Refine
Has Fulltext
- yes (9)
Is part of the Bibliography
- yes (9)
Year of publication
- 2021 (9) (remove)
Document Type
- Doctoral Thesis (9) (remove)
Language
- English (9) (remove)
Keywords
- Abbildungseigenschaften (1)
- Algebraische Zahlentheorie (1)
- Augmented Lagrangian (1)
- Belyi map (1)
- Belyi-Funktionen (1)
- Bilinear differential games (1)
- CSIDH (1)
- Cardinality Constraints (1)
- Cayley graph (1)
- Differential Games (1)
- Elliptische Kurve (1)
- Ensemble optimal control (1)
- Euler equations (1)
- Explicit Computation (1)
- Explizite Berechnung (1)
- Finite-Elemente-Methode (1)
- Finite-Volumen-Methode (1)
- Fixpunktsatz (1)
- Fluid (1)
- Formoptimierung (1)
- Functions with Primitive (1)
- Funktion von beschränkter Variation (1)
- Funktionen mit Stammfunktion (1)
- Galois-Theorie (1)
- Gestaltoptimierung (1)
- Gleichmäßige Konvergenz (1)
- Graph (1)
- Graph spectrum (1)
- Gravitationsfeld (1)
- Homicidal Chauffeur game (1)
- Integral graph (1)
- Integralgleichung (1)
- Kombinatorik (1)
- Kryptologie (1)
- Linearer Operator (1)
- Liouville and transport equations (1)
- Mapping Properties (1)
- Nichtlinearer Operator (1)
- Operatortheorie (1)
- Optimale Kontrolle (1)
- Optimierung (1)
- Optimierung / Nebenbedingung (1)
- Optimierungsproblem (1)
- Pontryagin maximum principle (1)
- Post-Quantum-Kryptografie (1)
- Regularisation Methods (1)
- Regularisierungsverfahren (1)
- Relaxation method (1)
- Schur ring (1)
- Sequential Quadratic Hamiltonian scheme (1)
- Shape Optimization (1)
- Totale Variation (1)
- Zero-divisor graph (1)
- elliptic curves (1)
- finite volume methods (1)
- isogeny-based cryptography (1)
- optimization (1)
- total variation (1)
- well-balancing (1)
- Überlagerung <Mathematik> (1)
Institute
- Institut für Mathematik (9) (remove)
Theoretical and numerical investigation of optimal control problems governed by kinetic models
(2021)
This thesis is devoted to the numerical and theoretical analysis of ensemble optimal control problems governed by kinetic models. The formulation and study of these problems have been put forward in recent years by R.W. Brockett with the motivation that ensemble control may provide a more general and robust control framework for dynamical systems. Following this formulation, a Liouville (or continuity) equation with an unbounded drift function is considered together with a class of cost functionals that include tracking of ensembles of trajectories of dynamical systems and different control costs. Specifically, $L^2$, $H^1$ and $L^1$ control costs are taken into account which leads to non--smooth optimization problems. For the theoretical investigation of the resulting optimal control problems, a well--posedness theory in weighted Sobolev spaces is presented for Liouville and related transport equations. Specifically, existence and uniqueness results for these equations and energy estimates in suitable norms are provided; in particular norms in weighted Sobolev spaces. Then, non--smooth optimal control problems governed by the Liouville equation are formulated with a control mechanism in the drift function. Further, box--constraints on the control are imposed. The control--to--state map is introduced, that associates to any control the unique solution of the corresponding Liouville equation. Important properties of this map are investigated, specifically, that it is well--defined, continuous and Frechet differentiable. Using the first two properties, the existence of solutions to the optimal control problems is shown. While proving the differentiability, a loss of regularity is encountered, that is natural to hyperbolic equations. This leads to the need of the investigation of the control--to--state map in the topology of weighted Sobolev spaces. Exploiting the Frechet differentiability, it is possible to characterize solutions to the optimal control problem as solutions to an optimality system. This system consists of the Liouville equation, its optimization adjoint in the form of a transport equation, and a gradient inequality. Numerical methodologies for solving Liouville and transport equations are presented that are based on a non--smooth Lagrange optimization framework. For this purpose, approximation and solution schemes for such equations are developed and analyzed. For the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov--Tadmor method, a Runge--Kutta scheme, and a Strang splitting method are discussed. Stability and second--order accuracy of these resulting schemes are proven in the discrete $L^1$ norm. In addition, conservation of mass and positivity preservation are confirmed for the solution method of the Liouville model. As numerical optimization strategy, an adapted Krylow--Newton method is applied. Since the control is considered to be an element of $H^1$ and to obey certain box--constraints, a method for calculating a $H^1$ projection is presented. Since the optimal control problem is non-smooth, a semi-smooth adaption of Newton's method is taken into account. Results of numerical experiments are presented that successfully validate the proposed deterministic framework. After the discussion of deterministic schemes, the linear space--homogeneous Keilson--Storer master equation is investigated. This equation was originally developed for the modelling of Brownian motion of particles immersed in a fluid and is a representative model of the class of linear Boltzmann equations. The well--posedness of the Keilson--Storer master equation is investigated and energy estimates in different topologies are derived. To solve this equation numerically, Monte Carlo methods are considered. Such methods take advantage of the kinetic formulation of the Liouville equation and directly implement the behaviour of the system of particles under consideration. This includes the probabilistic behaviour of the collisions between particles. Optimal control problems are formulated with an objective that is constituted of certain expected values in velocity space and the $L^2$ and $H^1$ costs of the control. The problems are governed by the Keilson--Storer master equation and the control mechanism is considered to be within the collision kernel. The objective of the optimal control of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. Existence of solutions of the optimal control problem is proven and a Keilson--Storer optimality system characterizing the solution of the proposed optimal control problem is obtained. The optimality system is used to construct a gradient--based optimization strategy in the framework of Monte--Carlo methods. This task requires to accommodate the resulting adjoint Keilson--Storer model in a form that is consistent with the kinetic formulation. For this reason, we derive an adjoint Keilson--Storer collision kernel and an additional source term. A similar approach is presented in the case of a linear space--inhomogeneous kinetic model with external forces and with Keilson--Storer collision term. In this framework, a control mechanism in the form of an external space--dependent force is investigated. The purpose of this control is to steer the multi--particle system to follow a desired mean velocity and position and to reach a desired final configuration in phase space. An optimal control problem using the formulation of ensemble controls is stated with an objective that is constituted of expected values in phase space and $H^1$ costs of the control. For solving the optimal control problems, a gradient--based computational strategy in the framework of Monte Carlo methods is developed. Part of this is the denoising of the distribution functions calculated by Monte Carlo algorithms using methods of the realm of partial differential equations. A standalone C++ code is presented that implements the developed non--linear conjugated gradient strategy. Results of numerical experiments confirm the ability of the designed probabilistic control framework to operate as desired. An outlook section about optimal control problems governed by non--linear space--inhomogeneous kinetic models completes this thesis.
This thesis aims at providing efficient and side-channel protected implementations of isogeny-based primitives, and at their application in threshold protocols. It is based on a sequence of academic papers.
Chapter 3 reviews the original variable-time implementation of CSIDH and introduces several optimizations, e.g. a significant improvement of isogeny computations by using both Montgomery and Edwards curves. In total, our improvements yield a speedup of 25% compared to the original implementation.
Chapter 4 presents the first practical constant-time implementation of CSIDH. We describe how variable-time implementations of CSIDH leak information on private keys, and describe ways to mitigate this. Further, we present several techniques to speed up the implementation. In total, our constant-time implementation achieves a rather small slowdown by a factor of 3.03.
Chapter 5 reviews practical fault injection attacks on CSIDH and presents countermeasures. We evaluate different attack models theoretically and practically, using low-budget equipment. Moreover, we present countermeasures that mitigate the proposed fault injection attacks, only leading to a small performance overhead of 7%.
Chapter 6 initiates the study of threshold schemes based on the Hard Homogeneous Spaces (HHS) framework of Couveignes. Using the HHS equivalent of Shamir’s secret sharing in the exponents, we adapt isogeny based schemes to the threshold setting. In particular, we present threshold versions of the CSIDH public key encryption and the CSI-FiSh signature scheme.
Chapter 7 gives a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Recent compact isogeny-based protocols, namely B-SIDH and SQISign, both require large primes that lie between two smooth integers. Finding such a prime can be seen as a special case of finding twin smooth integers under the additional stipulation that their sum is a prime.
This thesis is devoted to a theoretical and numerical investigation of methods to solve open-loop non zero-sum differential Nash games. These problems arise in many applications, e.g., biology, economics, physics, where competition between different agents appears. In this case, the goal of each agent is in contrast with those of the others, and a competition game can be interpreted as a coupled optimization problem for which, in general, an optimal solution does not exist. In fact, an optimal strategy for one player may be unsatisfactory for the others. For this reason, a solution of a game is sought as an equilibrium and among the solutions concepts proposed in the literature, that of Nash equilibrium (NE) is the focus of this thesis. The building blocks of the resulting differential Nash games are a dynamical model with different control functions associated with different players that pursue non-cooperative objectives. In particular, the aim of this thesis is on differential models having linear or bilinear state-strategy structures. In this framework, in the first chapter, some well-known results are recalled, especially for non-cooperative linear-quadratic differential Nash games. Then, a bilinear Nash game is formulated and analysed. The main achievement in this chapter is Theorem 1.4.2 concerning existence of Nash equilibria for non-cooperative differential bilinear games. This result is obtained assuming a sufficiently small time horizon T, and an estimate of T is provided in Lemma 1.4.8 using specific properties of the regularized Nikaido-Isoda function. In Chapter 2, in order to solve a bilinear Nash game, a semi-smooth Newton (SSN) scheme combined with a relaxation method is investigated, where the choice of a SSN scheme is motivated by the presence of constraints on the players’ actions that make the problem non-smooth. The resulting method is proved to be locally convergent in Theorem 2.1, and an estimate on the relaxation parameter is also obtained that relates the relaxation factor to the time horizon of a Nash equilibrium and to the other parameters of the game. For the bilinear Nash game, a Nash bargaining problem is also introduced and discussed, aiming at determining an improvement of all players’ objectives with respect to the Nash equilibrium. A characterization of a bargaining solution is given in Theorem 2.2.1 and a numerical scheme based on this result is presented that allows to compute this solution on the Pareto frontier. Results of numerical experiments based on a quantum model of two spin-particles and on a population dynamics model with two competing species are presented that successfully validate the proposed algorithms. In Chapter 3 a functional formulation of the classical homicidal chauffeur (HC) Nash game is introduced and a new numerical framework for its solution in a time-optimal formulation is discussed. This methodology combines a Hamiltonian based scheme, with proximal penalty to determine the time horizon where the game takes place, with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time. The resulting numerical optimization scheme has a bilevel structure, which aims at decoupling the computation of the end-time from the solution of the pursuit-evader game. Several numerical experiments are performed to show the ability of the proposed algorithm to solve the HC game. Focusing on the case where a collision may occur, the time for this event is determined. The last part of this thesis deals with the analysis of a novel sequential quadratic Hamiltonian (SQH) scheme for solving open-loop differential Nash games. This method is formulated in the framework of Pontryagin’s maximum principle and represents an efficient and robust extension of the successive approximations strategy in the realm of Nash games. In the SQH method, the Hamilton-Pontryagin functions are augmented by a quadratic penalty term and the Nikaido-Isoda function is used as a selection criterion. Based on this fact, the key idea of this SQH scheme is that the PMP characterization of Nash games leads to a finite-dimensional Nash game for any fixed time. A class of problems for which this finite-dimensional game admits a unique solution is identified and for this class of games theoretical results are presented that prove the well-posedness of the proposed scheme. In particular, Proposition 4.2.1 is proved to show that the selection criterion on the Nikaido-Isoda function is fulfilled. A comparison of the computational performances of the SQH scheme and the SSN-relaxation method previously discussed is shown. Applications to linear-quadratic Nash games and variants with control constraints, weighted L1 costs of the players’ actions and tracking objectives are presented that corroborate the theoretical statements.
The present thesis deals with optimisation problems with sparsity terms, either in the constraints which lead to cardinality-constrained problems or in the objective function which in turn lead to sparse optimisation problems. One of the primary aims of this work is to extend the so-called sequential optimality conditions to these two classes of problems. In recent years sequential optimality conditions have become increasingly popular in the realm of standard nonlinear programming. In contrast to the more well-known Karush-Kuhn-Tucker condition, they are genuine optimality conditions in the sense that every local minimiser satisfies these conditions without any further assumption. Lately they have also been extended to mathematical programmes with complementarity constraints. At around the same time it was also shown that optimisation problems with sparsity terms can be reformulated into problems which possess similar structures to mathematical programmes with complementarity constraints. These recent developments have become the impetus of the present work. But rather than working with the aforementioned reformulations which involve an artifical variable we shall first directly look at the problems themselves and derive sequential optimality conditions which are independent of any artificial variable. Afterwards we shall derive the weakest constraint qualifications associated with these conditions which relate them to the Karush-Kuhn-Tucker-type conditions. Another equally important aim of this work is to then consider the practicability of the derived sequential optimality conditions. The previously mentioned reformulations open up the possibilities to adapt methods which have been proven successful to handle mathematical programmes with complementarity constraints. We will show that the safeguarded augmented Lagrangian method and some regularisation methods may generate a point satisfying the derived conditions.
We compute genus-0 Belyi maps with prescribed monodromy and strictly verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the rational rigidity criterion yielding polynomials with prescribed Galois groups over Q(t). We also give an explicit version of a theorem of Magaard, which lists all sporadic groups occurring as composition factors of monodromy groups of rational functions.
This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.
Functions of bounded variation are most important in many fields of mathematics. This thesis investigates spaces of functions of bounded variation with one variable of various types, compares them to other classical function spaces and reveals natural “habitats” of BV-functions. New and almost comprehensive results concerning mapping properties like surjectivity and injectivity, several kinds of continuity and compactness of both linear and nonlinear operators between such spaces are given. A new theory about different types of convergence of sequences of such operators is presented in full detail and applied to a new proof for the continuity of the composition operator in the classical BV-space. The abstract results serve as ingredients to solve Hammerstein and Volterra integral equations using fixed point theory. Many criteria guaranteeing the existence and uniqueness of solutions in BV-type spaces are given and later applied to solve boundary and initial value problems in a nonclassical setting.
A big emphasis is put on a clear and detailed discussion. Many pictures and synoptic tables help to visualize and summarize the most important ideas. Over 160 examples and counterexamples illustrate the many abstract results and how delicate some of them are.
In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974.
We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree.
Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs.
Besides that, we study spectra of zero-divisor graphs over finite commutative rings.
Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.
Fluids in Gravitational Fields – Well-Balanced Modifications for Astrophysical Finite-Volume Codes
(2021)
Stellar structure can -- in good approximation -- be described as a hydrostatic state, which which arises due to a balance between gravitational force and pressure gradient. Hydrostatic states are static solutions of the full compressible Euler system with gravitational source term, which can be used to model the stellar interior. In order to carry out simulations of dynamical processes occurring in stars, it is vital for the numerical method to accurately maintain the hydrostatic state over a long time period. In this thesis we present different methods to modify astrophysical finite volume codes in order to make them \emph{well-balanced}, preventing them from introducing significant discretization errors close to hydrostatic states. Our well-balanced modifications are constructed so that they can meet the requirements for methods applied in the astrophysical context: They can well-balance arbitrary hydrostatic states with any equation of state that is applied to model thermodynamical relations and they are simple to implement in existing astrophysical finite volume codes. One of our well-balanced modifications follows given solutions exactly and can be applied on any grid geometry. The other methods we introduce, which do no require any a priori knowledge, balance local high order approximations of arbitrary hydrostatic states on a Cartesian grid. All of our modifications allow for high order accuracy of the method. The improved accuracy close to hydrostatic states is verified in various numerical experiments.