Mathematical Topics
Analysis of Partial Differential Equations and Evolutionary EquationsPartial differential equations offer a powerful and versatile framework for the continuum description of phenomena in nature and technology with complex coupling and dependencies. At the Weierstrass Institute this research has three essential focuses: (a) Rigorous mathematical analysis of general evolution equations in terms of existence, uniqueness and regularity of different types of solutions, (b) Development of variational approaches using the toolbox of the calculus of variations, (c) Regularity results for solutions of elliptic and parabolic partial differential equations. [>> more]
Free boundary problems for partial differential equationsFree boundary problems for partial differential equation describe problems such that a partial differential equation is considered on a domain depending on the solution to the equation. [>> more]
Interacting stochastic particle systemsIn the mathematical modeling of many processes and phenomena in the Sciences and Technology one employs systems with many random particles and interactions. In this context, we define the term "particle systems" very broadly and also include point processes with percolation properties and random graph structures as well as Gibbs interactions. It also includes random movements of these particles, such as those that occur in spatial models for communication. At WIAS, many macroscopic properties of these particle systems are investigated that arise from the microscopic rules, such as phase transitions (condensation, percolation, crystallization) and critical properties such as rescaling limits. [>> more]
Modeling, analysis and numerics of phase field modelsA diffuse phase field model is a mathematical model for describing microstructural phenomena and for predicting morphological evolution on the mesoscale. It is applied to a wide variety of material processes such as solidification, coarsening in alloys, crack propagation and martensitic transformations. [>> more]
Multi-scale modeling, asymptotic analysis, and hybrid modelsTo understand the interplay between different physical effects one often needs to consider models involving several length scales. The aim in this mathematical topic is the derivation of effective models for the efficient description of the processes. The understanding of the transfer between different scales relies on mathematical methods such as homogenization, asymptotic analysis, or Gamma convergence. The generated effective models are coupled partial differential equations combining volume and interfacial effects. [>> more]
Numerical Methods for PDEs with Stochastic DataModels of real-world phenomena inevitably include uncertainties which influence the solutions in a nonlinear way. Numerical methods for PDE with stochastic data enable to quantify such uncertainties of the solution in dependence of the stochastic input data. Due to the high problem complexity, modern compression techniques are mandatory. [>> more]
Numerical methods for problems from fluid dynamicsA main research field is the development, analysis, improvement and application of numerical methods for equations coming from CFD. The spatial discretization of the equations is based on finite element and finite volume methods. A focus of research is on so-called physically consistent methods, i.e., methods where important physical properties of the continuous problem are transferred to the discrete problem. [>> more]
Optimal Transport: Statistics, Numerics, and Partial Differential EquationsThe theory of Optimal Transport has been immensely influential in connecting partial differential equations, geometry, and probability. On the one hand, research at WIAS is focused on applying methods and tools from Optimal Transportation Theory to problems in statistics, such as semi-supervised and unsupervised learning, clustering, text classification, as well as in image retrieval, clustering, segmentation, and classification by developing and analyzing new numerical algorithms and schemes. On the other hand, the theory of optimal transport is extended, e.g., towards unbalanced optimal transport and connections to evolutionary partial differential equations via gradient systems. [>> more]
Optimal control of partial differential equations and nonlinear optimizationMany processes in nature and technics can only be prescribed by partial differential equations,e.g. heating- or cooling processes, the propagation of acoustic or electromagnetic waves, fluid mechanics. Additionally to challenges in modeling, in various applications the manipulation or controlling of the modeled system is also of interest in order to obtain a certain purpose... [>> more]
Statistical InferenceThe term statistical inference summerizes methods to extract information from observed data in order to characterize properties in populations. This involves statistical modeling, estimation and uncertainty assessment of parameters, testing of hypothesis. [>> more]
Statistical inverse problemsIn many applications the quantities of interest can be observed only indirectly, or they must be derived from other measurements. Often the measurements are noisy and the reconstruction of the quantities of interest from noisy measurements is unstable. [>> more]
Systems of partial differential equations: modeling, numerical analysis and simulationThe mathematical modelling of many scientific and technological problems leads to (initial) boundary value problems with systems of partial differential equations (PDEs). [>> more]
Variational methodsMany physical phenomena can be described by suitable functionals, whose critical points play the role of equilibrium solutions. Of particular interest are local and global minimizers: a soap bubble minimizes the surface area subject to a given volume and an elastic body minimizes the stored elastic energy subject to given boundary conditions. At WIAS, methids from the calcuus of variations are applied and further developed to solve problems in physics and technology such as continuum mechanics, quantum mechanics, and optimal control. [>> more]
Archive
Further mathematical topics where the institute has expertise in:
Nonlinear kinetic equationsKinetic equations describe the rate at which a system or mixture changes its chemical properties. Such equations are often non-linear, because interactions in the material are complex and the speed of change is dependent on the system size as well as the strength of the external influences. [>> more]
