Modules

41 results

A1.3 Mathematics Modules on Special Topics41

Nonconvex Global OptimizationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn methods of global optimization for nonconvex and possibly nonsmooth objective functions and how to apply them. Beginning with (stochastic) gradient methods and simulated annealing, you walk toward multi‑particle methods such as Particle Swarm Optimization (PSO) and Consensus‑Based Optimization (CBO) and understand their global convergence properties.6 ECTSruns this semesterCIT4130019Probabilistische Techniken und Algorithmen in der DatenanalyseNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn probabilistic techniques and algorithms used in data analysis for dimensionality reduction and data recovery from incomplete information. In the end, you will understand the basics of random matrices, assess randomized algorithms, and apply and analyze methods such as JL embeddings, compressed sensing, and randomness-based matrix/tensor reconstruction.6 ECTSruns this semesterMA4803Random Graphs and NetworksNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou 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 semesterMA5436Statistical LearningNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn the basics and advanced methods of supervised and unsupervised statistical learning. In the end you will be able to understand and apply models for regression and classification, use techniques for dimensionality reduction and cluster analysis, and follow probabilistic formulations of learning problems while designing new algorithms.6 ECTSruns this semesterMA4802Advanced Numerical Linear AlgebraNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn methods of numerical linear algebra with applications in data assimilation, model reduction and the solution of matrix equations. In the end you will be able to formulate and interpret variational and statistical data assimilation problems, understand and implement common model reduction procedures for linear systems, and apply theory and numerical methods for Lyapunov and Sylvester equations.1 ECTSno date this semesterMA5922
36 more in A1.3 Mathematics Modules on Special TopicsAdvanced Topics in Uncertainty QuantificationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn methods for uncertainties quantification for sensitivity analyses, estimation of rare events, and the Bayesian approach to inverse problems. In the end you will be able to select appropriate numerical and statistical solution procedures and critically assess their foundations as well as their limitations.6 ECTSno date this semesterCIT4130021Approximation AlgorithmsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn how to design and analyze efficient approximation algorithms for combinatorial optimization problems. In the end you can assess the running time and approximation guarantees of algorithms, apply known techniques (e.g. Greedy, LP-Rounding, Primal-Dual), and prove limits of approximability.9 ECTSno date this semesterCIT4100003Compressed SensingNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn how to reconstruct sparse signals from a few structured linear measurements and which measurement matrices guarantee reliable recovery. In the end you can formulate important recovery statements, apply the Restricted Isometry Property (RIP) to random matrices, and use the underlying probabilistic proof techniques.5 ECTSno date this semesterMA5352Computational Complexity in OptimizationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn the fundamentals of complexity theory with a focus on optimization problems: formal models, classes such as P and NP, NP-completeness and typical NP-hard optimization tasks. In the end you will be able to assess the difficulty of optimization problems, apply appropriate modeling guidelines, and analyze practical examples.5 ECTSno date this semesterMA5222Computational Convexity - Optimal ContainmentNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou deal with algorithmic questions on convex problems in arbitrary dimensions and, to some extent, in generalized normed spaces. In the module you will learn typical problems such as optimal containment, the underlying concepts of convex analysis and techniques from linear optimization, as well as their algorithmic solution and analysis.9 ECTSno date this semesterMA5206Computational Integer ProgrammingNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you learn the computational methods for solving mixed-integer optimization problems (MIP). You understand fundamental algorithms such as Simplex, Branch-and-Bound and Cutting-Plane-Separation as well as practical improvements and heuristics that make MIP solvers applicable to real problems. By the end you can explain these procedures, justify their correctness, and use your modeling knowledge to improve MIP models.3 ECTSno date this semesterMA8034Computational Inverse ProblemsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical and numerical methods for solving mainly linear inverse problems. In the end you will know the theory (ill-posedness, regularization, SVD, generalized Tikhonov) and common numerical procedures (direct and iterative regularization) as well as criteria for stability, convergence and stopping.6 ECTSno date this semesterMA4302Convex Duality and Applications in Mass Transport and Calculus of VariationsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn foundations of convex duality and how they are applied in variational problems and in the theory of optimal mass transport. In the end you will be able to understand Legendre/Fenchel duality, dual formulations of optimization problems and central results of mass transport theory as well as explain simple numerical procedures for it.3 ECTSno date this semesterMA5910Diskrete Geometrie: GitterpolytopeNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou deal with convex, rational polyhedra and their relation to lattices. In the module you will learn central results such as Ehrhart theory (the connection between volume and the number of lattice points) as well as aspects of the geometry of numbers and unimodular triangulations, and you can apply these methods to examples from combinatorial geometry, integer optimization, and algebraic geometry.9 ECTSno date this semesterMA5215Elliptic CurvesNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the foundations of the theory of elliptic curves: from algebraic curves and their invariants to the arithmetic structure of the points on elliptic curves. In the end you will know definitions, important theorems (e.g. Bezout, Mordell, Hasse) and you can compute invariants, differentials, the genus and the group structure for curves in Weierstrass form.9 ECTSno date this semesterMA5114First Order Mean Field GamesNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the foundations and current methods of First Order Mean Field Games (MFG) with a focus on variational principles, congestion or congestion models and connections to dynamic optimal transport. By the end you will understand existence and uniqueness questions, regularity tools and variants such as dense- or time-minimal constrained MFG.3 ECTSno date this semesterMA5912First Order Primal-Dual Optimization MethodsNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn modern First-Order Primal-Dual optimization methods. You will understand how simple iterative schemes with primal-dual decompositions can be combined to design efficient, structure-exploiting algorithms for large-scale problems, and you will be able to apply and further investigate these methods.5 ECTSno date this semesterCIT413065Foundations of Data AnalysisNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn mathematical methods of data analysis, in particular linear and nonlinear procedures for data dimensionality reduction as well as techniques for reconstructive restoration of structured signals. By the end you will be able to apply and assess singular value decomposition, random matrices, compressive methods and manifold-based procedures.8 ECTSno date this semesterMA4800Fundamentals of Optimization for Machine LearningNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn fundamentals and advanced techniques of optimization, both convex and nonconvex as well as combinatorial and continuous, with a focus on applications in machine learning. In the end you will be able to understand optimization problems from ML research and approach research questions in this area.5 ECTSno date this semesterCIT413031Geometrie und Topologie für die DatenanalyseNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn fundamental methods of geometric and topological data analysis. This includes constructions such as Voronoi and Delaunay diagrams, Alpha Shapes, as well as concepts from topology such as simplicial complexes and homology and their application to filtrations and persistent homology to investigate the topological properties of point sets.6 ECTSno date this semesterMA4804Graphical Models in StatisticsNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn statistical models whose conditional independencies are described by graphs. The focus is on continuous distributions (in particular the multivariate normal distribution), undirected graphical models (Gaussian models) and directed acyclic graphs (Bayesian networks). By the end you will be able to represent dependency structures in data with graphical models and select appropriate model classes.9 ECTSno date this semesterMA5439High-dimensional StatisticsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn methods for the analysis of data where the number of variables is large. After the module you can apply procedures for controlling false discoveries in large-scale multiple testing, use sparse regression methods (e.g., Lasso) and methods for group-specific or low-dimensional structured signals, as well as use regularization approaches for matrix parameters and graph models.5 ECTSno date this semesterMA5442Identification of Artificial Neural Networks: from the Analysis of one Neuron to Deep Neural NetworksNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical methods for the analysis and identification of feed-forward neural networks — from individual neurons to flat networks up to deep networks. The focus is on linear algebra, probability (in particular concentration inequalities) and optimization to determine weights and activation functions efficiently and robustly.6 ECTSno date this semesterMA5929Introduction to Stochastic Differential Equations: Theory and NumericsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn the fundamentals of stochastic differential equations (SDEs) as well as their numerical treatment. By the end you will know the most important stochastic processes, the construction of stochastic integrals, and solution methods for SDEs, and you can analytically solve simple SDEs and apply numerical schemes to approximate general SDEs.3 ECTSno date this semesterMA5950Mathematical Foundations of ImagingNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical foundations and models of imaging — from Fourier series and -transforms through frame theory and time-frequency analysis to wavelet transforms and variational methods. In the end you can compare different imaging models, apply their mathematical analysis, and employ suitable tools to solve concrete imaging problems.9 ECTSno date this semesterMA5063Mathematical Foundations of Machine LearningNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the mathematical foundations of modern machine learning methods: structure, learning algorithms and approximation properties of neural networks, theory and application of kernel methods in reproducing kernel Hilbert spaces as well as qualitative aspects such as loss functions, risk and complexity questions. By the end you will be able to construct networks, discuss their approximation properties, apply kernel methods and assess the statistical efficiency of procedures.6 ECTSno date this semesterMA4801Mathematische Grundlagen der Neuronalen NetzeNo ratings for this module yet.A1.3.1 Related to the Study ProgramThe module conveys selected mathematical foundations for the analysis of artificial neural networks. You will learn how approximation properties, stability with respect to input perturbations, and the learnability of networks are studied using various mathematical tools. In the end you will understand the central theoretical results and the analytical methods used.6 ECTSno date this semesterMA5913Methods for Inverse ProblemsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical foundations and numerical methods for solving inverse problems. The focus is on regularization techniques for linear and nonlinear problems as well as iterative solution methods (gradient-, Newton-, and Kaczmarz-type) and practical aspects such as adaptive discretization in PDE-based reconstruction tasks.3 ECTSno date this semesterMA5931Modern Approximation TheoryNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn tools of modern approximation theory: s-numbers (such as approximation, Gelfand and Kolmogorov numbers), entropy numbers as well as methods and bounds for sparse reconstruction (e.g., Prony methods, Restricted Isometry Property, iterative hard thresholding, CoSaMP). In the end you can apply these concepts to tasks for the approximation of vectors, functions and operators and you will recognize fundamental lower bounds in sampling theory.6 ECTSno date this semesterMA5952Moderne Methoden der Nichtlinearen Optimierung (2)No ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn selected modern methods of nonlinear optimization, such as convex methods, nonsmooth optimization, interior-point methods, semidefinite programming, robustness approaches, duality, or globalization techniques. In the end, you will be able to read current research articles on the treated topics and follow the presented methods independently further.5 ECTSno date this semesterMA4505Nonsmooth OptimizationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn methods and concepts of nonsmooth optimization and how to apply them. The module conveys fundamentals of nonsmooth analysis, numerical procedures for minimization and for handling nonsmooth equations, as well as their convergence properties. At the end you will be able to select appropriate methods for concrete nonsmooth models and to assess their behavior theoretically.5 ECTSno date this semesterCIT4130020Numerical Methods for Uncertainty QuantificationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn how to formulate, analyze, and numerically approximate elliptic boundary value problems with random coefficients. This includes modeling and sampling of random fields as well as numerical methods such as Monte Carlo, stochastic collocation, and stochastic Galerkin methods.6 ECTSno date this semesterMA5348Operator TheoryNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the fundamentals of spectral theory of operators, in particular spectra of operators, spectral theory for normal operators and functional calculus for different classes of operators. In the end you will be able to analyze and apply these tools to integral and differential operators.9 ECTSno date this semesterMA5012Optimal TransportNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the foundations of the mathematical theory of Optimal Transport (OT). By the end you will understand central concepts such as Monge and Kantorovich formulations, the Wasserstein distances and their role in analysis, physics, economics and machine learning, as well as methods for handling high-dimensional, multi-criterion problems.9 ECTSno date this semesterMA5934Optimal Transport, Numerics and SamplingNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn the theory of optimal transport and methods for its numerical treatment. You will understand the fundamental models (Monge, Kantorovich), important properties of Wasserstein spaces and gain an overview of numerical procedures and applications in Data Science.3 ECTSno date this semesterMA5933Optimale Steuerung gewöhnlicher Differentialgleichungen 1No ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn fundamental concepts and methods of optimal control for ordinary differential equations. In the end you will be able to formulate necessary optimality conditions (e.g., Euler–Lagrange, Legendre–Clebsch), distinguish different types of constraints and control restrictions, and convert control problems into boundary-value forms suitable for numerical treatment.5 ECTSno date this semesterMA3312Population GeneticsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical models of population genetics (e.g., Wright–Fisher, Moran, Kingman coalescent) and how evolutionary and ecological forces shape genomic variation. In the end you will be able to formulate models, analyze them, and interpret results as well as patterns in genomic polymorphism, including interspecific interactions such as cooperation or coevolution.6 ECTSno date this semesterMA5615Probability on GraphsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou 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 Related to the Study ProgramYou 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 semesterMA5306Scheduling: Theory and AlgorithmsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn models and algorithms for the assignment of tasks to scarce resources. The module covers classical and modern scheduling problems (including stochastic, online, robust) as well as methods for their modeling, analysis and solution, so that you can, in the end, assess complexity, design exact or approximate algorithms and prove their quality.5 ECTSno date this semesterCIT413053TUM Data Innovation LabNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn the TUM Data Innovation Lab you work in your Master's program in small, interdisciplinary teams on real data-driven projects from science or industry. In the end you will be able to process, analyze and visualize data, implement numerical solutions, and present your results both technically and in an understandable way.10 ECTSno date this semesterMA8113