inverse_lecture33 - t tr'cl D epartment o f P hysics l...

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Department of Physics lUnrve'^sil '.1 ..{ ill. .'rck ttr"'cl) PHYSICS 707 SC Inverse Problems 1S/1A/2004 Lecture 7 of MCMC/Bayes topi,c BlcrnRra (svNrHerrc oala) A population of bacteria reproduce and diffuse through their nutrient medium, which covers the interval 0 I r ( tr. The population density u(r,t) evolves according to a Fisher modei, 0u /r. , ^0'u 6l:ru\^-u)+uU, for r K and D real valued parameters of the growth and diffusion process. The region outside [0,I] is lethal, so that the boundary conditions are z(0, t) : g, u(L,t) : g. The initial density of bacteria is known fairly precisely, The forward evolution is represented in Figure 1. FrcuRn 1. The time evolution of the population density (time runs into the page). The interval length is ,L : 10, diffusivity D :1.5, per-capita reproduction r 1 and the carrying capacity K :2. After a time T the growth and diffusion are stopped and the population density is measured at M points {r6,d 1,2. ..M, along the interval. The measured density values do not of course exactly coincide with the true density values at the measured points but have iid Gaussian noise with std s. The observation model is [email protected]): u(ra,T) * ei, 6r - ly'(0, s2). The data is plotted in Figure 2. t ol\L<r<0.8L u(I,Ul : ( ^ ' I u zotnerwlse.
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FtcuRn 2. The data (r-axis is r, spatial position, y axis is z' population density), is measured in the final time slice. The observation noise is s : 0.03, and the data is gathered at time T :2. How could this data {Ad,rt]1,i: L,2. ..M be used to estimate the pa,rameterc r,K and, D? What practical biological considerations might inform our choice of prior for
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inverse_lecture33 - t tr'cl D epartment o f P hysics l...

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