Modules

14 results

A1.3 Stochastics14

Large DeviationsNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn the theory of large deviations, which describes how unlikely events occur in probability models and how quickly their probabilities decay. In the end you will be able to apply fundamental techniques of large-deviation theory and understand applications such as for sums of random variables and empirical distributions.5 ECTSruns this semesterMA5417Random Graphs and NetworksNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn models of random graphs and how they describe real networks. The focuses are Erdős–Rényi models, generalized random graphs, the configuration model and preferential-attachment models as well as proof techniques for the number of components, degree distribution and distances.5 ECTSruns this semesterMA5436Branching Random WalksNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn the basics and advanced aspects of branching random walks. In the end you will know the classical theory of branching processes and Galton‑Watson trees as well as the connections to partial differential equations and relevant application areas.3 ECTSno date this semesterCIT413035Concentration of MeasureNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn fundamental inequalities and techniques of Concentration of Measure (e.g., Chernoff, Hoeffding, Bernstein inequalities, isoperimetry, Johnson–Lindenstrauss, Martingale methods, Poincaré and Gromov–Milman theorems). In the end you will be able to understand these inequalities and apply them to simple examples from applications.5 ECTSno date this semesterCIT4100004Conformal Mapping and ProbabilityNo ratings for this module yet.A1.3.1 Modules in Probability TheoryThe module covers conformal mappings and their connection with stochastic processes in the plane. You will learn how conformal methods can solve probabilistic problems, and gain insight into two-dimensional critical systems and their scaling limits (e.g., percolation, loop-erased random walks, Ising model) as well as SLE and loop ensembles.5 ECTSno date this semesterMA5432
9 more in A1.3 StochasticsMarkov ProcessesNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn theory and methods of continuous-time Markov processes: continuous-time Markov chains, the Markov property, Feller processes as well as properties of transition semigroups and their generators. In the end you can analyze the long-term behavior of processes, apply ergodic theorems and use them in examples (e.g. queueing theory, interacting particle systems, time series).9 ECTSno date this semesterMA4408Mathematics of Reinforcement LearningNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn the mathematical foundations of reinforcement learning: Markov decision processes (MDPs), tabular RL methods such as Monte Carlo, Temporal Difference, SARSA and Q-Learning, as well as basics of stochastic approximation to analyze the convergence of these algorithms. In the end you can formulate dynamic decision problems under uncertainty as MDPs and solve them with tabular RL algorithms.6 ECTSno date this semesterCIT413036Perfect Simulation for Chains of Finite and Infinite OrderNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn about models of stochastic processes beyond finite Markov chains, in particular chains of infinite order, their properties and how to simulate them exactly (perfect simulation). In the end you will be able to assess existence and uniqueness questions, classify examples and apply perfect-simulation algorithms for such processes.5 ECTSno date this semesterCIT413043Probability on GraphsNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn stochastic models on graphs, e.g. Random Walks, percolation and random graphs. In the end you will be able to analyse Random Walks on networks, use the connection to electrical networks to show recurrence/transience, and understand phase transitions and random spanning trees.5 ECTSno date this semesterMA4406Random Matrices: Theory, Numerical Methods, and ApplicationNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will receive an introduction to large random matrices, their eigenvalue distributions and eigenvalue spacings. At the end of the module you will know the fundamental methods (e.g. Stieltjes transform, orthogonal polynomials, Fredholm determinants, free probability) and you will be able to compute limit distributions and apply them to models in applications.3 ECTSno date this semesterMA5306Self-interacting Random Walks and Statistical PhysicsNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn methods for the analysis of self-interacting random walks and their connections to statistical physics. In the end you will be able to understand and apply central models (e.g. edge-/vertex-reinforced processes), proofs of localization/delocalization, and connections to supersymmetric sigma models and random Schrödinger operators.3 ECTSno date this semesterMA5941Stochastic AnalysisNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou will learn the theory and fundamental applications of stochastic analysis. The focus is on Brownian motion (construction and properties), stochastic integrals and the Itô formula, as well as stochastic differential equations and methods such as Girsanov transformation and Donsker's invariance principle. In the end you will be able to formulate central statements and perform simple calculations with Itô integrals and SDEs.9 ECTSno date this semesterMA4405Theorie der ZufallsmatrizenNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou learn non-asymptotic deviation estimates for random variables and vectors (in particular subgaussian and subexponential as well as isotropic vectors) and apply these concepts to random matrices with independent rows or columns. In the end you can formulate statements about the extreme singular values of such matrices, compare them and apply them to concrete examples as well as applications such as dimensionality reduction and Compressed Sensing.5 ECTSno date this semesterMA5346Topics in the Theory of Markov ProcessesNo ratings for this module yet.A1.3.1 Modules in Probability TheoryYou engage with analytical aspects of Markov processes, in particular Itô diffusions (linear and nonlinear). By the end you will be able to understand Markov semigroups and generators, classify basic existence and uniqueness results for McKean-SDEs/McKean–Vlasov equations, and make statements about long-term behavior as well as phase transitions for nonlinear Fokker–Planck equations.3 ECTSno date this semesterMA5424