Bayesian Inversion Method Research Articles

A Hybrid Born Iterative Bayesian Inversion Method for Electromagnetic Imaging of Moderate-Contrast Scatterers With Piecewise Homogeneities

To tackle the problem of imaging moderate-contrast scatterers with piecewise homogeneities, a hybrid Born iterative (BI) Bayesian inversion method is proposed. First, we establish the nonlinear inverse scattering problem in the context of multiple measurement vector (MMV). Then, a computationally efficient BI formulation is used to account for the strong nonlinear relationship between the scattered field and the contrast. In each iteration, a regularization strategy that exploits the piecewise homogeneous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> priori information of the investigated domain is adopted to overcome the ill-posedness. Our proposed method can tackle the inverse scattering problem in its full nonlinearity and incorporate the available <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> priori information to stabilize the inversion procedure. Several representative tests are carried out to evaluate the performance of the proposed method. It is demonstrated that the proposed method outperforms other existing state-of-the-art algorithms in terms of accuracy, convergence, robustness, and computational efficiency.

Bridging time scales of faulting: From coseismic to postseismic slip of the Mw 6.0 2014 South Napa, California earthquake.

Transient fault slip spans time scales from tens of seconds of earthquake rupture to years of aseismic afterslip. So far, seismic and geodetic recordings of these two phenomena have primarily been studied separately and mostly with a focus on kinematic aspects, which limits our physical understanding of the interplay between seismic and aseismic slip. Here, we use a Bayesian dynamic source inversion method, based on laboratory-derived friction laws, to constrain fault stress and friction properties by joint quantitative modeling of coseismic and postseismic observations. Analysis of the well-recorded 2014 South Napa, California earthquake shows how the stressing and frictional conditions on the fault govern the spatial separation between shallow coseismic and postseismic slip, the progression of afterslip driving deep off-fault aftershocks, and the oblique ribbon-like rupture shape. Such inferences of stress and frictional rheology can advance our understanding of earthquake physics and pave the way for self-consistent cross-scale seismic hazard assessment.

Petrophysical inversion based on f-s-r amplitude-variation-with-offset linearization and canonical correlation analysis

The prediction of petrophysical properties, such as porosity and rock-fluid volumes, from partially stacked seismic data typically requires a rock-physics model that often is lithology dependent and difficult to calibrate. We adopt canonical correlation analysis (CCA) to infer the underlying relation between petrophysical properties and elastic attributes estimated from seismic data. We develop a two-step inversion approach: first, we predict elastic properties from partially stacked seismic data using a Bayesian linear inverse method based on an amplitude-variation-with-offset (AVO) linearization in terms of fluid, rigidity, and density factors, and then we predict petrophysical properties from the estimated AVO attributes using CCA. The novelty of our approach is the application of CCA to the fluid and rigidity factors, which avoids the calibration of an explicit rock-physics model by automatically deriving a linear relation in the lower dimensional space of the canonical variables. The parameterization of the linearization in terms of fluid, rigidity, and density factors maximizes the correlation with respect to the petrophysical properties of interest. Furthermore, the probabilistic approach is extended to the petrophysical inversion using Bayesian linear theory and the posterior distribution of petrophysical properties conditioned by seismic data is computed by combining the probability distributions obtained from seismic and petrophysical inversion to propagate the uncertainty from the seismic to the petrophysical domain. The inversion is validated on a synthetic case that finds high accuracy of our formulation. A case study with synthetic and real partially stacked seismic data also is presented and compared to a traditional inversion with an explicit rock-physics model.

Anthropogenic CO2 emission reduction during the COVID-19 pandemic in Nanchang City, China

China is the largest CO2 emitting country on Earth. During the COVID-19 pandemic, China implemented strict government control measures on both outdoor activity and industrial production. These control measures, therefore, were expected to significantly reduce anthropogenic CO2 emissions. However, large discrepancies still exist in the estimated anthropogenic CO2 emission reduction rate caused by COVID-19 restrictions, with values ranging from 10% to 40% among different approaches. Here, we selected Nanchang city, located in eastern China, to examine the impact of COVID-19 on CO2 emissions. Continuous atmospheric CO2 and ground-level CO observations from January 1st to April 30th, 2019 to 2021 were used with the WRF-STILT atmospheric transport model and a priori emissions. And a multiplicative scaling factor and Bayesian inversion method were applied to constrain anthropogenic CO2 emissions before, during, and after the COVID-19 pandemic. We found a 37.1–40.2% emission reduction when compared to the COVID-19 pandemic in 2020 with the same period in 2019. Carbon dioxide emissions from the power industry and manufacturing industry decreased by 54.5% and 18.9% during the pandemic period. The power industry accounted for 73.9% of total CO2 reductions during COVID-19. Further, emissions in 2021 were 14.3–14.9% larger than in 2019, indicating that economic activity quickly recovered to pre-pandemic conditions.

Source mechanism of the 2020 Mw 6.3 Nima earthquake derived from Bayesian inversions with InSAR observations: Insight into E-W extensional activity in the central Tibet

Normal faults and conjugate strike-slip faults have been considered to play an important role in response to the extension deformation of central Tibet. On 22 July 2020, a Mw 6.3 earthquake struck the Nima county, central Tibet, in China, which provides a rare opportunity to get insights into how the normal faults in central Tibet accommodates the east-west extension caused by the Indian-Eurasian convergence. In this study, the Sentinel-1 images are collected to measure the coseismic deformation associated with this 2020 event and image its slip distribution. To mitigate the atmospheric phase effects, the generic atmospheric correction online service (GACOS) model is used to correct the coseismic inteferograms. The final coseismic deformation results show mainly negative displacement with a maximum value of ∼ 30 cm in the line of sight (LOS) direction. After that, a Bayesian inversion method is used to invert the fault model. Our results reveal the optimal seismogenic fault of this event with a strike angle of 31.3°, a dip angle of 51.6°, and show that its slip distribution is dominated by normal slip with a maximum value of 2.55 m at a depth of 4.79–9.53 km, which suggest it’s a blind normal rupture with high east-trending dip angle. The total released geodetic moment is, equivalent to Mw 6.3. In addition, we analyze the Coulomb stress change due to the 2008 Gaize and this 2020 Nima events, suggesting the 2020 event should not be triggered by the 2008 event. Finally, we estimate an interseismic slip rate of 4.7±1.2 mm/yr on the Yibug Caka-RigainPun Co (YCRC) fault with published global navigation satellite system (GNSS) measurements. Given that these high frequent normal slip events but a low crustal extension rate of 4.7±1.2 mm/yr in this region, we speculate that the asthenosphere material upwelling should be also a possible reason for E-W extensional activities in central Tibet.

Asteroid Photometric Phase Functions From Bayesian Lightcurve Inversion

Photometry is an important tool for characterizing the physical properties of asteroids. An asteroid’s photometric lightcurve and phase curve refer to the variation of the asteroid’s disk-integrated brightness in time and in phase angle (the Sun-asteroid-observer angle), respectively. They depend on the asteroid’s shape, rotation, and surface scattering properties, and the geometry of illumination and observation. We present Bayesian lightcurve inversion methods for the retrieval of the asteroid’s phase function, the unambiguous phase curve of a spherical object with surface scattering properties equal to those of the asteroid. A collection of such phase functions can give rise to a photometric taxonomy for asteroids. In the inverse problem, first, there are four classes of lightcurves that require individual error models. The photometric observations can be absolute or relative and they can have dense or sparse cadence in comparison to the rotation period of the asteroid. Second, the observations extend over varying phase angle ranges, requiring different phase function models. Asteroid photometry from the European Space Agency Gaia space mission extends, typically, over a range of phase angles, where the phase curve tends to be linear on the magnitude scale. Photometry from ground-based observing programs can reach small phase angles, where the asteroids show an opposition effect, a nonlinear increase of brightness on the magnitude scale towards zero phase angle. We provide error models for all four classes of lightcurves and make use of linear or linear-exponential phase functions for phase angles below 50°. We apply the inverse methods to sparse absolute Gaia and dense relative ground-based lightcurves and obtain absolute magnitudes and phase functions, with uncertainties, for ∼500 asteroids. Finally, we assess the lightcurve inversion problem for dense absolute photometry with the help of a numerical simulation for a Gaussian-random-sphere asteroid.

Research on Prediction Method of Volcanic Rock Shear Wave Velocity Based on Improved Xu–White Model

Volcanic rock reservoirs have received extensive attention from scholars all over the world because of their geothermal, mineral, and oil and gas resources. Shear wave velocity is the essential information for AVO (amplitude variation with offset) analysis and the reservoir description of volcanic rocks. However, due to factors such as cost, technical reasons, and so on, shear wave velocity is not provided in many logging data. This paper proposes a shear wave velocity prediction method suitable for the conventional logging of volcanic rocks. Firstly, the Xu–White model is improved. The probability distributions formed by the prior information of the logging area are used to initialize the key petrophysical parameters in the model instead of the fixed parameter value to establish the statistical petrophysical model between the logging curve and shear wave velocity. Then, based on the Bayesian inversion method, the simulated P-wave velocity is matched with the actual P-wave logging data to calculate the key petrophysical parameters, and is then used for S-wave velocity prediction. The method is applied to the actual logging data of the No. 5 structure in Nanpu Sag, eastern China. The prediction effect of shear wave velocity is better than that of the conventional method, indicating the feasibility and effectiveness of this method. This study will provide more accurate shear wave velocity data for the exploration and development of volcanic reservoirs.

Bayesian rock-physics inversion with Kumaraswamy prior models

The prediction of rock and fluid volumetric fractions from elastic attributes often is referred to as petrophysical or rock-physics inversion because it requires rock-physics models to map petrophysical properties into geophysical variables, such as velocities and density. Bayesian approaches are suitable for rock-physics inverse problems because the solution, expressed in the form of a probability distribution, can represent the uncertainty of the model predictions due to the errors in the measured data. Bayesian inverse methods often rely on Gaussian prior distributions for their analytical tractability. However, Gaussian distributions are theoretically not applicable to rock and fluid volumetric fractions because, by definition, they are nonzero on the entire set of real numbers, whereas rock and fluid volumetric fractions are bounded between zero and one. The proposed rock physics inversion is based on a Bayesian approach that assumes Kumaraswamy probability density functions for the prior distribution to model double-bounded nonsymmetric continuous random variables between zero and one. The results of the Bayesian inverse problem are the pointwise probability distributions of the rock and fluid volumetric fractions conditioned on the seismic attributes. In the first application, the method is validated using synthetic well-log data for the soft sand and stiff rock-physics models with comparisons with several prior models. In the second application, the method is applied to a 2D real data set to obtain the posterior distribution, the maximum a posteriori, and the confidence intervals of porosity, mineral volumes, and fluid saturations. The most likely model of rock and fluid properties estimated from the posterior distribution assuming a Kumaraswamy prior model finds higher accuracies compared to the corresponding results obtained with a Gaussian prior model.

Subduction earthquakes controlled by incoming plate geometry: The 2020 M > 7.5 Shumagin, Alaska, earthquake doublet

In 2020, an earthquake doublet, a M7.8 on July 22nd and a M7.6 on October 19th, struck the Alaska-Aleutian subduction zone beneath the Shumagin Islands. This is the first documented earthquake doublet involving a megathrust event and a strike-slip event. The first event partially ruptured a seismic gap, which has not hosted large earthquakes since 1917, and the second event was unusual as it broke a trench-perpendicular fault within the incoming oceanic slab. We used an improved Bayesian geodetic inversion method to estimate the fault slip distributions of the major earthquakes using Interferometric Synthetic Aperture Radar (InSAR) wrapped phase and Global Navigation Satellite Systems (GNSS) offsets data. The geodetic inversions reveal that the Shumagin seismic gap is multi-segmented, and the M7.8 earthquake ruptured the eastern segment from 14 km down to 44 km depth. The coseismic slip occurred along a more steeply, 26∘ dipping segment, and was bounded up-dip by a bend of the megathrust interface to a shallower 8∘ dip angle connecting to the trench. The model for the M7.6 event tightly constrained the rupture depth extent to 19-39 km, within the depth range of the M7.8 coseismic rupture area. We find that the M7.6 event ruptured the incoming slab across its full seismogenic thickness, potentially reactivating subducted Kula-Resurrection seafloor-spreading ridge structures. Coulomb stress transfer models suggest that coseismic and/or postseismic slip of the M7.8 event could have triggered the M7.6 event. We conclude that the segmented megathrust structure and the location of intraslab fault structures limited the rupture dimensions of the M7.8 event and are responsible for the segmentation of the Shumagin seismic gap. Our study suggests that the western and shallower up-dip segments of the seismic gap did not fail and remain potential seismic and tsunami hazard sources. The unusual earthquake doublet provides a unique opportunity to improve our understanding of the role of the subducting lithosphere structure in the segmentation of subduction zones.

A novel trans-dimensional Bayesian inversion strategy for airborne time-domain electromagnetic data

The trans-dimensional Bayesian inversion method based on statistical theory regards the inversion parameters as random variable, it not only provides a reasonable inverse model based on the probability, but also offers the probability distribution and the uncertainty information of the inverse model parameters. However, for the conventional trans-dimensional Bayesian inversion, the model sampling efficiency and inversion convergence rate are influenced by several factors, and it requires sampling over a big model space, those limit the application of the trans-dimensional Bayesian inversion for airborne time-domain electromagnetic data. In this paper, we present a novel trans-dimensional Bayesian inversion strategy for airborne time-domain electromagnetic data, which uses a combining sampling update method to implement the model sampling. This method uses the block-wise updating method to run only one Markov chain with the initial state constructed by conductivity-depth imaging during the burn-in period, and adopts the component-wise updating method with an adaptive sampling step size to perform multiple Markov chains in parallel after the burn-in period ends. The effectiveness of the novel Bayesian inversion strategy was validated by both synthetic data and survey data. The experimental results showed that this inversion strategy can not only obtain better inversion results, but also shorten the burn-in period and reduce the total sampling times of the Markov chain.

Experimental evaluation and uncertainty quantification for a fractional viscoelastic model of salt concrete

This study evaluates and analyzes creep testing results on salt concrete of type M2. The concrete is a candidate material for long-lasting structures for sealing underground radioactive waste repository sites. Predicting operational lifetime and security aspects for these structures requires specific constitutive equations to describe the material behavior. Thus, we analyze whether a fractional viscoelastic constitutive law is capable of representing the long-term creep and relaxation processes for M2 concrete. We conduct a creep test to identify the parameters of the fractional model. Moreover, we use the Bayesian inversion method to evaluate the identifiability of the model parameters and the suitability of the experimental setup to yield a reliable prediction of the concrete behavior. Particularly, this Bayesian analysis allows to incorporate expert knowledge as prior information, to account for limited experimental precision and finally to rigorously quantify the post-calibration uncertainty.

Quantifying Uncertainties in Passive Microwave Remote Sensing of Soil Moisture via a Bayesian Probabilistic Inversion Method

Improving the accuracy of remotely sensed soil moisture (SM) is a challenging and popular topic. Quantifying the uncertainty in the SM inversion process and enhancing the confidence of SM retrieval are promising ways to address this challenge but have received little attention. We present a Bayesian probabilistic inversion algorithm that can simultaneously retrieve SM, surface roughness, and vegetation optical depth data and quantify the uncertainty in the inversion. The proposed algorithm is evaluated using airborne polarimetric L-band multibeam radiometer (PLMR) observations. We use three combinations, 3-angular observations at V-polarization (3CV), 3-angular observations at H-polarization (3CH) and 6-channel observations (6CA), to identify the optimal configuration for SM retrieval by taking advantage of the PLMR’s dual-polarization and multiple angles. Uncertainties are quantified by introducing multiple uncertainty quantification metrics into Bayesian posterior distributions of SM retrievals. The estimates are validated against multiscale ground-based measurements, including manual measurements and wireless sensor network (WSN) measurements, and the spatial representativeness of the ground-based reference regarding the validation of pixel-scale SM retrievals is discussed. The 6CA attempt yields the best SM estimates (correlation coefficient (R) ≥ 0.864, root mean square error (RMSE) ≤0.04 m3/m3, and unbiased RMSE (unRMSE) ≤ 0.035 m3/m3), while the 3CH attempt yields the lowest uncertainty. Additionally, dense manual measurements are more representative than sparsely distributed WSN measurements. Overall, combining dual-polarized observations yields the best SM estimates but introduces additional uncertainty. This study highlights uncertainties quantification in SM inversion and thus provides confidence in SM inversion, facilitating improved SM retrieval algorithms.

Environmental stress level to model tumor cell growth and survival.

Survival of living tumor cells underlies many influences such as nutrient saturation, oxygen level, drug concentrations or mechanical forces. Data-supported mathematical modeling can be a powerful tool to get a better understanding of cell behavior in different settings. However, under consideration of numerous environmental factors mathematical modeling can get challenging. We present an approach to model the separate influences of each environmental quantity on the cells in a collective manner by introducing the "environmental stress level". It is an immeasurable auxiliary variable, which quantifies to what extent viable cells would get in a stressed state, if exposed to certain conditions. A high stress level can inhibit cell growth, promote cell death and influence cell movement. As a proof of concept, we compare two systems of ordinary differential equations, which model tumor cell dynamics under various nutrient saturations respectively with and without considering an environmental stress level. Particle-based Bayesian inversion methods are used to quantify uncertainties and calibrate unknown model parameters with time resolved measurements of in vitro populations of liver cancer cells. The calibration results of both models are compared and the quality of fit is quantified. While predictions of both models show good agreement with the data, there is indication that the model considering the stress level yields a better fitting. The proposed modeling approach offers a flexible and extendable framework for considering systems with additional environmental factors affecting the cell dynamics.

An efficient Bayesian inversion method for seepage parameters using a data-driven error model and an ensemble of surrogates considering the interactions between prediction performance indicators

The Bayesian method has been increasingly applied to the inversion of seepage parameters owing to its superiority of considering the uncertainty in the inversion process. However, most of the current Bayesian inversion studies only focus on parameter uncertainty, ignoring the model structure error. In addition, existing research has mostly adopted a single machine learning algorithm or an ensemble of surrogates based on a single prediction performance indicator as a substitute for the seepage forward model and has not considered the interactions between multiple prediction performance indicators, thereby leading to poor accuracy. To address these issues, this study proposed an efficient Bayesian inversion method for seepage parameters using a data-driven error model and an ensemble of surrogates considering the interactions between prediction performance indicators. A data-driven error model based on Gaussian process regression was integrated into the Bayesian inversion model to modify the likelihood function for dealing with the model structure error. For determining the weight coefficients of the ensemble surrogates, the improved Dempster-Shafer (D-S) evidence theory based on the Hellinger distance and Deng entropy was proposed to fuse multiple prediction performance indicators and consider their interactions. Further, an ensemble of surrogates in conjunction with support vector regression, Kriging, and multivariate adaptive regression splines was constructed through weighted summation. The validity and accuracy of the proposed method were verified by applying it to a real hydropower station in China. The results showed that the proposed method can significantly improve the accuracy and efficiency of Bayesian inversion of seepage parameters. The proposed method therefore serves as a new basis for the inversion of seepage parameters and can be applied to parameter inversion in other related fields.

A Near-Real-Time Method for Estimating Volcanic Ash Emissions Using Satellite Retrievals

We present a Bayesian inversion method for estimating volcanic ash emissions using satellite retrievals of ash column load and an atmospheric dispersion model. An a priori description of the emissions is used based on observations of the rise height of the volcanic plume and a stochastic model of the possible emissions. Satellite data are processed to give column loads where ash is detected and to give information on where we have high confidence that there is negligible ash. An atmospheric dispersion model is used to relate emissions and column loads. Gaussian distributions are assumed for the a priori emissions and for the errors in the satellite retrievals. The optimal emissions estimate is obtained by finding the peak of the a posteriori probability density under the constraint that the emissions are non-negative. We apply this inversion method within a framework designed for use during an eruption with the emission estimates (for any given emission time) being revised over time as more information becomes available. We demonstrate the approach for the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions. We apply the approach in two ways, using only the ash retrievals and using both the ash and clear sky retrievals. For Eyjafjallajökull we have compared with an independent dataset not used in the inversion and have found that the inversion-derived emissions lead to improved predictions.

GPCE-Based Stochastic Inverse Methods: A Benchmark Study from a Civil Engineer’s Perspective

In civil and mechanical engineering, Bayesian inverse methods may serve to calibrate the uncertain input parameters of a structural model given the measurements of the outputs. Through such a Bayesian framework, a probabilistic description of parameters to be calibrated can be obtained; this approach is more informative than a deterministic local minimum point derived from a classical optimization problem. In addition, building a response surface surrogate model could allow one to overcome computational difficulties. Here, the general polynomial chaos expansion (gPCE) theory is adopted with this objective in mind. Owing to the fact that the ability of these methods to identify uncertain inputs depends on several factors linked to the model under investigation, as well as the experiment carried out, the understanding of results is not univocal, often leading to doubtful conclusions. In this paper, the performances and the limitations of three gPCE-based stochastic inverse methods are compared: the Markov Chain Monte Carlo (MCMC), the polynomial chaos expansion-based Kalman Filter (PCE-KF) and a method based on the minimum mean square error (MMSE). Each method is tested on a benchmark comprised of seven models: four analytical abstract models, a one-dimensional static model, a one-dimensional dynamic model and a finite element (FE) model. The benchmark allows the exploration of relevant aspects of problems usually encountered in civil, bridge and infrastructure engineering, highlighting how the degree of non-linearity of the model, the magnitude of the prior uncertainties, the number of random variables characterizing the model, the information content of measurements and the measurement error affect the performance of Bayesian updating. The intention of this paper is to highlight the capabilities and limitations of each method, as well as to promote their critical application to complex case studies in the wider field of smarter and more informed infrastructure systems.

Scalable, probabilistic echo inversion for multifrequency and broadband acoustic surveys

Identifying biological scatterers is a perennial challenge in fisheries acoustics. Most practitioners classify backscatter based on direct sampling and frequency-difference thresholds, then integrate at a single frequency. However, this approach struggles with species mixtures, and discards multi-frequency information when integrating. Inversion methods do not have these limitations, but are seldom used, because their species identifications are often ambiguous and their algorithms complicated to implement. We address these shortcomings with a probabilistic, Bayesian inversion method. Like other inversion methods, it handles species mixtures, uses all available frequencies, and extends naturally to broadband signals. Unlike prior approaches, it leverages Bayesian priors to rigorously incorporate information from direct sampling and biological knowledge, constraining the inversion and reducing ambiguity in species identification. Because it is probabilistic, it can be trusted to run automatically: it should not produce solutions that are both wrong and confident. Unlike some data-driven machine learning models, it is based on acoustical scattering processes, so its inferences are physically interpretable. Finally, the approach is straightforward to implement using existing Bayesian libraries, and is easily parallelized for large datasets. We present examples using simulations and field data from the Gulf of Alaska, and discuss possible extensions and applications of the method.

Bayesian Inversion of Soil Hydraulic Properties from Simplified Evaporation Experiments: Use of DREAM(ZS) Algorithm

There is an increasing interest in identifying soil hydraulic properties from simplified evaporation experiments. However, the conventional simplified evaporation method includes a deficit due to using the linear assumption and not accounting for uncertainty in parameters. A suggested alternative method is assessing the parameter uncertainties through inverse modeling. We examined the combination of a Bayesian inverse method, namely, DREAM(ZS), and a numerical simulation model, namely, HYDRUS-1D, for parameter inversion with data in simplified evaporation experiments. The likelihood function could be conditioned only on pressure head observations (single-objective (SO)), or on both pressure head and evaporation rate observations (multi-objective (MO)), with different treatments on the top boundary condition. Three synthetic numerical experiments were generated in terms of the soil types of sand, loam and clay to verify the inverse modeling method. The MO approach performed better than the SO approach and linear assumption when the stage 1 evaporation rate was kept constant. However, the SO inversion was more robust when oscillations existed in the potential evaporation rate. Then, the SO inverse modeling was adopted to investigate two real experiments on loamy-sand soils and compared with the linear assumption. The linear assumption could be reliable for wet conditions with stage 1 evaporation but was not always useable for a relatively dry condition, such as that with stage 2 evaporation. The inverse modeling could be more successful in capturing the whole evaporation process of soils when both stage 1 and stage 2 were involved.

Bayesian inversion for unified ductile phase-field fracture

The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the hat{R}{-}convergence tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

Bayesian inversion for a fractional Lotka-Volterra model: An application of Canadian lynx vs. snowshoe hares

The Bayesian statistical inversion method is implemented to obtain the parameter estimations for fractional LotkaVolterra models, including the derivative orders, based on analytical solutions obtained by the multi-step homotopy method. For the posterior distributions of the parameter of interest, we used Markov Chain Monte Carlo method through the JAGS package within R software. The posterior predictive model–checking method is implemented to select the best model for a real data set.