Hemanth Kolla

Saibal De, Zitong Li, Hemanth Kolla et al.

The Tucker decomposition, an extension of singular value decomposition for higher-order tensors, is a useful tool in analysis and compression of large-scale scientific data. While it has been studied extensively for static datasets, there are relatively few works addressing the computation of the Tucker factorization of streaming data tensors. In this paper we propose a new streaming Tucker algorithm tailored for scientific data, specifically for the case of a data tensor whose size increases along a single streaming mode that can grow indefinitely, which is typical of time-stepping scientific applications. At any point of this growth, we seek to compute the Tucker decomposition of the data generated thus far, without requiring storing the past tensor slices in memory. Our algorithm accomplishes this by starting with an initial Tucker decomposition and updating its components--the core tensor and factor matrices--with each new tensor slice as it becomes available, while satisfying a user-specified threshold of norm error. We present an implementation within the TuckerMPI software framework, and apply it to synthetic and combustion simulation datasets. By comparing against the standard (batch) decomposition algorithm we show that our streaming algorithm provides significant improvements in memory usage. If the tensor rank stops growing along the streaming mode, the streaming algorithm also incurs less computational time compared to the batch algorithm.

Saibal De, Reese E. Jones, Hemanth Kolla

Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the distributions of the high-dimensional primary solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter-to-solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. This work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data-driven sparse identification of nonlinear dynamics (SINDy) framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC-over-dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full-field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics.