Part 1 - Dimension Reduction for Rapid Localization of Parametrized Microstructures
In the first period, we successfully developed new approaches for (non-intrusive) model order reduction (MOR) capable of providing inexpensive surrogates for parametrized microscale representative volume elements as required in multi-scale simulations. This is a significant step towards our goal of developing dimension reduction methods capable of flexibly and efficiently dealing with complex, parametrized microstructures exhibiting nonlinear behaviour. With such methods, we are able to drastically reduce the computational cost to simulate a multi-scale component’s response for any parameter value within a prescribed range.
In particular, we developed a non-intrusive reduced basis method based on proper orthogonal decomposition and gaussian process regression capable of constructing inexpensive surrogates for parametrized microscale problems. The proposed method has three key features. First, the microscopic stress field can be fully recovered. Second, the method can accurately predict the stress field for a wide range of material parameters; furthermore, the derivatives of the effective stress with respect to the material parameters are available and can be readily utilized in solving optimization problems. Finally, it is more data efficient, i.e. requiring less training data, as compared to directly performing a regression on the effective stress. The method has been successfully applied to material and loading parametrizations (see [Guo, Rokos & Veroy, 2021]); a further publication applying the method to the more difficult case of geometric parametrizations has also been submitted to a top journal and is now in the final stages of revision. These works have been presented and disseminated in several contributed and invited talks in conferences and workshops.
Part 2 - Data Assimilation and Optimal Experimental Design in the Multi-Scale Setting
In this part, we have three main achievements in the first period.
(1) We successfully developed a reduced order methodology for ensemble Kalman inversion that even makes corrections for the errors introduced by the reduced basis surrogate. The method is applicable and relevant not only in materials, but also in other fields such as geophysics. This work has also been presented in several workshops and conferences. A manuscript describing this method is in the final stages of preparation and will be submitted to a journal by the end of the calendar year.
(2) In this period, we built upon our previous work on reduced basis and optimal experimental design methods for 3D-VAR (for stationary problems) and extended the methods to 4D-VAR (for time-dependent problems). We were able to develop efficiently computable reduced order models for space-time discretizations, which then allowed extension of the the 3D-VAR methodology also for time-dependent problems. Two manuscripts on this work are in the final stages of preparation and will be submitted to journals by the end of the calendar year.
(3) Building upon our previous work on optimal experimental design for a simpler setting, we developed a sequential method for optimal experimental design in the Bayesian (probabilistic) setting. In this first work, we focus on a single scale problem. The developed algorithm allows for correlated noise models even for large sets of admissible sensors and remains computationally efficient for high-dimensional forward models through MOR. A manuscript describing this work is in the final stages of preparation and will be submitted by the end of the year. This work has been presented by in several invited talks.
