Operator Upscaling and Adjoint State Method

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1 Operator Upscaling and Adjoint State Method Tetyana Vdovina, William Symes The Rice Inversion Project Rice University February 0, 009

2 Motivation Ultimate Goal: Use 3D elastic upscaling algorithm as a core in the inverse solver Principal Tool: Adjoint state method: gradient = forward + adjoint problems Major Question: How do we upscale the adjoint problem? Is the upscaling algorithm the same as for the forward problem? If it is, can we reuse as much of the existing code as possible? If it is not, what exactly changes and how can we reuse as much of the existing code as possible? T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method /9

3 Code Reusability Reasons to reuse as much of the code as possible: numerical solution of the elastic wave equation is complicated upscaling of the elastic wave equation is complicated adjoint state method is complicated Solution to the code reusability problem: TSOpt Figure: courtesy of Marco Enriquez Need to provide: forward and adjoint update functions simulator/tsopt and TSOpt/optimizer interfaces T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 3/9

4 Decisions 3D elastic upscaling vs. D acoustic upscaling Start with acoustics, since elastic upscaling is more complicated Optimize then discretize get adjoint for the continuous problem, discretize less complicated additional discretization error vs. Discretize then optimize get adjoint for the discrete problem directly more complicated no additional errors Use discretize then optimize approach T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 4/9

5 Forward Acoustic Upscaling Algorithm Pressure-acceleration formulation: 1 p c t + (u x + u y ) = w(t)δ(x, y), u x = p x, u y = p y boundary and initial conditions Upscale acceleration only: δu x, δu y are the subgrid components u c x, u c y are the coarse components After discretization mixed finite element method results in to a two-scale difference scheme T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 5/9

6 Forward Acoustic Upscaling Algorithm (cont.) Subgrid stencil Coarse-grid stencil i=0 i+1/ j+1/ m, l+1/ p n+1 = p n i+ 1,j+ 1 i+ 1,j+ p n 1 +c 1 i+ 1,j+ 1 i+ 1,j+ t D 1 x (δu x +ux c ) n i+ 1,j+ 1 + vertical component + source term p n+1 p n+1 (δu x ) n+1 i+ = 1 i,j+ 1,j+ 1 i 1,j+ 1, (ux c ) n+1 = m,l+ 1 j p n+1 1 p n+1,j+ 1 1,j+ 1, eqns for vertical component of subgrid and coarse accelerations T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 6/9

7 Adjoint Acoustic Upscaling Algorithm (cont.) Subgrid stencil Coarse-grid stencil i=0 i+1/ j+1/ m, l+1/ p n 1 = p n i+ 1,j+ 1 i+ 1,j+ p n+1 + t D 1 i+ 1,j+ 1 x (δu x + ux c ) n i+ 1,j+ 1 + vertical component + source term (δu x ) n 1 = c i+ 1,j+ p n 1 1 i,j+ 1 c i+ 1,j+ 1 i 1,j+ 1 p n 1 i 1,j+ 1 (ux c ) n 1 = c 1,j+ p n c m,l+ 1,j+ 1 1,j+ p n 1 1 1,j+ 1 j, eqns for vertical component of subgrid and coarse accelerations, T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 7/9

8 Implications For acoustic upscaling: Need to modify update functions for adjoint problem Modifications are minor For elastic upscaling: Need to modify update functions for adjoint problem Modifications are likely to be more serious than for acoustics ( ρ v ) 1 t, w ( = (λ + µ) u 1 x + λ u y + λ u 3 z, w ) x + four other inner products + source term Discretize: (λ + µ) u 1 x (λ + µ) (u 1 ) i+1,j,k (u 1 ) i,j,k i,j,k Take adjoint: (λ + µ) i+1,j,k (u 1 ) i+1,j,k (λ + µ) i,j,k (u 1 ) i,j,k Reconsider optimize then discretize approach? T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 8/9

9 Current and Future Work What have we done? Derived and implemented linearized and adjoint problems (parallel) for acoustic upscaling problem Built an interface between our simulator and TSOpt Verified that dot product test works up to machine precision! What are we doing? Building optimizer interface What is next? Do the same for 3D elastic upscaling T. Vdovina, TRIP, Rice University Upscaling & Adjoint State Method 9/9