Juliette Franzman

Chandrika Kamath, Juliette Franzman, Ravi Ponmalai

Solving inverse problems, where we find the input values that result in desired values of outputs, can be challenging. The solution process is often computationally expensive and it can be difficult to interpret the solution in high-dimensional input spaces. In this paper, we use a problem from additive manufacturing to address these two issues with the intent of making it easier to solve inverse problems and exploit their results. First, focusing on Gaussian process surrogates that are used to solve inverse problems, we describe how a simple modification to the idea of tapering can substantially speed up the surrogate without losing accuracy in prediction. Second, we demonstrate that Kohonen self-organizing maps can be used to visualize and interpret the solution to the inverse problem in the high-dimensional input space. For our data set, as not all input dimensions are equally important, we show that using weighted distances results in a better organized map that makes the relationships among the inputs obvious.

Chandrika Kamath, Juliette S. Franzman

Building an accurate surrogate model for the spatio-temporal outputs of a computer simulation is a challenging task. A simple approach to improve the accuracy of the surrogate is to cluster the outputs based on similarity and build a separate surrogate model for each cluster. This clustering is relatively straightforward when the output at each time step is of moderate size. However, when the spatial domain is represented by a large number of grid points, numbering in the millions, the clustering of the data becomes more challenging. In this report, we consider output data from simulations of a jet interacting with high explosives. These data are available on spatial domains of different sizes, at grid points that vary in their spatial coordinates, and in a format that distributes the output across multiple files at each time step of the simulation. We first describe how we bring these data into a consistent format prior to clustering. Borrowing the idea of random projections from data mining, we reduce the dimension of our data by a factor of thousand, making it possible to use the iterative k-means method for clustering. We show how we can use the randomness of both the random projections, and the choice of initial centroids in k-means clustering, to determine the number of clusters in our data set. Our approach makes clustering of extremely high dimensional data tractable, generating meaningful cluster assignments for our problem, despite the approximation introduced in the random projections.

Chandrika Kamath, Juliette S. Franzman, Brian H. Daub

Computer simulations, especially of complex phenomena, can be expensive, requiring high-performance computing resources. Often, to understand a phenomenon, multiple simulations are run, each with a different set of simulation input parameters. These data are then used to create an interpolant, or surrogate, relating the simulation outputs to the corresponding inputs. When the inputs and outputs are scalars, a simple machine learning model can suffice. However, when the simulation outputs are vector valued, available at locations in two or three spatial dimensions, often with a temporal component, creating a surrogate is more challenging. In this report, we use a two-dimensional problem of a jet interacting with high explosives to understand how we can build high-quality surrogates. The characteristics of our data set are unique - the vector-valued outputs from each simulation are available at over two million spatial locations; each simulation is run for a relatively small number of time steps; the size of the computational domain varies with each simulation; and resource constraints limit the number of simulations we can run. We show how we analyze these extremely large data-sets, set the parameters for the algorithms used in the analysis, and use simple ways to improve the accuracy of the spatio-temporal surrogates without substantially increasing the number of simulations required.