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Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting

Kathrin Hellmuth, Christian Klingenberg, Qin Li, Min Tang

TL;DR

The relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models, are discussed in the Bayesian framework and it is proved the asymptotic equivalence of the two posterior distributions.

Abstract

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller-Segel equation and a chemotaxis kinetic equation. These two equations describe the organism movement at the macro- and mesoscopic level respectively, and are asymptotically equivalent in the parabolic regime. How the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller-Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this one infers the chemotaxis response, which constitutes an inverse problem. \\ In this paper we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated to two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought after. We prove the asymptotic equivalence of the two posterior distributions.

Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting

TL;DR

The relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models, are discussed in the Bayesian framework and it is proved the asymptotic equivalence of the two posterior distributions.

Abstract

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller-Segel equation and a chemotaxis kinetic equation. These two equations describe the organism movement at the macro- and mesoscopic level respectively, and are asymptotically equivalent in the parabolic regime. How the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller-Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this one infers the chemotaxis response, which constitutes an inverse problem. \\ In this paper we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated to two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought after. We prove the asymptotic equivalence of the two posterior distributions.

Paper Structure

This paper contains 5 sections, 5 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Asymptotic analysis for kinetic chemotaxis equations and the Keller-Segel model
  3. Bayesian inverse problem setup
  4. Convergence of posterior distributions
  5. Summary and Discussion

Key Result

Theorem 1

Suppose $K_{\varepsilon}$ has the form of eqn:K_form with $(K_0,K_1)\in \mathcal{A}$ and suppose the initial condition $f_0\in C^{1,+}_c(\mathbb{R}^3\times V)$, then the solution $f_\varepsilon$ to the chemotaxis equation eq:chemotaxis satisfies the following:

Figures (3)

  • Figure 1: Two ways to compare the inverse problems: determining and comparing the tumbling kernels for both underlying chemotaxis and Keller Segel models (left) or determining the drift or diffusion coefficient for the Keller Segel model and the tumbling kernel for the chemotaxis model and calculating the corresponding drift and diffusion coefficients.
  • Figure 2: Measurement of the bacteria density (blue) at two different measuring times $t_j, t_{\tilde{j}}$. The location of the test functions is indicated by the support in space of the test functions $\chi_j, \chi_{\tilde{j}}$.
  • Figure 3: Two ways to determine the posterior distribution $\mu_{\mathrm{KS}}^y(D,\Gamma)$ from a prior $\mu_0(K_0,K_1)$ on the tumbling kernels.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof : Sketch of proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 4 more