L-curve-based regularization parameter selection regularization 10 4 10 3 10 2 10 3 10 2 10 1 5 10 1 10 0 5 10 0 10 1 10 1 10 1.4 10 1.5 10 1.6 10 1.7 10 1.8 10 1.9 misfit O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 36 / 67
Antarctic ice sheet inversion for basal sliding field: InSAR data Left: InSAR-based Antarctica ice surface velocity observations Right: Inferred basal sliding field O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babus ka Forum 37 / 67
Antarctic ice sheet inversion for basal sliding field: InSAR data InSAR-based Antarctica ice surface velocity observations O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 38 / 67
Antarctic ice sheet inversion for basal sliding field: InSAR data Reconstructed ice surface velocity field O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 38 / 67
Analysis of data misfit Hessian for a (nonlinear) elliptic coefficient inverse problem (P. Flath, 2013) Invert for log-coefficient field γ(x) in Poisson equation from observations of state u(x) within domain: 1 L min (b u(γ) d obs ) 2 dx γ M 2 where the map from γ to u is defined by solution of: d ( e γ du ) = 0 for x (0, L) dx dx Two cases of observation operator: 0 u(0) = u 0 u(l) = u L full observations: b(x) = 1 pointwise observations: b(x) = L n n j=1 δ(x x j) O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 52 / 67
Analysis of Hessian for elliptic coefficient inverse problem assume no noise, γ = γ 0 is true log-medium Fourier analysis of spectrum of data misfit Hessian, evaluated at γ 0, gives eigenvalues for both cases: full observations: λ i = (u L u 0) 2 π 2 i 2 n pointwise observations: λ i = (u L u 0) 2 4n 2 sin(iπ/2n) 2 Spectrum of data misfit Hessian for both full and pointwise observations Left: 6 observations Right: 500 observations O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 53 / 67
Analysis of Hessian for elliptic coefficient inverse problem Eigenfunctions of data misfit Hessian are cos(iπx/l) in full observations case, and piecewise-constant interpolants of the cosines in pointwise case Eigenfunctions 1, 2, 5, and 6 for full and pointwise (n = 6) observations O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 54 / 67
Spectrum of the prior-preconditioned data misfit Hessian 10 8 10 6 409,545 parameters 1,190,403 parameters eigenvalue 10 4 10 2 10 0 10 2 0 1000 2000 3000 4000 number Spectrum of Γ 1/2 pr F T ΓnoiseF 1 Γ 1/2 pr for Antarctica inverse problem with 410K and 1.19M basal sliding parameters (observed to decay like i 3 ) 4000 dominant modes, independent of parameter and data dimension intrinsic problem dimension depends on information content of data O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 59 / 67
Eigenvector 1 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 1 sheet ICES Babuška Forum 60 / 67
Eigenvector 2 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 2 sheet ICES Babuška Forum 60 / 67
Eigenvector 3 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 3 sheet ICES Babuška Forum 60 / 67
Eigenvector 4 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 4 sheet ICES Babuška Forum 60 / 67
Eigenvector 5 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 5 sheet ICES Babuška Forum 60 / 67
Eigenvector 6 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 6 sheet ICES Babuška Forum 60 / 67
Eigenvector 7 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 7 sheet ICES Babuška Forum 60 / 67
Eigenvector 8 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 8 sheet ICES Babuška Forum 60 / 67
Eigenvector 9 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 9 sheet ICES Babuška Forum 60 / 67
Eigenvector 10 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 10 sheet ICES Babuška Forum 60 / 67
Eigenvector 100 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 100sheet ICES Babuška Forum 60 / 67
Eigenvector 200 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 200sheet ICES Babuška Forum 60 / 67
Eigenvector 500 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 500sheet ICES Babuška Forum 60 / 67
E-vector 1000 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 1000sheet ICES Babuška Forum 60 / 67
E-vector 2000 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 2000sheet ICES Babuška Forum 60 / 67
E-vector 4000 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 4000sheet ICES Babuška Forum 60 / 67