Uncertainty quantification relies on the ability to draw representative samples from complex computational models of interest. In order to avoid a large number of full model evaluations, surrogate models provide an efficient approximation to the relationship between the input parameters and the output response. The GrahamBrady research group uses both adaptive sampling and machine learning to build these surrogate models.
Abstracts from a few relevant publications


Estimating microstructural feature distributions from images using a Bayesian approachMany microstructural characterizations methods collect data on a regular pixelized grid. This method of discretization introduces a form of measurement error which can be shown to be proportional to the resolution at which it is collected. Intuitively, measurements made from low resolution data are associated with higher error, but quanti fication of this error is typically not performed. This is reflected in international standards for measurements of grain size, which only provide a recommended minimum number of sample points per microstructural grain to ensure each grain is sufficiently resolved. In this work a new method for quantifying the relative uncertainty of such pixelized measurements is presented. Using a Bayesian framework and simulated data collection on features collected from a Voronoi tessellation, the distribution of true geometric properties given a particular set of measurements is computed. This conditional feature distribution provides a quantitative estimate for the relative uncertainty associated with measurements made at different resolutions. The approach is applied to measurements of size, aspect ratio, and perimeter of given microstructural components, such as grains. Size distributions are shown to be the least sensitive to sampling resolution, and evidence is presented which shows that the international standards provide an overly conservative minimum resolution for grain size measurement in microstructures represented by a Voronoi tessellation. Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysisThe presence of a highdimensional stochastic input domain with discontinuitiesposes major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on a loworder gPC approximation error to track discontinuities or nonsmoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the original highdimensional problem to a number of lowerdimensional subproblems. Specifically, this criterion checks all the existing interactions between input parameters of a specific problem based on the highdimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting input parameters. The efficiency of the approach is demonstrated using examples of both smooth and nonsmooth problems with number of input parameters up to 500, and the approach is compared against other existing algorithms. 
Wade, N. & GrahamBrady, L. (2023). “Estimating Microstructural Property Distributions from Image Data Using a Bayesian Framework,” Journal of Microscopy, accepted.
Bhaduri, A., He, Y., Shields, M., GrahamBrady, L., Kirby, R.M. (2018). “Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis,” Journal of Computational Physics, 371: 732750. 
Perturbationbased surrogate models for dynamic failure of brittle materials in a multiscale and probabilistic contextLocalization of failure in many materials is associated with the heterogeneity in the material microstructure. Multiscale models often address this heterogeneity by passing field variables back and forth between amacroscale model and subscale analyses at each integration point. Although this technique is often effective, it can be extremely costly to perform distinct microscale analyses for every integration point in the domain. The proposed work uses a perturbationbased approach, conceptually similar to in situ adaptive tabulation, which provides a straightforward surrogate model that can be orders of magnitude more efficient than the microscale model. The approach is demonstrated specifically for models of dynamic brittle failure, in which crack populations are tracked from one load step to the next. Furthermore, following an approach similar to that used in perturbationbased stochastic finite elements, this technique streamlines the process of probabilistic characterization of the instantaneous stress and the uniaxial compressive strength. Numerical examples show that the approach is accurate and highly efficient when considering random perturbations in both the underlying flaw population and the strain history in these brittle materials.

Liu, J. & GrahamBrady, L. (2016). “Perturbationbased surrogate models for dynamic failure of brittle materials in a multiscale and probabilistic context,” International Journal for Multiscale Computational Engineering, 14(3): 273290. 
Free energy calculation using space filled design and weighted reconstruction: A modified single sweep approachA modified single sweep approach is proposed for generating free energy landscapes. The approach replaces the use of temperatureaccelerated molecular dynamics (TAMD) to generate centres in collective variable (CV) space at which mean forces are computed using restrained molecular dynamics (MD) simulations with a sequential spacefilling design. This approach also modifies the radial basis function reconstruction step of the traditional single sweep approach and proposes a weighted reconstruction of the free energy surface using the previously generated mean forces. The modified approach is compared to the traditional single sweep (SS) approach on the (φ, ψ) dihedral freeenergy map of solvated alanine dipeptide (AD). It is found that the new approach results in a more accurate reconstructed free energy than does the traditional approach when compared to the directlycomputed reference free energy landscape. It is shown that the increased accuracy of the overall map stems from the improved 1dimensional space filling (projective) property of the proposed design compared to that of the TAMD generated centres. A further enhancement in the accuracy of the crucial lower energy regions is enabled by the introduction of weights in the reconstruction step that give more importance to lower energyvalued regions. 
Bhaduri, A., Gardner, J., Abrams, C., GrahamBrady, L. (2020). “Free energy calculation using space filled design and weighted reconstruction: A modified single sweep approach,” Molecular Simulations, 46(3):193206. 
Other relevant publicationsOlivier, A., Shields, M., GrahamBrady, L. (2021). “Probabilistic neural networks for uncertainty quantification in databased materials modeling,” Computer Methods in Applied Mechanics and Engineering, 386:114079. Bhaduri, A., Brandyberry, D., Shields, M., Geubelle, P., GrahamBrady, L. (2020). “On the usefulness of gradient information in surrogate modeling: Application to uncertainty propagation in composite material models,” Probabilistic Engineering Mechanics, 60,103024. Bhaduri, A. & GrahamBrady, L. (2018). “An efficient adaptive sparse grid collocation method through derivative estimation,” Probabilistic Engineering Mechanics, 51:1122. 