Spectrally-Matched Grids MOR approach for PDE discretization

1. Spectrally-Matched Grids MOR approach for PDE discretization

2. Spectrally Matched Grids (SMG) a.k.a. Finite-Difference Gaussian Quadrature Rules, or simply Optimal Grids

3. Outline Importance of DtN maps for remote sensing applications Philosophy: inversion of instability of inverse problems Finite-difference Spectrally Matched Grids (SMG) via matrix functions and MOR, spectral superconvergence of second order schemes Applications to forward and inverse problems for PDEs Finite-element SMG (if time permits)

4. Models in Oil & Gas Geophysics

5. PDE problems in oil exploration Large scale frequency or time domain PDE problems (Maxwell, elasticity, etc.) in unbounded domains Point (localized) sources and receivers, no need for global accuracy

6. DtN map and truncation of computational domains

7. DtN map and truncation of computational domains

8. DtN map and inverse EIT problem Forward problem: uniqueness and stability Inverse problem: uniqueness and instability

9. Second order finite-difference approximation We consider conventional second order FD (a.k.a. network or FV) approximations, e.g., the 5-point scheme in 2D, etc.

10. Inconsistency of forward and inverse problems INVERSE P.: exponential ill-posedness (error growth); the number of inverted parameters must be small FORWARD P.: slow convergence of the second order FD approximation; grids must be large INCONSISTENCY: forward=inverse(inverse); must be exponentially well-posed, i.e., exponentially convergent, but it is only second order

11. Spectrally matched grids (SMG) SMG: find a grid such that the discrete DtN matches the continuum DtN (by solving the discrete inverse problem) [Dr.&Knizhnerman, SINUM 1999] SMG grid makes forward and inverse problems consistent: exponential instability of the inverse p.= exponential superconvergence of the forward p. for standard second order schemes Applications to both forward and inverse problems

12. Model two-point operator problem

13. 2D Laplace, 5-point FD scheme 5-point FD (network) with global second order of convergence For simplicity we assume fine y-grid and neglect the error due to y-approximation We want to optimize the x-grid for computation of the Neumann-to-Dirichlet map (NtD) at x=0

14. The key observation is, that the continuum NtD maps are Stieltjes-Markov functions of A, and the FD NtD maps are rational functions of A, so grids can be computed by the methods of rational approximation theory minimizing the approximation error on A�s spectrum Can be viewed as MOR obtaining by approximation of transfer function

15. Construction of SMG, outline Represent the NtD as f(A) (transfer function) Approximate f(A) by a rational function in partial fraction form Find a three-term recursion for the partial fraction via Gaussian quadrature Convert the recursion to the finite-difference scheme The FD NtD approximation as good as the rational approximation on A�s spectrum, i.e., exponential convergence with rate weakly dependent on A!

16. NtD map as transfer function of operator

17. Resolvent form of the continuum transfer function

18. ROM via rat. approximation and quadratures

19. Finite-difference interpretation

20. Uniqueness and stability of the FD scheme Q. Is the FD scheme uniquely defined and stable?

21. Spectral Galerkin Equivalence

23. Resolvent representation of operator functions

24. NtD map via FD approximation

25. Convergence

26. SMG, applications Q. Why do we need to use SMG instead of just using rational approximants to compute f(A)b? A.1 Easy to apply within framework of conventional FD solvers for the approximation of the NtD map of a subdomain, e.g., for efficient truncation of computational domain (PML), domain decomposition, etc. A.2 For variable coefficient problems when the NtD can not be presented as f(A)b, but the SMGs still work A.3 Inverse problems

27. Example: plane wave propagation

28. Example: 1D wave propagationSMG + 1 node

29. Optimal grid,piece-wise constant coeff. 

30. 3D anisotropic problem, equidistant grid

31. 2D anis. problem, equidist. grid half plane

32. 2d anis.prob., quidist. grid one quarter

49. Variable coefficients So far I discussed constant or piece-wise constant coefficients What about general variable coefficients? Q. How sensitive are the SMGs to coefficient perturbations? To answer, we consider discrete inverse problems

50. Inverse Sturm-Liouville spectral problem

51. Finite-difference interpretation

52. Discrete inverse spectral problem

53. First try: let us solve the discrete inverse problem using true continuum data; then compute discrete conductivity using the equidistant grids

55. How to fix the discrete inversion

57. Asymptotic independence of SMGs on coefficientsExtensions to multidimensional problems Direct problems: very successful application to 3D anisotropic Maxwell�s system, 2-3 orders acceleration, wide applications in geophysics for oil explorations [S.Davydycheva et al, Geophysics, 2003]. Inverse problems (more recent): very good results for 2D EIT problems on planar graphs [F.Guevara Vasquez]; promising results for 2.5D problems (2D conductivity, 3D sources) [A.Mamonov]

58. SMG for FE? Optimal refinements with hp-elements: optimal convergence order but require larger stencils SMG for the Galerkin piece-wise linear FE: impossible with linear elements because the error of the NtD map is the square of the energy norm of the global error Can SMG be obtained via goal oriented adaptation? No, it works well for the hp-FE, but can not improve convergence order for Galerkin h-formulations Very promising recent results for Bubnov-Galerkin piece-wise linear FE with midpoint integration rules by Guddati et al., 2003. SMG with exponential convergence, limited to constant (or piece-wise constant) problems

60. Conclusions Rational approximant of the NtD map can be obtained via the second order FD scheme with SMG. Exponential superconvergence at prescribed surfaces or points The SMGs are designed via connection of the Gaussian quadratures and rational approximants. Applications: discretization of exterior problems in geophysics, domain decomposition, inverse problems. Significant effects New research: SMG for FE