A model reduction approach to numerical inversion for parabolic partial differential equations

Transcription

1 A model reduction approach to numerical inversion for parabolic partial differential equations Liliana Borcea Alexander V. Mamonov 2, Vladimir Druskin 3, Mikhail Zaslavsky 3 University of Michigan, Ann Arbor 2 University of Houston 3 Schlumberger Doll Research, Boston Liliana Borcea (University of Michigan) Model Reduction for Inversion / 29

2 Problem formulation Controlled Source Electromagnetics Inverse problem: determine underground resistivity r(x) from measurements of the magnetic field H(t, x) at the surface. Model: diffusion Maxwell system (negligible displacement current) [ ] r(x) H(x, t) = t H(x, t) Setup: x = (x, z), r := r(x) for x Ω R n, simply connected with boundary B = B A B I. Measurements are on accessible B A. Polarization H(x, t) = u(x, t)e z and the boundary conditions n u(, t) = 0 and BA u(, t) = 0. BI Excitation is in initial conditions and measurements are u(, t) Liliana Borcea (University of Michigan) Model Reduction for Inversion 2 / 29 BA

3 Problem formulation Generic parabolic problem t u(x, t) = (r(x) u(x, t)), x Ω, t > 0 Boundary conditions u(, t) BI = 0, n u(, t) = 0. BA Initial conditions u(x, 0) = u o (x) supported on B A. The sources/receivers are on B A and give knowledge of the measurement operator M(u o ) = u(x, t) for all u o supported in B A and t > 0. BA Inverse problem is to find r(x) given M. Liliana Borcea (University of Michigan) Model Reduction for Inversion 3 / 29

4 Problem formulation Model reduction The measurement operator M is nonlinear in the unknown r(x), but it depends linearly on the initial conditions u o. Suffices to consider integral kernel of mapping from u o to M(u o ). In -D we have Ω = [0, ], with B A = {0} and B I = {} and u(0, t) = u o M(δ(x)) with scalar response y(t) := M(δ). In 2-D B A is a line segment and y ij (t) := M(δ(x j )), x i, x j B A. xi Laplace transform of y(t) is the transfer function of the system. Reduced model rational interpolation of transfer function. Liliana Borcea (University of Michigan) Model Reduction for Inversion 4 / 29

5 Problem formulation Model reduction motivation Benefit for forward problem: Reduced model is small (fast computations) and yet approximates well the transfer function. We may relate the model to a finite difference spatial discretization of the PDE on a special grid, with few points. Benefit for inversion: The nonlinear map R from the function space of the unknown resistivity r(x) to the low-dimensional space of the discrete resistivities may be used to precondition the inversion. We can adapt the size of the reduced model to the noise level (regularization) so we get a natural stability-resolution trade-off. Liliana Borcea (University of Michigan) Model Reduction for Inversion 5 / 29

6 Model order reduction and inversion -D case. Semi-discrete case (for simplicity). Discretize Ω = [0, ] on a fine grid with spacing h = N+ t u(t) = A(r)u(t), u(0) = h e Differential operator (r ) is replaced by the matrix A(r) = D T diag(r)d, r R N + Inverse problem: given y(t; r) = e T u(t) find r RN +. Forward model reduction: we know r and seek a reduced model that approximates y. Inverse model reduction: obtain a reduced model from y, then find r which has this reduced model. Liliana Borcea (University of Michigan) Model Reduction for Inversion 6 / 29

7 Model reduction Model order reduction and inversion Transfer function of full model A(r) R N N, b R N G(s; r) = 0 dt y(t; r)e st = b T (si A(r)) b, s > 0, b = e h Reduced model (A m, b m ) with A m R m m, b m R m and m N m G m (s) = b T m(si m A m ) c j b m = s + θ j with θ j = eigenvalues of A m, and c j = (b T mz j ) 2, where z j = normalized eigenvectors. j= Rational interpolation s k G m (σ j ) = s k G(σ j ) at nodes σ j [0, + ), l for j =,..., l, k = 0,..., 2M j and m = M j. j= Liliana Borcea (University of Michigan) Model Reduction for Inversion 7 / 29

8 Model order reduction and inversion Projection-based model reduction Grimme 997 showed that interpolation is equivalent to projection-based model reduction: A m = V T AV R m m, b m = V T b R m, V R N m has orthonormal columns spanning rational Krylov space } K m (σ) = span {(σ j I A) k b j =,..., l; k =,..., M j The choice of the points σ j matter as we shall see! Liliana Borcea (University of Michigan) Model Reduction for Inversion 8 / 29

9 Model order reduction and inversion Connection to finite differences The matrix A m is dense in general, so how do we connect the reduced model to a finite difference scheme? We have the (Stieltjes) continued fraction representation G m (s) = κ s + κ κ m s + κ m G m (s) = W (s) the response of second-order difference scheme ( Wj+ W j W ) j W j sw j = 0, j =,..., m κ j κ j κ j where r(x) is encoded in {κ j, κ j }. Liliana Borcea (University of Michigan) Model Reduction for Inversion 9 / 29

10 Model order reduction and inversion Optimization formulation Data is d(t) = y(t; r true ) + N (t) where N (t) = noise and discretization error. Reconstruction is r = arg min r R N + 2 Q(d(t)) Q(y(t; r)) 2 2 Instead of minimizing misfit between d(t) and y(t; r) = [F(r)] (t), where F is the forward map, we work with R(r) = Q F(r) where Q is used as a preconditioner. We seek r that gives the reduced model calculated for the measurements d(t). Liliana Borcea (University of Michigan) Model Reduction for Inversion 0 / 29

11 Model order reduction and inversion The map R(r) = Q(y( ; r)) = Q F(r). R : r (a) A(r) (b) V (c) (d) A m {(c j, θ j )} m (e) j= {(log κ j, log κ j )} m j= (a) is definition A(r) = D T diag(r)d. (b) is calculation of orthonormal basis of rational Krylov subspace projection matrix V. (c) is definition A m = V T A(r)V and b m = V T b. (d) here θ m are the eigenvalues of A m and c j = (b T mz j ) 2, where z j are the eigenvectors. (e) Conversion of rational interpolant to continued fraction (Lanczos) and then logarithm of the coefficients. Here all the steps are stable! Liliana Borcea (University of Michigan) Model Reduction for Inversion / 29

12 Model order reduction and inversion The chain of mappings in Q(d(t) Q : d(t) (a) G(s) (b) G m (s) (c) {(c j, θ j )} m (d) j= {(log κ j, log κ j )} m j=. (a) is Laplace transform of measured data. (b) is rational (Padé) interpolation. This is the only ill-conditioned step and we regularize by reducing m until we get positive continued fraction coefficients. (c) Calculate the poles θ j and residues c j of the rational G m (s). (d) Calculate the continued fraction coefficients (Lanczos iteration). Liliana Borcea (University of Michigan) Model Reduction for Inversion 2 / 29

13 Model order reduction and inversion The map R(r) = Q F(r) and discrete resistivities Change coordinates x ξ so that r /2 x = ξ and w(ξ, s) = Laplace transform of u(x, t) satisfies Reduced model corresponds to ( Wj+ W j κ j κ j r /2 ξ (r /2 ξ w) sw = 0 W j W j κ j ) sw j = 0...., m The discrete resistivity is defined by κ 2 j and κ 2 j up to scaling by mesh steps, which we do not need to know. In the optimization we work with the log of κ j and κ j, so grid factors cancel out. R takes r to the (log of ) discrete resistivities. If we interpolated these grid values in Ω, we would expect that R(r) would be close to the identity for smooth r i.e., Q is like inverse of F. Liliana Borcea (University of Michigan) Model Reduction for Inversion 3 / 29

14 Model order reduction and inversion Optimal grids calculated from κ o j and κ o j for r o m = m = Compare three choices of matching conditions: (, ) Moment matching at zero, K m (0) = span { A b, A 2 b,..., A m b } (, ) Interpolation K m (σ) = span { (σ j I A) b j =,..., m } at geometrically spaced nodes σ j = σ ( + 2/m) j (, ) Interpolation at faster growing nodes σ Liliana Borcea (University of Michigan) Model Reduction for Inversion 4 / 29

15 Model order reduction and inversion Conditioning of the Jacobian DR cond(dr) ( ) N ( ) N κ κ Rows and r k k= r k k= are almost collinear when grid points get too close poor conditioning m Condition number growth for moment matching at zero ( ), interpolation at σ ( ), interpolation at σ ( ) Liliana Borcea (University of Michigan) Model Reduction for Inversion 5 / 29

16 Model order reduction and inversion Sensitivity functions κ k r (blue) and κ k r (red) k =,... m = 6 from left to right, top to bottom Liliana Borcea (University of Michigan) Model Reduction for Inversion 6 / 29

17 Reconstruction Regularization and numerical results We solve min r Q(d( )) R(r) 2 with Gauss-Newton iteration. Jacobian DR R 2m N has a large null space 2m N. At each iteration the Gauss-Newton update is in the 2m dimensional range of the pseudoinverse DR. We can improve results with prior info, by adding correction δr that minimizes regularization term P(r + δr) over δr null(dr). E.g. weighted discrete H seminorm P(r) = 2 W/2 Dr 2 2. Liliana Borcea (University of Michigan) Model Reduction for Inversion 7 / 29

18 Regularization and numerical results Numerical experiments: D setup Synthetic noisy data d j = y(t j ; r true ) + N (t j ), j =,..., N T Noise model N = ɛ diag(χ,..., χ NT )y, with independent χ k Gaussian mean zero, variance. Reduced model size m chosen based on noise level ɛ m 3 4 ɛ Initial guess r, five Gauss-Newton iterations n GN = 5. Liliana Borcea (University of Michigan) Model Reduction for Inversion 8 / 29

19 Regularization and numerical results Numerical results: smooth resistivity m = 3, ɛ = m = 4, ɛ = e e True resistivity r true (quadratic) Reconstruction r (2) after one Gauss-Newton iteration Reconstruction r (6) after five Gauss-Newton iterations Liliana Borcea (University of Michigan) Model Reduction for Inversion 9 / 29

20 Regularization and numerical results Numerical results: smooth resistivity m = 3, ɛ = m = 4, ɛ = e e True resistivity r true (linear + Gaussian) Reconstruction r (2) after one Gauss-Newton iteration Reconstruction r (6) after five Gauss-Newton iterations Liliana Borcea (University of Michigan) Model Reduction for Inversion 20 / 29

21 Regularization and numerical results Numerical results: piecewise constant resistivity m = 4, ɛ = m = 5, ɛ = e e True resistivity r true (jump of contrast 2) Reconstruction r (2) after one Gauss-Newton iteration Reconstruction r (6) after five Gauss-Newton iterations Liliana Borcea (University of Michigan) Model Reduction for Inversion 2 / 29

22 2-D case Philosophy -D problem is formally determined: D unknown r(x) and D data y(t). 2-D problem is overdetermined: 2D unknown r(x) but 3D data y ij (t), where (i, j) are source-detector pairs. We proceed as in -D and construct separately reduced models (i.e., maps R ij ) for data subsets indexed by i, j =,..., N d. The maps R ij are coupled by their dependence on the same r(x) and are all used in inversion. Liliana Borcea (University of Michigan) Model Reduction for Inversion 22 / 29

23 2-D case Semi-discretization on fine grid with N points For the j source in B A (a line on the surface) we have t u (j) (t) = A(r)u (j) (t), u (j) (0) = b (j). The response matrix y ij (t) is the restriction of the solution u (j) (t) = e A(r)t b (j) to the support of the i th receiver y ij (t) = b (i)t e A(r)t b (j). Its Laplace transform is the transfer function G (ij) (s). The reduced model is (A (ij) m, b (ij) m ) and its transfer function is ( ) G (ij) m (s) = b (i)t m si m A (ij) (j) m b m = m q= c (ij) q s + θ (ij) q Liliana Borcea (University of Michigan) Model Reduction for Inversion 23 / 29

24 2-D case Model reduction We can do this for any (i, j), but we are interested in using the continued fraction representation of G (ij) m (s). {( Coefficients κ (ij) l )}, κ (ij) m l c (ij) q = l= (b (i)t m are guaranteed to be positive if ) ) (b (j)t > 0 z (ij) q m z (ij) q Thus, we work only with the diagonal G (jj) m (s). We have as before R j (r) = Q ( y jj ( ; r) ) ( = log κ (jj) l, log κ (jj) l We solve optimization r = arg min r R N + 2 N d Q(d j (t)) R j (r)) 2 2. j= ) m l= Liliana Borcea (University of Michigan) Model Reduction for Inversion 24 / 29

25 2-D case Sensitivity functions κjj k r (left) and κjj k r (right) k =,... m = 5 (top to bottom) for a single source/receiver. Liliana Borcea (University of Michigan) Model Reduction for Inversion 25 / 29

26 2-D case Reconstructions: two checkerboard inclusions Top: true r(x, y). Bottom: reconstruction after a single Gauss-Newton iteration. Constant initial guess r 0 (x, y). Sensors:. Liliana Borcea (University of Michigan) Model Reduction for Inversion 26 / 29

27 2-D case Reconstructions: two layered inclusions Top: true r(x, y). Bottom: reconstruction after a single Gauss-Newton iteration. Constant initial guess r 0 (x, y). Sensors:. Liliana Borcea (University of Michigan) Model Reduction for Inversion 27 / 29

28 2-D case Reconstructions: single skewed inclusion Top: true r(x, y). Bottom: reconstruction after a single Gauss-Newton iteration. Constant initial guess r 0 (x, y). Sensors:. Liliana Borcea (University of Michigan) Model Reduction for Inversion 28 / 29

29 Conclusion End notes All the details are in: A model reduction approach to numerical inversion for a parabolic partial differential equation. L. Borcea, V. Druskin, A.V. Mamonov and M. Zaslavsky, Inverse Problems 30 (204), 250 (33pp). Still plenty left to be done. The higher dimensional problem treatment is not optimal. Dealing with one source and multiple receivers means doing matrix rational approximation... 3-D problems? Liliana Borcea (University of Michigan) Model Reduction for Inversion 29 / 29