Computing the homology of basic semialgebraic sets in weak exponential time

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data)

Algorithms for computing norms and characteristic polynomials on general Drinfeld modules

We provide two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms work for Drinfeld modules of any rank, defined over any base curve. When the base curve is $\mathbb P^1_{\mathbb F_q}$, we do a thorough study of the complexity, demonstrating that our algorithms are, in many cases, the most asymptotically performant. The first family of algorithms relies on the correspondence between Drinfeld modules and Anderson motives, reducing the computation to linear algebra over a polynomial ring. The second family, available only for the Frobenius endomorphism, is based on a new formula expressing the characteristic polynomial of the Frobenius as a reduced norm in a central simple algebra

Multiscale Modeling and Simulation of Deformation Accumulation in Fault Networks

Strain accumulation and stress release along multiscale geological fault networks are fundamental mechanisms for earthquake and rupture processes in the lithosphere. Due to long periods of seismic quiescence, the scarcity of large earthquakes and incompleteness of paleoseismic, historical and instrumental record, there is a fundamental lack of insight into the multiscale, spatio-temporal nature of earthquake dynamics in fault networks. This thesis constitutes another step towards reliable earthquake prediction and quantitative hazard analysis. Its focus lies on developing a mathematical model for prototypical, layered fault networks on short time scales as well as their efficient numerical simulation. This exposition begins by establishing a fault system consisting of layered bodies with viscoelastic Kelvin-Voigt rheology and non-intersecting faults featuring rate-and-state friction as proposed by Dieterich and Ruina. The individual bodies are assumed to experience small viscoelastic deformations, but possibly large relative tangential displacements. Thereafter, semi-discretization in time with the classical Newmark scheme of the variational formulation yields a sequence of continuous, nonsmooth, coupled, spatial minimization problems for the velocities and states in each time step, that are decoupled by means of a fixed point iteration. Subsequently, spatial discretization is based on linear and piecewise constant finite elements for the rate and state problems, respectively. A dual mortar discretization of the non-penetration constraints entails a hierarchical decomposition of the discrete solution space, that enables the localization of the non-penetration condition. Exploiting the resulting structure, an algebraic representation of the parametrized rate problem can be solved efficiently using a variant of the Truncated Nonsmooth Newton Multigrid (TNNMG) method. It is globally convergent due to nonlinear, block Gauß–Seidel type smoothing and employs nonsmooth Newton and multigrid ideas to enhance robustness and efficiency of the overall method. A key step in the TNNMG algorithm is the efficient computation of a correction obtained from a linearized, inexact Newton step. The second part addresses the numerical homogenization of elliptic variational problems featuring fractal interface networks, that are structurally similar to the ones arising in the linearized correction step of the TNNMG method. Contrary to the previous setting, this model incorporates the full spatial complexity of geological fault networks in terms of truly multiscale fractal interface geometries. Here, the construction of projections from a fractal function space to finite element spaces with suitable approximation and stability properties constitutes the main contribution of this thesis. The existence of these projections enables the application of well-known approaches to numerical homogenization, such as localized orthogonal decomposition (LOD) for the construction of multiscale discretizations with optimal a priori error estimates or subspace correction methods, that lead to algebraic solvers with mesh- and scale-independent convergence rates. Finally, numerical experiments with a single fault and the layered multiscale fault system illustrate the properties of the mathematical model as well as the efficiency, reliability and scale-independence of the suggested algebraic solver

Arithmetical problems in number fields, abelian varieties and modular forms

La teoria de nombres, una àrea de la matemàtica fascinant i de les més antigues, ha experimentat un progrés espectacular durant els darrers anys. El desenvolupament d'una base teòrica profunda i la implementació d'algoritmes han conduït a noves interrelacions matemàtiques interessants que han fet palesos teoremes importants en aquesta àrea. Aquest informe resumeix les contribucions a la teoria de nombres dutes a terme per les persones del Seminari de Teoria de Nombres (UB-UAB-UPC) de Barcelona. Els seus resultats són citats en connexió amb l'estat actual d'alguns problemes aritmètics, de manera que aquesta monografia cerca proporcionar al públic lector una ullada sobre algunes línies específiques de la recerca matemàtica actual.Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Numerical Methods for Partial Differential Equations

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248