S. Mishra

M. Hadi Amini, R. Jaddivada, S. Mishra et al.

In this paper, we investigate two decomposition methods for their convergence rate which are used to solve security constrained economic dispatch (SCED): 1) Lagrangian Relaxation (LR), and 2) Augmented Lagrangian Relaxation (ALR). First, the centralized SCED problem is posed for a 6-bus test network and then it is decomposed into subproblems using both of the methods. In order to model the tie-line between decomposed areas of the test network, a novel method is proposed. The advantages and drawbacks of each method are discussed in terms of accuracy and information privacy. We show that there is a tradeoff between the information privacy and the convergence rate. It has been found that ALR converges faster compared to LR, due to the large amount of shared data.

M. Longo, S. Mishra, T. K. Rusch et al.

We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input data space for deep and shallow Neural Networks with holomorphic activation functions such as tanh. These novel training points are proved to facilitate higher-order decay (in terms of the number of training samples) of the underlying generalization error, with consistency error bounds that are free from the curse of dimensionality in the input data space, provided that DNN weights in hidden layers satisfy certain summability conditions. We present numerical experiments for DtO maps from elliptic and parabolic PDEs with uncertain inputs that confirm the theoretical analysis.