BiCONTROL

Bilevel control of sequential parameter identification

Variational approaches based on optimal control have proven successful for optimal design of single excitation pulses in MRI. However, measurements are usually taken in a sequence, either due to a time-varying parameter (e.g., MRI of the beating heart) or due to sequential data acquisition (e.g., phase-encoding in static MRI). Typically, such measurements are designed once and then repeated identically, with enough time between them to return the system to a “steady-state”. Optimizing instead the full sequence with reconstruction quality in mind will lead to faster and more robust measurements as each pulse is adapted individually to its position in the sequence.

The mathematical goal of this subproject is to derive optimality conditions and efficient numerical algorithms for the optimal design parameters by formulating the controlled reconstruction problem as a bilevel problem, where the convex nonsmooth lower-level problem is the parameter identification problem for a given design, and the upper-level problem minimizes the reconstruction error to a given ground truth. For this, we will make use of modern tools of variational analysis, in particular properties like metric subregularity and calmness. For the numerical solution, we consider both primal-dual first-order methods and semismooth Newton-type methods based on coderivatives of the lower-level optimality conditions.

We also aim at applying these results to optimal control of the Bloch–Torrey equation for optimal design of measurement sequences following an online approach that exploits new data as it is generated. The sequential measurement situation can be treated as an alternating model predictive control or receding horizon control problem. Here we will ensure robustness of the design parameter over a range of model parameters by integrating an ensemble Kalman filter approach.

The project will primarily cooperate with IdCONTROL in the context of online control and identification; with BiLEARN and OptOP in the context of bilevel optimization, and with GenMRI on integrating the developed approaches into MRI sequence.

Team