From Model to Computation
Simulations are based on mathematical models that are initially consist of differential equations. To prepare them for being implemented on computers, in a process called discretization, we convert the differential equations into algebraic equations systems so they can be evaluated numerically. This process results in discretization errors.
We have made it our ambitious goal to quantify these errors along with uncertainties in the system description as precisely as possible and thus make them easier to manage.
Quantifying discretization errors helps us develop better and more precise solution methods for systems of equations. In doing so, we further our highest priority goal, which is to develop self-adaptive computer systems that can deal expertly with multiple scales while also coupling various physical processes.
Multiple scales and physical processes call for a variety of mathematical solution strategies. A deformation in a material, for example, is calculated on a molecular level differently than is a flow phenomenon on a macroscopic level. Our goal is to resolve these “tower of Babel” differences by developing a common mathematical language. In turn, this is expected to enable simulations that can run self-adaptively with the aid of error indicators, i.e., that choose autonomously where and when to operate, on which scales, and when to switch on or shut off physical processes so as to optimally simulate an entire process.
In addition, self-adaptive simulations will help speed up individual solution strategies considerably. Together with methods for model validation and model reduction we develop in cooperation with researchers from Research Area C, self-adaptive systems will help develop simulations which are highly efficient and responsive to future real-world challenges.