Anil Damle

Linear Hamilton Jacobi Bellman Equations in high dimensions

The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finite-horizon, average cost, and first-exit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively.

Compressed Representation of Kohn-Sham Orbitals via Selected Columns of the Density Matrix.

Given a set of Kohn-Sham orbitals from an insulating system, we present a simple, robust, efficient, and highly parallelizable method to construct a set of optionally orthogonal, localized basis functions for the associated subspace. Our method explicitly uses the fact that density matrices associated with insulating systems decay exponentially along the off-diagonal direction in the real space representation. We avoid the usage of an optimization procedure, and the localized basis functions are constructed directly from a set of selected columns of the density matrix (SCDM). Consequently, the core portion of our localization procedure is not dependent on any adjustable parameters. The only adjustable parameters present pertain to the use of the SCDM after their computation (for example, at what value should the SCDM be truncated). Our method can be used in any electronic structure software package with an arbitrary basis set. We demonstrate the numerical accuracy and parallel scalability of the SCDM procedure using orbitals generated by the Quantum ESPRESSO software package. We also demonstrate a procedure for combining the orthogonalized SCDM with Hockneys algorithm to efficiently perform Hartree-Fock exchange energy calculations with near-linear scaling.

Fast Spatial Gaussian Process Maximum Likelihood Estimation via Skeletonization Factorizations

Maximum likelihood estimation for parameter fitting given observations from a Gaussian process in space is a computationally demanding task that restricts the use of such methods to moderately sized datasets. We present a framework for unstructured observations in two spatial dimensions that allows for evaluation of the log-likelihood and its gradient (i.e., the score equations) in

A Technique for Updating Hierarchical Skeletonization-Based Factorizations of Integral Operators

\tilde O(n^{3/2})

Simultaneous numerical optimization for data association and parameter estimation

time under certain assumptions, where

A Recursive Skeletonization Factorization Based on Strong Admissibility

n

Pole Expansion for Solving a Type of Parametrized Linear Systems in Electronic Structure Calculations

is the number of observations. Our method relies on the skeletonization procedure described by Martinsson and Rokhlin [J. Comput. Phys., 205 (2005), pp. 1--23] in the form of the recursive skeletonization factorization of Ho and Ying [Comm. Pure Appl. Math., 69 (2015), pp. 1415--1451]. Combining this with an adaptation of the matrix peeling algorithm of Lin, Lu, and Ying [J. Comput. Phys., 230 (2011), pp, 4071--4087] for constructing

A Geometric Approach to Archetypal Analysis and Nonnegative Matrix Factorization

\mathcal{H}

Computing Localized Representations of the Kohn--Sham Subspace Via Randomization and Refinement

-matrix representations of black-box operators, we obtain a framework that can be used in the context of any first-order o...

Wireless pressure sensing rock climbing handhold and dynamic method of customized routing

We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and update the initial factorization to reflect this change with asymptotic complexity that is poly-logarithmic in the total number of unknowns and linear in the number of perturbed unknowns. We apply our method to the recursive skeletonization factorization and hierarchical interpolative factorization and demonstrate scaling results for a number of different two-dimensional (2D) problem setups.