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### Uncertainty Quantification

The integration of the PRISM Uncertainty Quantification (PUQ) software into nanoHUB and the Rappture toolkit provides nanoHUB users with powerful tools for uncertainty propagation, statistical model calibration and data analysis, and validation of simulations that will eventually enable predictions with quantified confidence. Users can now propagate uncertainties in inputs and quantify how they affect outputs. The beauty of this integration is that UQ is automatically available to the vast majority of nanoHUB tools (those built using the Rappture toolkit) without changing the underlying deterministic code: all the tasks involved are performed automatically by the cyberinfrastructure.

The overall approach for uncertainty propagation in nanoHUB is based on the method of collocation followed by the the construction of surrogate models (also called response surfaces) through which input distributions can be propagated in a computationally efficient manner.

As shown in the figure, users can easily specify input variables in terms of distributions by clicking on the distribution button that is automatically added to all real-valued inputs. Given the specified distributions of input parameters and the number of simulations the user is willing to perform, PUQ selects optimal collocation points using Smolyak sparse grids. Using the Rappture submit command the deterministic code underlying the tool is executed for each set of the collocation points.

Once the simulations finish, the results are then used to construct Reduced Order Models (ROM) using generalized polynomial chaos and to compute the sensitivity of each output to the uncertain inputs, see bottom-right panel in the figure. Finally the ROM is used to propagate the distributions of input parameters and predict a distribution of outputs using Monte Carlo techniques.

## Uncertainty Quantification (UQ) and UQ-Enabled Tools

**Thermoelectric Power Factor Calculator for Superlattices**

Using ballistic NEGF, the fortran code calculates the Seebeck coefficient and electrical conductivity from IV characteristics on a three-layer superlattice (film) structure. A temperature difference is imposed on the device, which produces a current (Seebeck effect). A bias is applied such that the net current in the device is zero. This applied bias is the Seebeck voltage. The slope of the IV curve at the Seebeck voltage is the electrical conductivity.

**Bayesian calibration**

Many models have parameters that are difficult or impossible to measure. In the case of a curvefit or regression, we are trying to determine parameters that make the model fit the data. In other cases, we have an estimate of a parameter, but we wish to use observed data to improve our estimate.

This tool has examples illustrating common uses.

**Continuous casting of weld filler wire**

Simulate 1D metal casting of small diameter wire using finite volumes

**Gaussian Process Regression**

Perform Gaussian process regression in x-y data. The code makes use of the excellent GPy package.

**nanoPLASTICITY**

This tool uses a phase field approach to simulate plastic deformation in nano-crystalline materials. It captures the competing grain-boundary and dislocation-mediated deformation mechanisms that govern plastic deformation in these materials. The model is based on a multi phase field approach in which dislocations and grain boundary sliding are represented by means of scalar phase fields described in “The role of grain boundary energetics on the maximum strength of nanocrystalline Ni”, Koslowski, Lee and Lei, Journal of the Mechanics and Physics of Solids, 59 1427-1436, 2011.

The tool enables users to quantify how uncertainties in the input parameters (materials properties such as elastic constants, Peierls energy barrier for dislocation glide and activation barrier for grain boundary sliding) affect the prediction of the yield stress. In addition, it provides a sensitivity analysis that quantifies the relative importance of each input variable. In order to achieve this, the phase field simulation code is orchestrated by the PRISM Uncertainty Quantification (PUQ) tool that enables users to select various state-of-the-art methods for uncertainty propagation.