Resources for Blockkurs on Markov Chain Monte Carlo for Inverse Problems in PDEs

Instructors: Colin Fox

  • Some Reading
  • Compute 1 (for m-files, right click to 'Save As' to get formatting correct, and add .m extension (my silly Wiki does not allow .m extensions))
    • mcmc.m.m is a basic Metropolis-Hastings algorithm, with acceptance prob. for exponential distribution
    • mcgaus.m MH for sampling a Gaussian in 1-dim
    • mcgaus_demo.m Traces of Gaussian sampler for different window sizes
  • Compute 2 (for m-files, right click to 'Save As' to get formatting correct)
    • Ulli Wollf's UWerr MatLab code and documentation
    • Code to plot IACT as function of window using mcgauss.m
    • Code to sample inverse heat problem
    • heatfem.m that builds FEM mass and stiffness matrices for 1-dim heat problem
    • Write a sampler for the inverse heat-conductivity problem in (constant) D (or use my awful code, above)
    • Tune the window size for your RWM sampler, or better still (challenge question) plot a graph of IACT as function of window w to find the optimal w
    • Consider error in the final time T, T~Unif[1.8,2.2]. Perform joint inference for D and T, and say whether inference for D is significantly altered.
    • Python code: Andres' python code zip file plus IP text
  • Compute 3
    • heatIPsvals.m plot the singular values for the inverse-source problem in the 1-dim heat equation
    • DJac.m function to return Jacobian matrix for inverse coefficient problem, built using secant approximation
    • robfem.m that builds FEM matrix for 1-dim operator -cu'' + alpha u
    • Task: Evaluate the Jacobian (linearized forward map) for D(x)->u(x,T), using the secant method in DJac, and FEM program in heatfem (or your own). What is the effective rank of this map? What happens to the rank as the discretization of D (and u) is refined?
  • Some good books on FEM are Hackbusch and Ern and Guermond
  • Lecture 4 slides
  • Compute 4
  • HGS MathComp Curriculum page