Modelling, Analysis and Optimization with Differential Equations
Applied analysis links mathematics and scientific computing to engineering and natural sciences. Modelling in the applied sciences typically leads to systems of nonlinear ordinary or partial differential equations, or variational problems. The nonlinearities often reflect complex phenomena which result in oscillations, concentrations, or singularities in the mathematical model. Several research projects of the applied analysis groups in Berlin are motivated by mathematical problems in continuum mechanics, including fluid dynamics, nonlinear elasticity, and
chemical reaction systems, and semiconductor models.
Most problems involve the interaction of processes on different time and length scales, leading to so-called multiscale systems. Resolution on all scales is usually impossible, which requires deriving hierarchies of models to describe the problem on the various scales. Applied analysis aims at an explanation or better understanding of the physical phenomena by applying methods from real and functional analysis to prove existence and finer properties of solutions of the partial differential equations or variational problems.
Berlin research groups
Nonlinear Evolution EquationsThe focus of our research is on nonlinear partial differential equations describing time-dependent processes. In particular, we are interested in the mathematical description of complex fluids and of nonlocal continuum mechanics. We study the existence and uniqueness of generalized solutions. This is often related to an analysis of suitable discretization methods. |
TU Berlin |
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| Maximilian Engel | Random and Multiscale Dynamical Systems Engel's research is focused on Dynamical Systems, i.e., the mathematical description of time-dependent behaviour, and its relations to Probability, Analysis and Geometry. Firstly, he is working on problems in random dynamical systems, in particular chaotic random attractors and their parameter-dependent change, called bifurcations. Furthermore, he investigates multiscale systems with non-hyperbolic singularities, analyzing discretization problems for ODEs, SDEs and PDEs. Additionally, he is working on models of stochastic dynamics in applications, with an increased focus on chemical reaction, learning and atmospheric dynamics. |
FU Berlin |
Stochastic homogenization of randomly perforated domainsMartin Heida mostly works on stochastic homogenization of randomly perforated domains, searching for sufficient and necessary conditions on critical geometric quantities that allow to proof compactness results for the involved function spaces. He also works on developing algorithms for high dimensional Voronoi meshes and implementing them in Julia code. |
TU Berlin | |
Analysis of Partial Differential Equations Partial Differential Equations (PDEs) are essential for modeling diverse phenomena across multiple disciplines. Ensuring well-posedness guarantees the accuracy of these models. Deriving effective models e.g. via calculus of variations simplifies complex PDEs while retaining key behavior. PDEs are closely linked to stochastic processes, and studying the properties of PDEs, in particular concerning regularity of solutions, is crucial for developing efficient numerical approximation schemes and optimization algorithms. Overall, PDEs are integral to understanding complex systems in various scientific and engineering domains. |
HU Berlin |
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Analysis of Non-Smooth Systems in Materials ModelingMany problems in engineering, physical or geoscientific applications lead to systems of coupled nonlinear partial differential equations, where some of its components feature non-smooth terms resulting in subdifferential inclusions, resp. variational inequalities. Often, different time and length scales are involved in the model, for example if some of the processes are confined to material interfaces. Such variational formulations with non-smooth terms arise in the modeling of very different applications, for example, for fracture processes in elastic solids, but also for geological flows in the earth’s mantle or for sea ice dynamics. Marita Thomas studies the well-posedness of such systems, scaling laws and approximation methods as well as the properties of solutions using tools from nonlinear PDE theory, calculus of variations and variational convergence methods. |
FU Berlin |
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Thermodynamic Modeling and Analysis of Phase Transitionstext will follow |
TU Berlin | |
Analysis of Elliptic Equations with Unbalanced Growth Elliptic equations with unbalanced growth have numerous applications in physics and engineering, for example to describe models for strongly anisotropic materials in the context of homogenization and elasticity theory, but also in the study of duality theory and the Lavrentiev gap phenomenon. Well-known operators of this class of problems are the double phase operator and the logarithmic double phase operator. Boundedness, concentration, existence, multiplicity, and uniqueness of solutions are of particular interest in the analysis of such equations. |
TU Berlin | |
Direct and Inverse Problems in Wave PropagationThe phenomenon of wave propagation is very common in nature and real-world applications, such as the medical imaging, non-destructive testing or nano-technology. From the mathematical point of view, the waves are modelled by partial differential equations. To understand these physical phenomenon, the PDEs are analysed and the numerical methods are developed to simulate the processes. To study inverse problems, optimisation techniques or fast imaging methods are proposed. This topic involves several different areas in mathematics, such as the analysis of PDEs, numerical analysis, optimization, etc. |
TU Berlin | |
| Barbara Zwicknagl | Calculus of Variations and Pattern Formation The main focus of our research lies on the analysis of variational problems arising in the natural and materials sciences, in particular pattern forming systems such as shape-memory alloys, magnetic compounds, or biological membranes. Mathematically, this typically leads to nonconvex or nonlocal variational problems. We study existence and uniqueness of solutions as well as refined properties, including scaling laws for the minimal energies. |
HU Berlin |
Links to other areas
There are natural, strong collaborative ties to the following groups:
Geometric partial differential equations and integrable systems in RTA 1:
Thomas Walpuski, Chris Wendl
Stochastic differential equations and large deviations principles in RTA 3:
Peter Friz, Wolfgang König, Nicolas Perkowski, Wilhelm Stannat
Numerical analysis of partial differential equations in RTA 6:
Carsten Carstensen, Volker John
Core Courses
Core Courses: Functional analysis; Partial Differential Equations; Modelling with PDEs.
The BMS Core Courses in Area 7 and details about their content can be found under "Course Program":
Area 7 - Core Courses
Advanced Courses
Topics for typical advanced courses include:
Nonlinear PDEs; Nonlinear Functional Analysis; Multidimensional Caluclus of Variations; Functions of Bounded Variation.
The current advanced courses are listed under "Course Program":
Area 7 - Advanced Courses
