Identification of reaction rate parameters from uncertain spatially distributed concentration data using gradient-based PDE constrained optimization

A promising approach to quantify reaction rate parameters is to formulate and solve inverse problems by
minimizing the deviation between simulation and measurement. One major challenge may become the non-
uniqueness of the recovered parameters due to the ill-posed problem formulation, which requires sophisticated
approaches such as regularization. This study investigates the feasibility of using spatially distributed reference
data, i.e., concentration distributions of reactive flows, which could be obtained by magnetic resonance imaging
(MRI), instead of isolated points or integral values to recover reaction rate parameters. We propose a combined
framework of computational fluid dynamics (CFD) and gradient-based optimization methods, which minimizes
the difference between the simulated concentration distribution and a given data set by automatic iterative
parameter adjustments. The forward problem is formulated as a coupled system of reaction-advection-diffusion
equations (RADE), which is solved by the lattice Boltzmann method (LBM). Therefore, a system of non-linear
partial differential equations (PDE) acts as optimization constraints, limiting the possible outcomes of the
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inverse problem. A benchmark test case using a CFD simulation as reference data confirms the validity of the
presented method by successfully identifying up to three a priori set reaction parameters reversely. With it,
initial relative errors could be reduced from around 150% to 10−3 % in 13 optimization steps corresponding to 37
simulations. Even after reducing the accessible reference data from 2D concentration distributions to 1D outflow
concentration distribution or by adding noise signals onto the reference data with a signal-to-noise ratio (SNR)
of 5, our framework successfully recovered the parameters with a relative error of ≈ 1%. Both, the chosen LBM
and optimization algorithms are implemented in the open-source library OpenLB.