Numerical Analysis, Optimization, Scientific Computing
The mathematical description of many processes in nature and industry leads to complex models, like systems of partial differential equations or integral equations. Besides equations, models might contain (physical) constraints. It is usually not possible to calculate an analytic solution of such models. In practice, however, approximations of these solutions are needed. RTA 6 is devoted to methods for computing such approximations.
Numerical mathematics comprises the development, analysis, and improvement of numerical methods. Numerical analysis studies, e.g., the accuracy of numerical methods by deriving rigorous estimates of the error of a computed solution to the analytic solution. Modern techniques study the impact of the problem's coefficients on the error bounds. Another topic of numerical analysis is the sensitivity of computed solutions on perturbations of the data. For performing numerical analysis a profound knowledge on the mathematical model is needed. In this respect there are connections to other mathematical disciplines, like the analysis of partial differential equations (RTA 7).
In practice it is often of utmost interest to specify a concrete model by finding an optimal setup, which can be described as minimizer of a cost functional. Usually there are additional constraints that have to be taken into account. To investigate such optimization problems from the point of view of mathematical analysis, e.g., proving the existence of a solution, and to propose and analyze appropriate numerical methods are topics of continuous optimization. As example, the training of neural networks, as it is part of RTA 8, belongs to the field of continuous optimization.
The time for computing numerical solutions is often of great importance in practice. Efficient simulations can be performed, on the one hand, by applying appropriate methods, like adaptive methods or methods based on reduced order models (ROMs). On the other hand, efficiency can be achieved by utilizing modern hardware, like (massively) parallel computers or GPUs. Both approaches can be also combined. The field of scientific computing is devoted to the exploration (implementation and improvement) of numerical methods, often for complex problems from applications.
Numerical simulations of concrete problems often have to incorporate data, e.g., from experiments. In addition, models contain usually data, like physical coefficients or initial conditions. Topics investigated in RTA 8 might be helpful for appropriately incorporating available data in numerical simulations. In addition, data are usually erroneous or they are incomplete, like initial or boundary conditions that are known only at a few points, but not in the complete domain or the whole boundary, respectively. Techniques that come from stochastics, see RTA 3, might be used to quantify the uncertainty of numerical solutions in dependency on the uncertainty of data. .
Berlin research groups
Optimal Control and Model Reduction of Large-Scale Dynamical Systems Control theory deals with the analysis and design of theoretical and numerical methods for systems described by dynamical equations, such as ordinary or partial differential equations, that are subject to external forcing terms ("controls"). Finding "optimal" or "robust" controls for large-scale systems (e.g., spatially semi-discretized PDEs) often suffers from the so-called curse of dimensionality which prevents designing real-time capable algorithms. The study of fast and efficient (numerical) methods that try to mitigate this curse of dimensionality is a central goal within the research field of model reduction. |
TU Berlin |
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Biophysics-based data assimilation in medicineThis area focuses on the combination of available medical data (e.g., measurements and images) with mathematical models for the dynamics of biological flows and tissues, in order to characterize, non-invasively, different biomarkers that can be used by clinicians to characterize different pathologies. Research topics concern numerical solution of PDEs for flow and tissue mechanics, reduced-order modeling for efficient computation, and definition of robust algorithm for the solution of inverse problems. |
WIAS | |
Adaptive mesh-refining in nonstandard finite element methodologiesThe area of research is two-fold with focus on the mandatory adaptive mesh-refining in computational partial differential equations and its (asymptotic) optimal convergence rates. That is a broad field that ranges from minimal properties of the newest-vertex bisection in triangulations to advanced a posteriori error analysis with various ongoing generalisations of the axioms of adaptivity from 2014. The second focus is on nonstandard finite element techniques such as various nonconforming, mixed, and least squares finite elements on the classical side and, of course, some postmodern methodologies like hybrid-high order (HHO), discontinuous Petrov-Galerkin (DPG), virtual elements (VEM), and various discontinuous Galerkin schemes partly with automatic stabilisation with or without appropriate smoothers also in discrete nonlinear terms, to name a few. Traditional applications arise in the computational solid mechanics with phase transition or elastoplacticity, although there is a focus on the numerical treatment of eigenvalue problems with differential operators as well as the computational calculus of variations. One exciting aspect is on the particular advantages of non-standard schemes in nonlinear problems like guaranteed lower eigenvalue or energy bounds even in non convex minimisation. |
HU Berlin |
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Numerical Methods, Modeling and Charge Transport ApplicationsMany physical applications can be described by models based on partial differential equations. For example, the underlying mathematical models may rely on nonlinear drift-diffusion, hyperelastic material laws, inverse PDE problems, Helmholtz operators, localized landscape theory and atomistic coupling. These models are then analyzed and solved numerically via physics-preserving finite volume methods, data-driven techniques and meshfree methods. Examples for charge transport applications are perovskite solar cells, memristors, bent nanowires, quantum wells, lasers as well as doping reconstruction. |
WIAS | |
Optimization of Complex SystemsThe research area comprises optimal control of complex dynamical systems, in particular concerning problems with spatio-temporal processes. Examples include optimization and control of hybrid systems and switching systems involving partial differential equations, and systems on heterogeneous domains such as physical network flow. A particular focus lies on characterizations of optimal solutions, algorithm design for solvers, and applications. |
HU Berlin |
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Nonlinear Optimization and Inverse Problemstext will follow |
WIAS | |
Optimization with partial differential equationsMotivated by practical applications, this area is concerned with the analysis and solver design for optimization or optimal control problems subject to partial differential equations (PDEs) or variational inequalities (VIs). Moreover, equilibrium problems such as generalized Nash equilibrium problems with PDEs are considered. Concerning the theory, a delicate interplay of PDE resp. VI solution (state) regularity and the topology underlying further constraints (on controls and/or states) is studied and the derivation of tight stationarity conditions and, if applicable, second conditions is focused on. Solution algorithms are addressed first in function space and then upon discretization with the aim of mesh independence. In connection with inequality constraints semi smooth Newton methods are studied. |
HU Berlin |
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Optimal control of coupled multifield problemsHömberg is interested in modelling, analysis and optimal control of coupled multi-scale and multi-field problems. These occur in various applications. In recent years, his group has been working on systems related to industrial manufacturing, such as induction heat treatments, where the Maxwell equations are coupled with an energy balance and further model equations for the changing microstructure. Another exciting topic they are currently working on is multiscale topology optimization for 3D printing, in collaboration with the 3D printing laboratory at the TU. Problems with different time scales arise, for example, in the optimal design of electrical microgrids. These consist of photovoltaic systems coupled with a battery and a diesel engine as a back-up, and serve as an energy supply in rural areas without a stable connection to a power grid. |
TU Berlin and WIAS | |
Dimension Reduction and Adaptive Regularization A key challenge for the efficient handling of high-dimensional data is to identify and exploit low-dimensional structures. We investigate adaptive techniques for that purpose. |
TU Berlin | |
| Volker John | Numerical Methods for Partial Differential EquationsMain scientific interests of Prof. John include the development and analysis of physically consistent discretizations of (initial) boundary value problems from fluid mechanics, like convection-diffusion or incompressible flow problems, and the use of these methods in applications, like in hemodynamics. An aspect of the research is the exploration of techniques from machine learning for solving partial differential equations. |
FU Berlin and WIAS |
Numerical Linear Algebra The area of Numerical Linear Algebra is concerned with the numerical solution of linear algebraic problems, for example linear systems and eigenvalue problems. Another important task is the computation of matrix functions. The computational problems occur in applications throughout the natural sciences and engineering. They typically result from the discretization of differential or integral equations that are used to model real world phenomena. In large scale applications the numerical solution methods must be adapted to the particular problem structures, and hence the area requires numerous different and often interdisciplinary tools. |
TU Berlin | |
Numerical methods for partial differential equationsThis research group focuses on developing new numerical methods for solving partial differential equations that describe convection dominated flow problems. One main goal is to develop new methods that allow to simulate flow around complex geometries more easily by using a cut cell approach. Methods of choice are finite volume and discontinuous Galerkin schemes. We cover the whole range from mathematical proofs on simplified models to simulations on bigger clusters. Interests include also PDE-constrained optimization and two-phase flow. |
TU Berlin | |
Applied Linear Algebra One important aspect of Applied Linear Algebra is the theoretical investigation and numerical solution of standard, generalized and polynomial eigenvalue problems. Of particular interest are eigenvalue problems whose underlying matrices, matrix pencils or matrix polynomials carry an additional symmetry structure. |
TU Berlin | |
| Caren Tischendorf | Applied Mathematicstext will follow |
HU Berlin |
Nonsmooth optimization & analysisNonsmooth regularization functionals are a cornerstone of many modern areas of applied mathematics including, e.g., variational approaches to inverse problems or machine learning. This is, in particular, due to the fact that the correct choice of the regularizer as well as of a suitable variable space promotes desirable structural features in the solutions of the associated minimization problem. The main topic of the group is a novel perspective on nonsmooth minimization based on the study of the convex geometry of the sublevel sets of the regularizer, in particular their extremal points. A precise characterization of the latter allows for the derivation of meaningful first and second order optimality conditions as well as the design of efficient numerical discretization schemes and fast solution algorithms. Application areas include, e.g., inverse problems and optimal (feedback) control problems with constraints given by partial differential equations. |
HU Berlin |
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Scientific Computing, Mathematical OptimizationFor the efficient solution of nonlinear optimization problems, structure exploitation is indispensable. Corresponding approaches include for example adjoint-based methods and approaches that take nonsmoothness into account. The required derivative information is provided by algorithmic differentiation. The solution of optimization problems from real-world applications like the optimal shape of gas turbines or the parameter identification fpr piezoceramics are based on strong software components. |
HU Berlin |
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Scientific ComputingScientific Computing comprises the development and analysis of numerical methods for partial differential equations and optimization, and their application to challenging simulation, optimization, and inverse problems from natural, engineering, and life sciences. It is concerned with efficiency and reliability of discretization and solution methods as well as with their implementation. |
ZIB |
Links to other areas
There are natural, strong collaborative ties to the following research areas:
RTA 3: incorporating noisy data in simulations, performing uncertainty quantification of computed solutions
RTA 7: applying tools and results from analysis of partial differential equations for analysis of numerical methods
RTA 8: using techniques from machine learning in simulations, developing data-driven algorithms
Core Courses
The BMS Core Courses in Area 6 and details about their content can be found under "Course Program":
Area 6 - Core Courses
Advanced Courses
Topics for advanced courses include: Model Reduction, Optimal Control of PDEs, Advanced Finite Element Methods, Computational Photonics, Uncertainty Quantification, Control Theory, Energy Based Modeling, Theory and Numerics of Non-Smooth Optimization, Numerical Analysis of PDEs, Variational Inequalities, Inverse Problems.
The current advanced courses are listed under "Course Program":
Area 6 - Advanced Courses
