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

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