L-curve-based regularization parameter selection
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1 L-curve-based regularization parameter selection regularization misfit O. Ghattas (UT-Austin) Bayesian inversion for Antarctic sheet ICES Babuška Forum 36 / 67
2 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
3 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
4 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
5 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
6 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
7 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
8 Spectrum of the prior-preconditioned data misfit Hessian ,545 parameters 1,190,403 parameters eigenvalue 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
9 Eigenvector 1 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 1 sheet ICES Babuška Forum 60 / 67
10 Eigenvector 2 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 2 sheet ICES Babuška Forum 60 / 67
11 Eigenvector 3 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 3 sheet ICES Babuška Forum 60 / 67
12 Eigenvector 4 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 4 sheet ICES Babuška Forum 60 / 67
13 Eigenvector 5 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 5 sheet ICES Babuška Forum 60 / 67
14 Eigenvector 6 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 6 sheet ICES Babuška Forum 60 / 67
15 Eigenvector 7 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 7 sheet ICES Babuška Forum 60 / 67
16 Eigenvector 8 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 8 sheet ICES Babuška Forum 60 / 67
17 Eigenvector 9 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesianeigenvector inversion for Antarctic 9 sheet ICES Babuška Forum 60 / 67
18 Eigenvector 10 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 10 sheet ICES Babuška Forum 60 / 67
19 Eigenvector 100 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 100sheet ICES Babuška Forum 60 / 67
20 Eigenvector 200 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 200sheet ICES Babuška Forum 60 / 67
21 Eigenvector 500 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 500sheet ICES Babuška Forum 60 / 67
22 E-vector 1000 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 1000sheet ICES Babuška Forum 60 / 67
23 E-vector 2000 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 2000sheet ICES Babuška Forum 60 / 67
24 E-vector 4000 of prior-preconditioned data misfit Hessian O. Ghattas (UT-Austin) Bayesian eigenvector inversion for Antarctic 4000sheet ICES Babuška Forum 60 / 67
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