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

Data Processing for Oscillatory Pumping Tests

Characterizing the subsurface is important for many hydrogeologic projects such as site remediation and groundwater resource exploration. Methods based on the analysis of conventional pumping tests have the notable disadvantage that at a certain distance, the signal is small relative to the noise due to the effects of recharge, pumping in neighboring wells, change in the level or adjacent streams, and other common disturbances. This work focuses on oscillatory pumping tests in which fluid is extracted for half a period, then reinjected. We discuss a major advantage of oscillatory pumping tests: small amplitude signals can be recovered from noisy data measured at observation wells and quantify the uncertainties in the estimates. We demonstrate results from a joint inversion of storativity and transmissivity. We conclude with an analysis of the duration of the initial transient, providing lower bounds on the length of elapsed time until the effects of the transient can be neglected

Frequency Dependent Hydraulic Properties Estimated from Oscillatory Pumping Tests in an Unconfined Aquifer

Oscillatory pumping tests were conducted at the Boise Hydrogeophysical Research Site. A periodic pressure signal is generated by pumping and injecting water into the aquifer consecutively and the pressure response is recorded at many points around the source. We present and analyze the data from the field test after applying Fourier analysis. We then match the data with a recently derived analytical solution for homogeneous formations to estimate the equivalent aquifer properties: conductivity K, specific storage Ss and specific yield Sy. The estimated values are shown to be in agreement with previous estimates conducted at this site. We observe variations in the estimated parameters with different oscillation periods of pumping. The trend of the parameters with changing period is discussed and compared to predictions by existing theory and laboratory experiments dealing with dynamic effective properties. It is shown that the results are qualitatively consistent with recent works on effective properties of formations of spatially variable properties in oscillatory flow. To grasp the impact of heterogeneity, a simple configuration is proposed, helping explain the observed increase in effective conductivity with decreasing period

Erratum to: Guidelines for the use and interpretation of assays for monitoring autophagy (3rd edition) (Autophagy, 12, 1, 1-222, 10.1080/15548627.2015.1100356

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