After modelling and simulation of a technological or economical process the ultimate goal is the computation of optimal process parameters and solving problems of optimal control in this respect. In this main application area, methods of PDE-constrained optimal control, control of differential algebraic equations, and stochastic optimal control are employed in a variety of applications in technology and economy. These applications may range from basic production processes such as welding and hardening to the design of diffractive structures and simulation tasks in process engineering industry to optimal decision in financial environments such as financial (energy) derivatives, energy production and storage.
Diffractive optical elements employ the controlled use of light propagation on microstructered interfaces. They perform functions unattainable with traditional elements and can be applied in diffractive measurement technique, spectroscopy, astronomy, optical communication techniques, and information processing. For the design of such elements, Maxwell`s equations have to be solved and special optimization algorithms are to be adapted.
In the area of procuction, there is a huge demand for planning and optimization tools that allow for an accelerated re-planning and reconfiguration of complex production facilities. In this context, the key words are "digital factory" or "Industry 4.0", which represents the transition to a connected, widely self-organized industrial production. This interconnectivity allows not only for the optimization of individual production steps, but also of the entire value chain.
Dynamic process simulation has become an indispensable tool for design, analysis, and operation of complex plants in industry. Here initial value problems for large systems of differential-algebraic equations (DAEs) have to be solved. The simulation concept developed at WIAS exploits the modular structure of the process models to use divide-and-conquer techniques for solving the DAE system with block-structured methods. The concept is implemented in the Simulator BOP and has been successfully used in different industrial applications.
Calibration of realistic stock and interest rate models is of prime importance. While standard Libor interest rate products such as caps and swaptions can be priced (quasi-) analytically in a simple Libor market model, it is impossible to match cap and swaption volatility smiles and skews observed in the markets using this model. Moreover, after the financial crisis the Libor rate could not be considered risk free anymore.
Mobile telephones and similar devices are in wide and intensive use and place high demands on the networks to which they connect. The behaviour of individual users is, for practical purposes, unpredictable and the load on the network varies randomly in space and time. We study the probability of extreme overloading events which cause significant degradation in the user experience. Most problems can be solved by the installation of sufficient networking equipment, but this has a significant financial cost. Our work allows informed decisions to be made balancing cost and service quality.
Nonlinear material models, multifunctional materials and hysteresis in connection with elasto-plastic processes
Many components in modern equipment rely on specific properties of so-called multifunctional materials. These materials are distinguished by the fact that therein properties like elastic deformability, thermal expansibility, magnetizability, or polarizability interact nontrivially like for instance in a piezo-crystal. At WIAS coupled models describing these properties are developed and analyzed.
Derivatives of complex structured financial instruments with early exercise features (e.g. American style products and energy swing options) are important instruments traded at financial and energy markets. Such products give the holder the right to exercise a cash-flow or energy unit or a stream of cash-flows (energy units) at any date before expiry. For example, an investor who holds a certain loan may buy an option to swap his floating interest rate payments with fixed rate payments specified in the contract.
Main Application Areas
- Partial Differential Equations
- Numerical Mathematics and Scientific Computing
- Nonlinear Optimization and Inverse Problems
- Stochastic Algorithms and Nonparametric Statistics
- Thermodynamic Modeling and Analysis of Phase Transitions
- Nonsmooth Variational Problems and Operator Equations
- Probabilistic methods for mobile ad-hoc networks