Anderson akselerasjon. Disse to forskjellige akselerasjonsmetodene har kompatible styrker i at overrelaksering akselererer når man er langt fra løsningen (som er tilfellet når sprekken propagerer), og Anderson akselerasjon fungerer bra når man er nærme løsningen. For å veksle mellom de to metodene har et kriterium basert på residualfeilen blitt brukt. Resultatet er en pålitelig akselerasjonsmetode som alltid akselererer og ofte er svært effektiv. Det siste modellen kalles Cahn-Larché ligningene og er også en fasefeltmodell, men denne beskriver elastisitet i et medium bestående av to elastiske materialer som kan bevege seg basert på overflatespenningen mellom dem. Dette problemet er spesielt utfordrende å løse da det verken er lineært eller konvekst. For å håndtere dette har en ny måte å behandle tidsavhengigheten til det underliggende koblede problemet på blitt utviklet. Dette leder til et diskret system som er ekvivalent med et konvekst minimeringsproblem, som derfor er velegnet til å løses med de fleste numeriske optimeringsmetoder, også splittemetoder. Den nye modellen som har blitt utviklet er en utvidelse av Cahn-Larché ligningene og har fått navnet Cahn-Hilliard-Biot. Dette er fordi ligningene utgjør en fasefelt modell som beskriver flyt i et deformerbart porøst medium med to poroelastiske materialer. Disse kan forflytte seg basert på overflatespenning, elastisk spenning, og poretrykk, og det er tenkt at modellen kan anvendes i forbindelse med kreftsvulstmodellering.There are many applications where the study of coupled physical processes is of great importance. These range from the life sciences with flow in deformable human tissue to structural engineering with fracture propagation in elastic solids. In this doctoral dissertation, there is a twofold focus on coupled problems. Firstly, robust and efficient solution strategies, with a focus on iterative decoupling methods, have been applied to several coupled systems of equations. Secondly, a new thermodynamically consistent coupled system of equations is proposed. Solution strategies are developed for three different coupled problems; the quasi-static linearized Biot equations that couples flow through porous materials and elastic deformation of the solid medium, variational phase-field models for brittle fracture that couple a phase-field equation for fracture evolution with linearized elasticity, and the Cahn-Larché equations that model elastic effects in a two-phase elastic material and couples an extended Cahn-Hilliard phase-field equation and linearized elasticity. Finally, the new system of equations that is proposed models flow through a two-phase deformable porous material where the solid phase evolution is governed by interfacial forces as well as effects from both the fluid and elastic properties of the material. In the work that concerns the quasi-static linearized Biot equations, the focus is on the fixed-stress splitting scheme, which is a popular method for sequentially solving the flow and elasticity subsystems of the full model. Using such a method is beneficial as it allows for the use of readily available solvers for the subproblems; however, a stabilizing term is required for the scheme to converge. It is well known that the convergence properties of the method strongly depend on how this term is chosen, and here, the optimal choice of it is addressed both theoretically and practically. An interval where the optimal stabilization parameter lies is provided, depending on the material parameters. In addition, two different ways of optimizing the parameter are proposed. The first is a brute-force method that relies on the mesh independence of the scheme's optimal stabilization parameter, and the second is valid for low-permeable media and utilizes an equivalence between the fixed-stress splitting scheme and the modified Richardson iteration. Regarding the variational phase-field model for brittle fracture propagation, the focus is on improving the convergence properties of the most commonly used solution strategy with an acceleration method. This solution strategy relies on a staggered scheme that alternates between solving the elasticity and phase-field subproblems in an iterative way. This is known to be a robust method compared to the monolithic Newton method. However, the staggered scheme often requires many iterations to converge to satisfactory precision. The contribution of this work is to accelerate the solver through a new acceleration method that combines Anderson acceleration and over-relaxation, dynamically switching back and forth between them depending on a criterion that takes the residual evolution into account. The acceleration scheme takes advantage of the strengths of both Anderson acceleration and over-relaxation, and the fact that they are complementary when applied to this problem, resulting in a significant speed-up of the convergence. Moreover, the method is applied as a post-processing technique to the increments of the solver, and can thus be implemented with minor modifications to readily available software. The final contribution toward solution strategies for coupled problems focuses on the Cahn-Larché equations. This is a model for linearized elasticity in a medium with two elastic phases that evolve with respect to interfacial forces and elastic effects. The system couples linearized elasticity and an extended Cahn-Hilliard phase-field equation. There are several challenging features with regards to solution strategies for this system including nonlinear coupling terms, and the fourth-order term that comes from the Cahn-Hilliard subsystem. Moreover, the system is nonlinear and non-convex with respect to both the phase-field and the displacement. In this work, a new semi-implicit time discretization that extends the standard convex-concave splitting method applied to the double-well potential from the Cahn-Hilliard subsystem is proposed. The extension includes special treatment for the elastic energy, and it is shown that the resulting discrete system is equivalent to a convex minimization problem. Furthermore, an alternating minimization solver is proposed for the fully discrete system, together with a convergence proof that includes convergence rates. Through numerical experiments, it becomes evident that the newly proposed discretization method leads to a system that is far better conditioned for linearization methods than standard time discretizations. Finally, a new model for flow through a two-phase deformable porous material is proposed. The two poroelastic phases have distinct material properties, and their interface evolves according to a generalized Ginzburg–Landau energy functional. As a result, a model that extends the Cahn-Larché equations to poroelasticity is proposed, and essential coupling terms for several applications are highlighted. These include solid tumor growth, biogrout, and wood growth. Moreover, the coupled set of equations is shown to be a generalized gradient flow. This implies that the system is thermodynamically consistent and makes a toolbox of analysis and solvers available for further study of the model.Doktorgradsavhandlin

A note on the penalty parameter in Nitsche's method for unfitted boundary value problems

Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. From the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors