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1. Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty

   https://doi.org/10.5194/tc-15-1731-2021  

   Babaniyi, Olalekan; Nicholson, Ruanui; Villa, Umberto; Petra, Noémi (January 2021, The Cryosphere)

   null (Ed.)

   Abstract. We consider the problem of inferring the basal sliding coefficientfield for an uncertain Stokes ice sheet forward model from syntheticsurface velocity measurements. The uncertainty in the forward modelstems from unknown (or uncertain) auxiliary parameters (e.g., rheologyparameters). This inverse problem is posed within the Bayesianframework, which provides a systematic means of quantifyinguncertainty in the solution. To account for the associated modeluncertainty (error), we employ the Bayesian approximation error (BAE)approach to approximately premarginalize simultaneously over both thenoise in measurements and uncertainty in the forward model. We alsocarry out approximative posterior uncertainty quantification based ona linearization of the parameter-to-observable map centered at themaximum a posteriori (MAP) basal sliding coefficient estimate, i.e.,by taking the Laplace approximation. The MAP estimate is found byminimizing the negative log posterior using an inexact Newtonconjugate gradient method. The gradient and Hessian actions to vectorsare efficiently computed using adjoints. Sampling from theapproximate covariance is made tractable by invoking a low-rankapproximation of the data misfit component of the Hessian. We studythe performance of the BAE approach in the context of three numericalexamples in two and three dimensions. For each example, the basalsliding coefficient field is the parameter of primary interest whichwe seek to infer, and the rheology parameters (e.g., the flow ratefactor or the Glen's flow law exponent coefficient field) representso-called nuisance (secondary uncertain) parameters. Our resultsindicate that accounting for model uncertainty stemming from thepresence of nuisance parameters is crucial. Namely our findingssuggest that using nominal values for these parameters, as is oftendone in practice, without taking into account the resulting modelingerror, can lead to overconfident and heavily biased results. We alsoshow that the BAE approach can be used to account for the additionalmodel uncertainty at no additional cost at the online stage. 

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2. A Non-Convex Approach To Joint Sensor Calibration And Spectrum Estimation

   https://doi.org/10.1109/SSP.2018.8450691  

   Cho, Myung; Liao, Wenjing; Chi, Yuejie (June 2018, 2018 IEEE Statistical Signal Processing Workshop (SSP))

   Blind sensor calibration for spectrum estimation is the problem of estimating the unknown sensor calibration parameters as well as the parameters-of-interest of the impinging signals simultaneously from snapshots of measurements obtained from an array of sensors. In this paper, we consider blind phase and gain calibration (BPGC) problem for direction-of-arrival estimation with multiple snapshots of measurements obtained from an uniform array of sensors, where each sensor is perturbed by an unknown gain and phase parameter. Due to the unknown sensor and signal parameters, BPGC problem is a highly nonlinear problem. Assuming that the sources are uncorrelated, the covariance matrix of the measurements in a perfectly calibrated array is a Toeplitz matrix. Leveraging this fact, we first change the nonlinear problem to a linear problem considering certain rank-one positive semidefinite matrix, and then suggest a non-convex optimization approach to find the factor of the rank-one matrix under a unit norm constraint to avoid trivial solutions. Numerical experiments demonstrate that our proposed non-convex optimization approach provides better or competitive recovery performance than existing methods in the literature, without requiring any tuning parameters. 

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3. Ensemble Kalman inversion for spatially varying rheological parameters in a stress-driven model of post-seismic deformation

   https://doi.org/10.1093/gji/ggaf208  

   Fukuda, Junichi; Barbot, Sylvain (June 2025, Geophysical Journal International)

   SUMMARY Geodetic observations of post-seismic deformation due to afterslip and viscoelastic relaxation can be used to infer fault and lithosphere rheologies by combining the observations with mechanical models of post-seismic processes. However, estimating the spatial distributions of rheological parameters remains challenging because it requires solving a nonlinear inverse problem with a high-dimensional parameter space and potentially computationally expensive forward model. Here we introduce an inversion method to estimate spatially varying fault and lithospheric rheological parameters in a mechanical model of post-seismic deformation using geodetic time series. The forward model combines afterslip and viscoelastic relaxation governed by a velocity-strengthening frictional rheology and a power-law Burgers rheology, respectively, and incorporates the mechanical coupling between coseismic slip, afterslip and viscoelastic relaxation. The inversion method estimates spatially varying fault frictional parameters, viscoelastic constitutive parameters and coseismic stress change. We formulate the inverse problem in a Bayesian framework to quantify the uncertainties of the estimated parameters. To solve this problem with reasonable computational costs, we develop an algorithm to estimate the mean and covariance matrix of the posterior probability distribution based on an ensemble Kalman filter. We validate our method through numerical tests using a 2-D forward model and synthetic post-seismic GNSS time-series. The test results suggest that our method can estimate the spatially varying rheological parameters and their uncertainties reasonably well with tolerable computational costs. Our method can also recover spatially and temporally varying afterslip, viscous strain and effective viscosities and can distinguish the contributions of afterslip and viscoelastic relaxation to observed post-seismic deformation. 

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4. Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

   https://doi.org/10.1088/1361-6420/ad602e  

   Alexanderian, Alen; Nicholson, Ruanui; Petra, Noemi (July 2024, Inverse Problems)

   We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximuma posterioriprobability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages. 

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5. Adjoint-based inversion for stress and frictional parameters in earthquake modeling

   https://doi.org/10.1016/j.jcp.2024.113447  

   Stiernström, Vidar; Almquist, Martin; Dunham, Eric M (December 2024, Journal of Computational Physics)

   We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction and, for forward problems involving fault normal stress changes, nonzero fault opening, with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge–Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments. 

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