A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method

Cross-Correlation of Planck CMB Lensing with DESI-Like LRGs

Cross-correlations between the lensing of the cosmic microwave background (CMB) and other tracers of large-scale structure provide a unique way to reconstruct the growth of dark matter, break degeneracies between cosmology and galaxy physics, and test theories of modified gravity. We detect a cross-correlation between DESI-like luminous red galaxies (LRGs) selected from DECaLS imaging and CMB lensing maps reconstructed with the Planck satellite at a significance of $S/N = 27.2$ over scales $\ell_{\rm min} = 30$, $\ell_{\rm max} = 1000$. To correct for magnification bias, we determine the slope of the LRG cumulative magnitude function at the faint limit as $s = 0.999 \pm 0.015$, and find corresponding corrections on the order of a few percent for $C^{\kappa g}_{\ell}, C^{gg}_{\ell}$ across the scales of interest. We fit the large-scale galaxy bias at the effective redshift of the cross-correlation $z_{\rm eff} \approx 0.68$ using two different bias evolution agnostic models: a HaloFit times linear bias model where the bias evolution is folded into the clustering-based estimation of the redshift kernel, and a Lagrangian perturbation theory model of the clustering evaluated at $z_{\rm eff}$. We also determine the error on the bias from uncertainty in the redshift distribution; within this error, the two methods show excellent agreement with each other and with DESI survey expectations.Comment: 18 pages, 14 figures, 6 tables; final version accepted for publicatio

Estimation of historical groundwater contaminant distribution using the adjoint state method applied to geostatistical inverse modeling

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/95609/1/wrcr10022.pd