SpectrallyMatched Grids MOR approach for PDE discretization

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Spectrally-Matched Grids MOR approach for PDE
discretization
Vladimir Druskin, Schlumberger-Doll
Research Contributors Sergey Asvadurov (former
SDR), Liliana Borcea (Rice), Murthy Guddati
(NCSU), Fernando Guevara Vasquez(Stamford), David
Ingerman (former MIT), Leonid Knizhnerman, Shari
Moskow (Drexel)
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Spectrally Matched Grids (SMG) a.k.a.
Finite-Difference Gaussian Quadrature Rules, or
simply Optimal Grids

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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)

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Models in Oil Gas Geophysics
Single-well EM logging
Cross-well EM Acoustic
Surface EM Seismic
Surfaces sources receivers
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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

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DtN map and truncation of computational domains
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DtN map and truncation of computational domains
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DtN map and inverse EIT problem

• Forward problem uniqueness and stability
• Inverse problem uniqueness and instability

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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.

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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 forwardinverse(inverse) must be
  exponentially well-posed, i.e., exponentially
  convergent, but it is only second order

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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

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Model two-point operator problem
l
0
Laplace equation, Agt0
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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 x0

l
0
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• 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 As spectrum
• Can be viewed as MOR obtaining by approximation
  of transfer function

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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 As spectrum, i.e., exponential
  convergence with rate weakly dependent on A!

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NtD map as transfer function of operator
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Resolvent form of the continuum transfer function
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ROM via rat. approximation and quadratures
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Finite-difference interpretation
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Uniqueness and stability of the FD scheme

• Q. Is the FD scheme uniquely defined and stable?

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Spectral Galerkin Equivalence
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SMG for bounded domains, examples K10
primary
dual
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Resolvent representation of operator functions
Sp(A)
sp( )
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NtD map via FD approximation
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Convergence
Spectral convergence with speed similar to the
optimal hp-FE but with the cost of the second
order FD!
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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

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Example plane wave propagation
Sound soft
Space optimal
x
Time equidistant
t
Sound hard
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Example 1D wave propagationSMG 1 node
1 node
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Optimal grid,piece-wise constant coeff.
1. Approximation in external infinite domains and
optimal discrete PML Tensor produc of 1D grids.
Exponential convergence of the external DtN maps
Asvadurov et al., SINUM 2002.
2. Domain decomposition for skewed checkerboard
models. Tensor product of nonorthogonal 1d
optimal grids, optimal hp-element convergence
rate at the corners. Asvadurov et al., JCP
2000, 2002 Dr.Knizhnerman, Num.Alg., 2002
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3D anisotropic problem, equidistant grid
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2D anis. problem, equidist. grid half plane
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2d anis.prob., quidist. grid one quarter
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Acoustic logging problem
Computational domain -2,2X-2,2 m
Isotropic fluid, speed 1.
For EQ/OP computations optimal grid starts at
x0.3
Group of receivers at (x,y) (0, 0.1)(0,1.9)
Anisotropic solid, matrix A 2, 1,1,2
Pressure source at (x,y) (0,0)
Fluid borehole -0.25,0.25X-2,2
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Time t0.1 sec
Fully EQ grid
EQ/OP grid
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Time t0.2 sec
Fully EQ grid
EQ/OP grid
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Time t0.3 sec
Fully EQ grid
EQ/OP grid
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Time t0.4 sec
Fully EQ grid
EQ/OP grid
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Time t0.5 sec
Fully EQ grid
EQ/OP grid
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Time t0.6 sec
Fully EQ grid
EQ/OP grid
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Time t0.7 sec
Fully EQ grid
EQ/OP grid
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Time t0.8 sec
Fully EQ grid
EQ/OP grid
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Time t0.9 sec
Fully EQ grid
EQ/OP grid
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Time t1.0 sec
Fully EQ grid
EQ/OP grid
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Time t1.1 sec
Fully EQ grid
EQ/OP grid
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Time t1.2 sec
Fully EQ grid
EQ/OP grid
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Time t1.3 sec
Fully EQ grid
EQ/OP grid
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Time t1.4 sec
Fully EQ grid
EQ/OP grid
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Time t1.5 sec
Fully EQ grid
EQ/OP grid
accurate
large dispersion
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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

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Inverse Sturm-Liouville spectral problem
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Finite-difference interpretation
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Discrete inverse spectral problem
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• First try let us solve the discrete inverse
  problem using true continuum data then compute
  discrete conductivity using the equidistant grids

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How to fix the discrete inversion
Borcea et al., CPAM, 2005
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Asymptotic independence of SMGs on
coefficientsExtensions to multidimensional
problems

• Direct problems very successful application to
  3D anisotropic Maxwells 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

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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

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Literature (incomplete)
1. Gaussian spectral rules for the three-point
second differences I. A two-point positive
definite problem in a semiinfinite domain, Dr.,
L.Knizhnerman, SINUM, 37, No 2, (2000),
403-426. 2.Optimal finite-difference grids and
rational approximations of square root. I.
Elliptic problems,. D.Ingerman Dr. and L.
Knizhnerman, Comm. Pure. Appl. Math., LIII,
(2000), 1-27 3.Gaussian spectral rules for second
order finite-difference schemes, with
L.Knizhnerman, Numerical Algorithms, 25
pp.139-159, 2000. 4.Application of the difference
Gaussian rules to solution of hyperbolic
problems, S.Asvadurov, Dr., and L.Knizhnerman,
J. Comp. Phys., 158, 116-135 (2000) 5.Application
of the difference Gaussian rules to solution of
hyperbolic problems II, Global Expansion,
S.Asvadurov, Dr.and L.Knizhnerman, J. Comp.
Phys., 175, 24-49 (2002) 6.Three-point
finite-difference schemes, Pade and the spectral
Galerkin method. 1. One-sided impedance
approximation , Dr. and S.Moskow, Math.Comp., 71,
N239, 995-1019, (2002) 7.Optimal
finite-difference grids for direct and inverse
Sturm-Liouville problems, L.Borcea and
Dr.,Inverse Problems, 18, (2002), 979-1001  8. On
optimal finite-difference approximation of PML,
S.Asvadurov, Dr., M.Guddati and L.Knizhnerman
SINUM, 41, N1, (2003), 287-305 9. An efficient
finite-difference scheme for electromagnetic
logging in 3D anisotropic inhomoheneous media,
S.Davydycheva, Dr. and T.Habashy, Geophysics,
2003, 68, No.5, pp.1525-1537 10. On the
continuum limit of a discrete inverse spectral
problem on optimal finite difference grids,
L.Borcea Dr. and L.Knizhnerman, Comm.Pure Appl.
Math., 58, N9, 1231-1279, 2005. 11. On the
sensitivity of Lanczos recursion to the spectrum,
Dr., L.Borcea, and L.Knizhnerman, Lin. Alg.Appl.,
396, 2005, 103-125
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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