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October  2019, 12(5): 1109-1130. doi: 10.3934/krm.2019042

## Kinetic methods for inverse problems

 RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany

* Corresponding author: Giuseppe Visconti

Received  November 2018 Revised  February 2019 Published  July 2019

The Ensemble Kalman Filter method can be used as an iterative numerical scheme for parameter identification ornonlinear filtering problems. We study the limit of infinitely large ensemble size and derive the corresponding mean-field limit of the ensemble method. The solution of the inverse problem is provided by the expected value of the distribution of the ensembles and the kinetic equation allows, in simple cases, to analyze stability of these solutions. Further, we present a slight but stable modification of the method which leads to a Fokker-Planck-type kinetic equation. The kinetic methods proposed here are able to solve the problem with a reduced computational complexity in the limit of a large ensemble size. We illustrate the properties and the ability of the kinetic model to provide solution to inverse problems by using examples from the literature.

Citation: Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
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##### References:
Left: vector field of the ODE system (16) with $(y, G) = (2, 1)$. Red lines are the nullclines. Right: trajectory behavior around the equilibrium $(\frac{y}{G}, \frac{y^2}{G^2}) = (2, 4)$
Left: vector field of the ODE system (24) with $(y, G) = (2, 1)$. Red lines are the nullclines. Right: trajectory behavior around the equilibrium $\tilde{F}_1^+ = (2, 8)$
Elliptic problem - Test case 1 with $\gamma = 0.01$. Top row: plots of the residual $r$, the projected residual $R$ and the misfit $\vartheta$ for $M = 250,500, 1000$. Bottom row: plots of the noisy data, the reconstruction of $p(x)$ and the reconstruction of the control $u(x)$ at final iteration for $M = 250,500, 1000$
Elliptic problem - Test case 1 with $\gamma = 0.1$. Top row: plots of the residual $r$, the projected residual $R$ and the misfit $\vartheta$ for $M = 250,500, 1000$. Bottom row: plots of the noisy data, the reconstruction of $p(x)$ and the reconstruction of the control $u(x)$ at final iteration for $M = 250,500, 1000$
Elliptic problem - Test case 1. Left: spectrum of $\boldsymbol{{ \mathcal{C}}}(t) G^T \boldsymbol{{\Gamma}} G$ for the initial data with $\gamma = 0.01$ and $\gamma = 0.1$. Right: adaptive $\epsilon$ and spectral radius of $\boldsymbol{{ \mathcal{C}}}(t) G^T \boldsymbol{{\Gamma}} G$ over iterations with $\gamma = 0.01$ and $\gamma = 0.1$
Elliptic problem - Test case 2 with $\gamma = 0.01$. Top row: plots of the residual $r$, the projected residual $R$ and the misfit $\vartheta$ for $J = 25, 25\cdot2^9$. Middle row: plots of the noisy data and of the reconstruction of $p(x)$ at final iteration for $J = 25, 25\cdot2^9$. Bottom row: plots of the reconstruction of the control $u(x)$ at final iteration for $J = 25, 25\cdot2^9$ and behavior of the relative error $\frac{\|\overline{\mathbf{{u}}}-\mathbf{{u}}\|_2^2}{\|\overline{\mathbf{{u}}}\|_2^2}$
Nonlinear problem. Top row: plots of the density estimation of the initial samples (left) and position of the samples at final iteration (right). Middle row: Marginals of $u_1$ (left) and $u_2$ (right) as relative frequency plot. Bottom row: residual errors $r$ and $R$ (left) and misfit error (right)
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