A Closed-Form Solution to Estimate Spatially Varying Parameters in Heat and Mass Transport

Abstract

This paper presents a closed-form solution to estimate space-dependent transport parameters of a linear one dimensional diffusion-transport-reaction equation. The infinite dimensional problem is approximated by a finite dimensional model by 1) taking a frequency domain approach, 2) linear parameterization of the unknown parameters, and 3) using a semidiscretization. Assuming full state knowledge, the commonly used output error criterion is rewritten as the equation error criterion such that the problem results in linear least squares. The optimum is then given by a closed-form solution, avoiding computational expensive optimization methods. Functioning of the proposed method is illustrated by means of simulation.

Fast simultaneous estimation of nD transport coefficients and source function in perturbation experiments

“…In conclusion, our proposed methodology provides new opportunities to study nD transport by estimating the sinks/sources and transport coefficients while providing transparency about regularization and interpolation with extremely low computational effort due to the closed-form solution that guarantees the global optimum for the selected optimization criteria. The low computational cost of the closed-form solution opens up a number of new significant opportunities: (i) machine learning like approaches to find the best parameterization of the transport coefficients as demonstrated in 16 ; (ii) fast estimation of multidimensional transport coefficients, for which we present the first results in this letter. Moreover, as the source no longer needs to be localized, since the transport parameters and sources/sinks can be estimated simultaneously, adding a source to regions with a low signal-to-noise can significantly improve the quality of the coefficient estimation.…”

“…To estimate the transport coefficients {D, V, K, P}, we reformulate the methodology presented in 16 to be applied to a more general nD setting. As the model, gradient and Laplacian operators are linear, we can rewrite (1) in the frequency domain as…”

“…This is in contrast to the classical transport codes where the choice of basis functions directly limits the solution space. Moreover, due to the low computational cost of the closed form solution, many different parametrizations can be evaluated, such that a machine learning approach (as shown in 16 ) can be applied to find a suitable parametrization.…”

Fast simultaneous estimation of nD transport coefficients and source function in perturbation experiments

“…To estimate the transport coefficients , we reformulate the methodology presented in Ref. 16 to be applied to a more general n D setting. As the model, gradient and Laplacian operators are linear, we can rewrite ( 1 ) in the frequency domain as where and are the Fourier transform of and , respectively.…”

“…Similar to Refs. 7 , 16 , ( 8 ) can be seen as a linear regression model where and are a concatenation of the considered data points for and , respectively. Then, the optimal solution in the least square sense is given by where is the Hermitian transpose of .…”