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 theeffectivemechanical 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 aunifiedformulation 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. Astep-wiseBayesian 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 $$\hat{R}{-}convergence$$R^-convergence tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

Spatial prediction of precipitation with high resolution is a challenging task in regions with strong climate variability and scarce monitoring. For this purpose, the quasi-continuous supply of information from satellite imagery is commonly used to complement in situ data. However, satellite images of precipitation are available at coarse resolutions, and require adequate methods for spatial downscaling and calibration. The objective of this paper is to introduce and evaluate a 2-step spatial downscaling approach for monthly precipitation applied to TRMM 3B43 (from $0.{25}^{\circ }\approx$ 27 km to 5 km resolution), resulting in 5 downscaled products for the period 01-2001/12-2011. The methodology was evaluated in 3 contrasting climatic regions of Ecuador. In step 1, bilinear resampling was applied over TRMM, and used as a reference product. The second step introduces further variability, and consists of four alternative gauge-satellite merging methods: (1) regression with in situ stations, (2) regression kriging with in situ stations, (3) regression with in situ stations and auxiliary variables, and (4) regression kriging with in situ stations and auxiliary variables. The first 2 methods only use the resampled TRMM data set as an independent variable. The last 2 methods enrich these models with auxiliary environmental factors, incorporating atmospheric and land variables. The results showed that no product outperforms the others in every region. In general, the methods with residual kriging correction outperformed the regression models. Regression kriging with situ data provided the best representation in the Coast, while regression kriging with in situ and auxiliary data generated the best results in the Andes. In the Amazon, no product outperformed the resampled TRMM images, probably due to the low density of in situ stations. These results are relevant to enhance satellite precipitation, depending on the availability of in situ data, auxiliary satellite variables and the particularities of the climatic regions.