Ralph C. Smith

John T. Holodnak, Ilse C. F. Ipsen, Ralph C. Smith

Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions but only on the numerical rank (intrinsic dimension) of E. Our approach resembles the low-rank approximation of kernel matrices from random features, but our accuracy measures are more stringent. In the context of parameter selection based on active subspaces, where S is computed via Monte Carlo sampling, we present a bound on the number of samples so that with high probability the angle between the dominant subspaces of E and S is less than a user-specified tolerance. This is a substantial improvement over existing work, as it is a non-asymptotic and fully explicit bound on the sampling amount n, and it allows the user to tune the success probability. It also suggests that Monte Carlo sampling can be efficient in the presence of many parameters, as long as the underlying function f is sufficiently smooth.

Sreeram Venkat, Ralph C. Smith, Carl T. Kelley

In the construction of reduced-order models for dynamical systems, linear projection methods, such as proper orthogonal decompositions, are commonly employed. However, for many dynamical systems, the lower dimensional representation of the state space can most accurately be described by a \textit{nonlinear} manifold. Previous research has shown that deep learning can provide an efficient method for performing nonlinear dimension reduction, though they are dependent on the availability of training data and are often problem-specific \citep[see][]{carlberg_ca}. Here, we utilize randomized training data to create and train convolutional autoencoders that perform nonlinear dimension reduction for the wave and Kuramoto-Shivasinsky equations. Moreover, we present training methods that are independent of full-order model samples and use the manifold least-squares Petrov-Galerkin projection method to define a reduced-order model for the heat, wave, and Kuramoto-Shivasinsky equations using the same autoencoder.