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In Bayesian inference, probabilistic information about models is posited a priori. This information, which may very well include features in the null space of the forward problem, affects both the computed models and the resulting resolution estimates. In Occam's inversion, on the other hand, the goal is to construct the smoothest model consistent...
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Earthquake source estimates are affected by many types of uncertainties, deriving from observational errors, modelling choices and our simplified description of the Earth’s interior. While observational errors are often accounted for, epistemic uncertainties, which stem from our imperfect description of the forward model, are usually neglected. In...
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... Model flatness and smoothness for higher dimensional cases can be defined similarly. Such flatness and smoothness priors for model parameters constrain the variation of derivations of model parameters and yield a smoother posterior estimation of model parameters (Scales et al., 1990;Gouveia and Scales, 1997;Lelievre and Farquharson, 2013). ...
... In parallel with the development of CSEM from an equipment point of view, major advances were also achieved regarding processing and interpretation of the acquired data. Initially this analysis was carried out in the data domain by the use of normalized magnitude versus offset (MVO) plots of the inverted model and to estimate its error (Jackson, 1972(Jackson, , 1979Parker, 1980;Oldenburg, 1983;Oldenburg, 1989, 1991;Sen et al., 1993;Gouveia & Scales, 1997;Alumbaugh & Newman, 2000;Berryman, 2000;Friedel, 2003;Günther, 2004;Routh, 2005;Routh & Miller , 2006;Gao et al., 2007;Xia et al., 2008;Oldenborger & Routh, 2009;Routh, 2009;Mordret et al., 2013;Kalscheuer et al., 2013;Grayver et al., 2014;Fichtner & Leeuwen, 2015;Mattsson, 2015;Bogiatzis et al., 2016;Chrapkiewicz et al., 2020;Ren & Kalscheuer, 2020). ...
Marine Controlled Source Electromagnetic (CSEM) is employed both in large-scale geophysical applications as well as within exploration of hydrocarbons and gas hydrates. Due to the diffusive character of the EM field only very low frequencies are used leading to inversion results with rather low resolution. In this paper, we calculate the resolution matrix associated with the inversion and derive the corresponding point spread functions (PSFs). The PSFs give information about how much the actual inversion has been blurred, and use of space-varying deconvolution can therefore further improve the inversion result. The actual deblurring is carried out by use of the nonnegative flexible conjugate gradient algorithm for least squares problem (NN-FCGLS), which is a fast iterative restoration technique. For completeness, we also introduce results obtained by use of a blind deconvolution algorithm based on maximum likelihood estimation (MLE) with unknown PSFs. The potential of the proposed approaches have been demonstrated using both complex synthetic data as well as field data acquired at the Wisting oil field in the Barents Sea. In both cases, the resolution of the final inversion result has improved and shows better agreement with the known target area.
... The corresponding inverse problem is that of finding the thermal conductivity field given the temperature field. Similar inverse problems are at the core of techniques that infer mechanical properties of tissue in elastography [2,3], X-ray tomography, inverse acoustic/electromagnetic scattering [4,5], waveform inversion in geophysics [6,7], data assimilation in climate modeling [8,9], and remote sensing in astronomy [10,11]. ...
... Mean. An estimate of the mean of the posterior distribution, p post X (x|ŷ), can be computed by using l(x) = x in Eq. (7). That is ...
Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is tackling their ill-posed nature. Bayesian inference provides a principled approach for overcoming this by formulating the inverse problem into a statistical framework. However, it is challenging to apply when inferring fields that have discrete representations of large dimensions (the so-called “curse of dimensionality”) and/or when prior information is available only in the form of previously acquired solutions. In this work, we present a novel method for efficient and accurate Bayesian inversion using deep generative models. Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems. Our statistical framework preserves the underlying physics and is demonstrated to yield accurate results with reliable uncertainty estimates, even in the absence of information about underlying noise model, which is a significant challenge with many existing methods. We demonstrate the effectiveness of proposed method on a variety of inverse problems which include both synthetic as well as experimentally observed data.
... Inverse problems arise in various areas of science and engineering, such as computerized tomography [1], seismology [2,3], climate-modeling [4,5], and astronomy [6]. While the forward/direct problem is generally well-studied and easier to solve, the inverse problem can be notoriously hard to tackle due to the lack of well-posedness [7]. ...
In this work, we train conditional Wasserstein generative adversarial networks to effectively sample from the posterior of physics-based Bayesian inference problems. The generator is constructed using a U-Net architecture, with the latent information injected using conditional instance normalization. The former facilitates a multiscale inverse map, while the latter enables the decoupling of the latent space dimension from the dimension of the measurement, and introduces stochasticity at all scales of the U-Net. We solve PDE-based inverse problems to demonstrate the performance of our approach in quantifying the uncertainty in the inferred field. Further, we show the generator can learn inverse maps which are local in nature, which in turn promotes generalizability when testing with out-of-distribution samples.
... Introduction. Inverse problems arise in various areas of science and engineering, such as computerized tomography [22], seismology [14,33], climate-modeling [18,17], and astronomy [8]. While the forward/direct problem is generally wellstudied and easier to solve, the inverse problem can be notoriously hard to tackle due to the lack of well-posedness [16]. ...
... The training dataset for this problem consists of 8,000 samples of (x,y) pairs. In order to construct each pair, we sample κ from a prior distribution and then solve for u, from (14) and (15), using the standard Bubnov-Galerkin approach implemented in FEniCS [4]. We use triangular elements to discretize the domain Ω and firstorder Lagrange shape functions to approximate the trial solutions and weighting functions. ...
... Using (14), and retaining only the first order terms, this yields ...
... The corresponding inverse problem is that of finding the thermal conductivity field given the temperature field. Similar inverse problems are at the core of techniques that infer mechanical properties of tissue [2,3], X-ray tomography, inverse acoustic/electromagnetic scattering [4,5], geophysics [6,7], climate modeling [8,9], and astronomy [10,11]. ...
... In this section we provide details about different modeling choices and hyper-parameters used in this study. In all numerical experiments we use WGAN-GP [7] for learning the prior density. The detailed architecture and associated hyper-parameters used in training of this model for different numerical experiments is provided in Table 2 and Figures 5 and 6. ...
Inverse problems are notoriously difficult to solve because they can have no solutions, multiple solutions, or have solutions that vary significantly in response to small perturbations in measurements. Bayesian inference, which poses an inverse problem as a stochastic inference problem, addresses these difficulties and provides quantitative estimates of the inferred field and the associated uncertainty. However, it is difficult to employ when inferring vectors of large dimensions, and/or when prior information is available through previously acquired samples. In this paper, we describe how deep generative adversarial networks can be used to represent the prior distribution in Bayesian inference and overcome these challenges. We apply these ideas to inverse problems that are diverse in terms of the governing physical principles, sources of prior knowledge, type of measurement, and the extent of available information about measurement noise. In each case we apply the proposed approach to infer the most likely solution and quantitative estimates of uncertainty.
... Inverse problems are some of the most interesting and important mathematical problems in science and engineering playing key role in many application domains such as geophysics [1][2][3][4], climate modeling [5,6], astrophysics [7,8], heat conduction [9,10], medical imaging [11,12], chemical kinetics [13], materials modeling [14], machine learning [15], disease diagnostics [16,17] and so on. In different domains, inverse problems appear in different names such as system identification, data assimilation, history matching, design sensitivity 1 analysis, or PDE-constrained optimization. ...
... Inverse problems refer to the process of inferring the latent parameters (the "cause") from a set of measured observations (the "effect") for a given system. Such problems arise in various areas of science and engineering such as geophysics [1][2][3][4], climate modeling [5], chemical kinetics [13], heat conduction [9], astrophysics [7,8], materials modeling [14], and the detection and diagnosis of disease [16,17]. Due to their enormous practical importance and associated algorithmic and computational challenges it has gained substantial research interest in recent years [55,56]. ...
... There are two popular and widely different approaches to tackle this ill-posedness: (1) deterministic approach based on regularization techniques, (2) statistical approach based on Bayesian inference. 14 ...
... Bayesian inference is a well-established technique for quantifying uncertainties in inverse problems that are constrained by physical principles [1,2,3]. It has found applications in diverse fields such as geophysics [4,5,6,7], climate modeling [8], chemical kinetics [9], heat conduction [10], astrophysics [11,12], materials modeling [13] and the detection and diagnosis of disease [14,15]. The two critical ingredients of a Bayesian inference problem are -an informative prior representing the prior belief about the parameters to be inferred and an efficient method for sampling from the posterior distribution. ...
Bayesian inference is used extensively to infer and to quantify the uncertainty in a field of interest from a measurement of a related field when the two are linked by a physical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and/or have prior distributions that are difficult to represent mathematically. In this manuscript we consider the use of Generative Adversarial Networks (GANs) in addressing these challenges. A GAN is a type of deep neural network equipped with the ability to learn the distribution implied by multiple samples of a given field. Once trained on these samples, the generator component of a GAN maps the iid components of a low-dimensional latent vector to an approximation of the distribution of the field of interest. In this work we demonstrate how this approximate distribution may be used as a prior in a Bayesian update, and how it addresses the challenges associated with characterizing complex prior distributions and the large dimension of the inferred field. We demonstrate the efficacy of this approach by applying it to the problem of inferring and quantifying uncertainty in the initial temperature field in a heat conduction problem from a noisy measurement of the temperature at later time.
... Performance comparisons of Tikhonov regularization and the optimal estimation method in the context of passive infrared/microwave atmospheric sounding were given by Eriksson [2000]; Steck and von Clarmann [2001]; Senten et al. [2012]. Besides, Gouveia and Scales [1997] explored the similarities and differences of both techniques for solving geophysical inverse problems in terms of mathematical fundamentals, resolution analysis, and error estimates. ...
Recently, several new generation instruments for far infrared and microwave remote sensing of the Earth's atmosphere have been launched, and enables us to observe the atmospheric composition based upon the thermal emission technique. These new technologies and observational data pave the way for more dedicated atmospheric research missions in the future. The impetus for my thesis is the growing interest in robust inversion algorithms for solving nonlinear inverse problems arising in atmospheric remote sensing. A retrieval code PILS (Profile Inversion for Limb Sounding) which allows for high resolution radiative transfer computations and the reconstruction of atmospheric parameters from infrared and microwave limb sounding measurements is presented. The employed forward model simulates physically realistic limb emission spectra in an efficient manner, by taking into account the instrument performance and the measurement characteristics. In particular, automatic differentiation (AD) techniques providing a rapid and reliable implementation of exact Jacobians, are a special optimization feature of the forward model. The inversion methodology is essentially based on a nonlinear least squares framework with adaptive (direct and iterative) numerical regularization approaches. The performance of these regularization techniques relies on the design of the regularization parameter choice methods and the a posteriori stopping rules. The characterization of the retrieval error, including the smoothing error, the noise error, and the model parameters error, assesses the accuracy of the regularized solution. An intercomparison between PILS and the retrieval code developed by the Netherlands Institute for Space Research (SRON), dealing with radiative transfer and inversion calculations with predefined input, aims to clarify the correctness and consistency of the implementations. Small differences in the forward model mainly result from continuum absorption and the integration of the radiative transfer equation. The possible causes of discrepancies in the retrieval results are the consequences of the different inversion methods employed (regularization, a priori information) and discretization. Retrieval results pertaining to trace gas retrievals from balloon-borne measurements by TELIS (TErahertz and submillimeter Limb Sounder) are discussed by analyzing both synthetic and real radiance spectra. A sensitivity study of hydroxyl radical (OH) retrieval is used to evaluate the inversion performance of PILS and to reveal the initial expectation of TELIS's measurement capabilities (e.g. critical error sources, data quality). Furthermore, retrieval results of ozone (O3), hydrogen chloride (HCl), carbon monoxide (CO), and OH from the winter flights during 2009-2011 are presented to assess the performance of the TELIS 1.8THz channel and to judge the reliability of PILS by comparing with the data products obtained by the TELIS 480-650 GHz channel and other limb sounders. These observations offer opportunities for the scientific community to make an extensive investigation into the stratospheric chemistry and dynamics, and to study the atmospheric environment over the polar region of the Northern Hemisphere.
... The primary advantages of particle filters lie in their general nature and wide applicability for problems even with high degrees of nonlinearity. These methods have been used for system identification in a wide variety of problems such as, climate modeling [26], geophysics [27,28], heat transfer [29,30] and structural health monitoring [31][32][33][34][35][36][37]. Particle filters require the solution of the forward problem for a large number of realizations for θ, generated using Monte Carlo simulations and evaluating their likelihood when compared with the measurement data. ...
... The collocation points can be chosen as the zeros of the Hermite polynomials and a Gauss-Hermite quadrature scheme can be used for evaluating the expectations in Eq. (28). This implies that the problem defined in Eq. (27) need to be solved deterministically corresponding to the collocation points defined in the probability space. For a differential equation with multiple random coefficients, the collocation grids need to be constructed using tensor products of the one dimensional grids. ...
... The results of the predictions for β α and β ǫ are shown in Figs. [25][26][27][28][29][30][31][32]. The summary of the analysis by all the four methods has been shown in Table 3. ...
This study focuses on the development of a computationally efficient algorithm for the offline identification of system parameters in nonlinear dynamical systems from noisy response measurements. The proposed methodology is built on the bootstrap particle filter available in the literature for dynamic state estimation. The model and the measurement equations are formulated in terms of the system parameters to be identified - treated as random variables, with all other parameters being considered as internal variables. Subsequently, the problem is transformed into a mathematical subspace spanned by a set of orthogonal basis functions obtained from polynomial chaos expansions of the unknown system parameters. The bootstrap filtering carried out in the transformed space enables identification of system parameters in a computationally efficient manner. The efficiency of the proposed algorithm is demonstrated through two numerical examples - a Duffing oscillator and a fluid structure interaction problem involving an oscillating airfoil in an unsteady flow.
