Exponentially Convergent Sparse Discretizations and Application to Near Surface Geophysics

Transcription

1 Exponentially Convergent Sparse Discretizations and Application to Near Surface Geophysics Murthy N. Guddati North Carolina State University November 9, 017

2 Outline Part 1: Impedance Preserving Discretization Part : Absorbing Boundary Conditions (1-sided DtN map) Joint work with Tassoulas, Druksin, Lim, Zahid, Savadatti, Thirunavukkarasu Part 3: Complex-length FEM for finite domains (-sided DtN map) Joint work with Druskin, Vaziri Astaneh Part 4: Inversion for Near-surface Geophysics Joint work with Vaziri Astaneh

3 Part1. Impedance Preserving Discretization

4 Model Problem 3D wave equation in free space u u u 1 u 0 x y z c t Fourier transform in t, y, z, with u Ue ik yik zit y z U x 0 ku k k y kz is complex valued c Exact solution: ikx ikx U Ae Be, where k is the horizontal wavenumber i( kxk yk zt ) ikx y z. U e u e is a plane/evanescent wave 4

5 Finite Element Solution on a Uniform Grid in x FE discretization of: U ku x 0 Element contribution matrix with uniform element size of h: k elem A B kh h B A 6 3 A B 1 kh 1 h 3 1 kh 1 h 6 Assembly results in the difference equation: BU AU BU 0 j1 j j1 5

6 Changing Mesh Size: Reflections A simple analysis using two uniform meshes with different element sizes (h, H), but the same material What happens when a right propagating wave hits the interface? Exact solution just passes through Finite element solution reflections due to impedance mismatch R Z Z H H Z Z h h Z Z H h : discrete impedance of left domain : discrete impedance of right domain 6

7 Computing Discrete Impedance (Half-space Stiffness) Basic idea: discrete half-space + finite element = discrete half-space A B U ZU A Z B U 0 A Z B B A Z h U B A Z 0 h 0 h 0 h 1 0 h U1 0 A B 1 kh 1 h 3 1 kh 1 h 6 Zh A B ik 1 kh 1 Error term Z h. depends on element size, resulting impedance mismatch when the element size changes, resulting in reflections 7

8 Optimal Integration for Minimizing Reflection Error Minimize the error in impedance by using generalized integration rules A B h h 4 k h k h Minimize the error term by choosing kh Zh A B ik Error term The error in impedance is completely eliminated! No more reflections Formally valid for more general nd order equations (anisotropic, viscoelasticity etc., electromagnetics etc. G, 006, CMAME) Linear elements + midpoint integration = Impedance Preserving Discretization 8

9 Part. Absorbing Boundary Conditions Perfectly Matched Discrete Layers

10 Perfectly Matched Discrete Layers Impedance Preserving Discretization of PML Perfectly Matched Layers (PML) (Berenger, 1994; Chew et.al. 1995) Step 1: Bend the domain into complex space Reduced reflection into the interior Interior (real x) PML Region (imaginary or complex x) RPML ikl e Step : discretize PMDL domain (in complex space) Impedance is no longer preserved; perfect matching is destroyed Requires a large number of carefully chosen PML layers Impedance preserving discretization comes to the rescue! Impedance is preserved/matched, irrespective of element length, small, large, real, complex Perfectly Matching Discrete Layers (PMDL) Discretize with 3-5 complex-length linear finite elements No discretization error, but truncation causes reflections. The reflection coefficient is derived as R PMDL jnlayer 1 iklj / j1 1 iklj / 10

11 PMDL vs PML: Effectiveness of Midpoint Integration PMDL with 3 layers PML with 3 layers 11

12 PMDL: Some More Old Results Impedance preservation property is valid for any equation that is linear and second order in space (G, CMAME, 006) Elastic and other complicated wave equations (G, Lim & Zahid, 007) PMDL with 5 layers PML with 5 layers Evanescent waves can be treated effectively Padded PMDL contains large real lengths with midpoint integration (Zahid & G, CMAME, 006) 1

13 Salient Features of PMDL Exponential convergence R jnlayer k k j j1 k k j Near optimal discretization Optimal: need staggered grids (with Druskin et al., 003) Links PML to rational ABCs Lindman, Engquist-Majda, Higdon and variants (e.g. CRBC) We started this from E-M/Higdon ABCs (G, Tassoulas, 000) Extensions to corners is straightforward Additional advantage: Provides solutions to some difficult cases Backpropagating waves: anisotropy PML for discrete/periodic media 13

14 PMDL for Backpropagating Waves Opposing signs of phase and group velocities Backpropagating waves grow in the PML region PML cannot work! (Bécache, Fauqueux and Joly, 003) Reduced Reflection into the interior PML result: radiation in anisotropic media Interior PML Region A counter-intuitive idea: make the reflections in PML region decay faster than the growth of the incident wave Works only with PMDL: needs impedance preserving discretization! Result from PMDL after the fix Savadatti & G (01), J Comp. Phys. 14

15 Anisotropic elasticity Tilted Elliptic Case Arbitrary parameters Ideal Slowness Stable parameters Savadatti & G (01), J Comp. Phys. 15

16 Anisotropic elasticity Non-elliptic Case Traditional mesh Two different coordinated materials Savadatti & G (01), J Comp. Phys. 16

17 PMDL for Periodic Media (after discretization) Periodic media has internal reflections and transmissions Constructive interference leads to long-range propagation PML s complex stretching spoils this balance and internal reflections and transmissions get mixed up! Basic Ideas (Discrete/Periodic PMDL): Periodic media = Discrete vector wave equation (vector size = ndof in a cell) Discrete vector equation = impedance preserving discretization of more complicated wave equation Apply PMDL on the complicated wave equation results in impedance matching for periodic media Open problem: stability for complex problems PML for Lattice Waves: 7% reflections w/0 PML layers Discrete PMDL: less than 1% error w/ 4 PMDL layers G & Thirunavukkarasu, JCP (009), Waves

18 Part 3. Two-Sided DtN Map Complex-length Finite Element Method

19 Facilitating the Approximation of -Sided DtN Map u Consider the equation: z A B Exact -sided DtN map: Kexact = B A u 0, z (0, L) By definition, exact DtN Map is impedance preserving: A B Z exact Consider impedance preserving discretization of the interval: A B Kexact =, A B Z B A exact Error in A and B would be similar since: A B Z A B exact Approximating two-sided map reduces to approximating one-sided map Better derivation based on Crank-Nicolson discretization of the propagator 19

20 exp(+il) exp(-il) 1D Helmholtz Equation z 0 u z u 0 z 1 st Order Form v u / z L u 0 i iu z v / i i 0 v / i Eigenvalues i z Downward waves: 0

21 Propagator Approximation approximate P n L 1 u 0 L... u 1 1 u1 u0 1 0 i u1 u0 L v v i 0 v v st Order Form v u / z, v v / i Crank- Nicolson L n ul u 0 i u z v i 0 v Crank- Nicolson Valid for both Downward AND upward waves! 1

22 Padé Approximant Imaginary L 1 L... j exp( ) Padé j d L d P d j j d 0 0 L n ( j 0,, n) Real Complex-Length Finite Element Method n j0 ( n j)! ( ) j x 0 Lj L / x j! ( n j)! j

23 Complex-Length FEM: Exponential Convergence f 1 L 1 Laplace Equation Helmholtz Equation (can t beat Nyquist limit)

24 Complex-Length FEM: Some Observations Exponentially convergent Piecewise linear interpolation sparse computation Edges do not move ( L j =L) can be combined with other types of meshes for other subdomains Mesh is not bent outside (Re(L j )>0) Order of elements do not matter! more on this later. With refinement, and proper ordering, mesh converges to a smooth curve on the complex plane n j0 j exp( ) Padé j d L d P d j j d 0 0 ( n j)! ( ) j x 0 Lj L / x j! ( n j)! j

25 Energy Conservation and Eigenvalue Problems Do complex lengths lead to energy absorption, like PML? No, due to conjugate pair of lengths decay grows back! -sided DtN Map is Hermitian Do complex lengths lead to complex eigenvalues of K with respect to M? No. Eigenvalues are real and positive! Eigenvectors are complex (K and M are complex symmetric) u z u 0 ( ) K M 0

26 Element Ordering u c exp( i x) 6

27 Part 4. Near Surface Geophysical Site Characterization using Guided Wave Inversion

28 Guided Wave Dispersion Low Frequency High Frequency Dispersion Curve 8

29 Spectral Analysis FFT Phase Velocity (m/s) = Frequency (1/s) Wavenumber (1/m) Experimental Dispersion Curve 9

30 Medium Characterization Inverse Identification Experimental Dispersion Curve Iteratively Minimize N experimantal predicted i i E c c i1 Initial Guess Optimization Scheme Gradient Based, e.g. Newton-like Methods Global Search, e.g. Genetic Algorithm Forward Problem: Predicted Dispersion Curve 30

31 Forward Modeling State of the Art 1 u u T u u 1 Drrr Drzr Dr u Drz Dzz Dz u r r r z z r z r 1 T u T u 1 Dr Dz Du Iu 0 r r z r Discretize z direction Hankel Transform r direction Quadratic Eigenvalue Problem det K 0 k c p k 31

32 Reducing the Problem Size: CFEM+PMDL Complex-Length FEM (Finite Layers) Perfectly Matched discrete Layers (Halfspace) 3

33 Forward Modeling: CFEM vs. FEM Error in dispersion curve 33

34 Inversion: Experimental Dispersion Curve Experimental Dispersion Curve Inverse Identification 40 Geophones 36 Geophones 1 Geophones 5 th Mode Experimental Dispersion Curve 4 th Mode 3 rd Mode nd Mode 1 st (fundamental) Mode 34

35 Challenge No analytical derivative Rough misfit function Minimize N experimantal predicted i i E c c i1 Existing approach: Finite Difference Method (FDM) Expensive: Multiple computations of dispersion curve Slow convergence: Oscillatory gradient 35

36 Proposed Derivative for Experimental Curve Analytical Derivative k L( K m j ) i m ( K k ) j L i R R Alternative Approach for Eigenvector R K 1 P k k k i k experimental 6 ( 10 ) 36

37 Proposed Derivative j 1 1 ( P K )( K / m j )( K P) 1 1 ( P K )( K / ki )( K P) eff. ki m Analytical vs. FDM Gradient 37

38 Inversion Results: Synthetic Examples Analytical Gradient FDM Gradient 38

39 Inversion Results: 14-Layer Soil Profile Analytical Gradient FDM Gradient Iterations (Existing) Iterations (Proposed) CPU Time (Proposed) CPU Time (Existing) s s Experimental data from: J Xia et al., J. Environ. Eng. Geophys., 5.3, 1-13 (000) 39

40 Conclusions Discretization that perfectly preserves the impedance is possible Linear FEM with midpoint integration preserves impedance Related to Crank-Nicolson discretization of the propagator Preserves the evanescence in PML region Absorbing Boundary Conds.: Perfectly Matched Discrete Layers (PMDL) Exponential convergence Link to other ABCs we can get the best of both worlds Facilitates stable ABCs for some backward propagating waves Formally extensible to discrete periodic media Open questions: Parameters of discretization for stability and accuracy 40

41 Conclusions (Contd.) Two-sided DtN Map Exponential convergence on the edges is possible with linear interpolation: Complex-length Finite Element Method (CFEM) Impedance preserving discretization is the key! Currently based on Padé approximant; could be further optimized Open questions: further theoretical understanding; extensions to variable coefficients and higher dimensions? Guided Wave Inversion Forward modeling: a good application of CFEM Approximate differentiation of the effective dispersion curve facilitates faster convergence and efficient gradient computation Future work: Bayesian and hybrid inversion 41

42 Thank you! Impedance Preserving Discretization G (006), Arbitrarily wide angle wave equations for complex media, CMAME Perfectly Matched Discrete Layers G, Tassoulas (000), Continued fraction absorbing boundary conditions for the wave equation, J Comp. Acoustics. Asvadurov, Druskin, G, Knizhnerman (003), On optimal finite-difference approximation of PML, SINUM. G, Lim (006), Continued fraction absorbing boundary conditions for convex polygonal domains, IJNME. Thirunavukkarasu, G (011), Absorbing boundary conditions for time-harmonic wave propagation in discretized domains, CMAME. Savadatti, G (01), Accurate absorbing boundary conditions for anisotropic elastic media. parts 1/, JCP. Complex-length FEM G, Druskin, Vaziri Astaneh (016), Exponential Convergence through Linear Finite Element Discretization of Stratified Subdomains, JCP. Vaziri Astaneh, G (016), Efficient Computation of Dispersion Curves for Multilayered Waveguides and Half-Spaces, CMAME. Guided Wave Inversion Vaziri Astaneh, G (016), Improved Algorithms for Inversion of Surface Waves Using Multistation Analysis, Geoph. J. Intl. This work is supported by National Science Foundation DMS , CMMI