The EMD algorithm, first proposed in [11], made more robust as well as more versatile in [13], is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time-frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies -see, e.g. [12]. On the other hand, the EMD algorithm contains heuristic and ad-hoc elements that make it hard to analyze mathematically.In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.

Markov state models (MSMs) have become the tool of choice to analyze large amounts of molecular dynamics data by approximating them as a Markov jump process between suitably predefined states. Here we investigate "Core Set MSMs," a new type of MSMs that build on metastable core sets acting as milestones for tracing the rare event kinetics. We present a thorough analysis of Core Set MSMs based on the existing milestoning framework, Bayesian estimation methods and Transition Path Theory (TPT). We show that Core Set MSMs can be used to extract phenomenological rate constants between the metastable sets of the system and to approximate the evolution of certain key observables. The performance of Core Set MSMs in comparison to standard MSMs is analyzed and illustrated on a toy example and in the context of the torsion angle dynamics of alanine dipeptide.

Electron repulsion integral tensor has ubiquitous applications in electronic structure computations. In this work, we propose an algorithm which compresses the electron repulsion tensor into the tensor hypercontraction format with O(nN 2 log N ) computational cost, where N is the number of orbital functions and n is the number of spatial grid points that the discretization of each orbital function has. The algorithm is based on a novel strategy of density fitting using a selection of a subset of spatial grid points to approximate the pair products of orbital functions on the whole domain.

a b s t r a c tWe develop a hierarchical matrix construction algorithm using matrix-vector multiplications, based on the randomized singular value decomposition of low-rank matrices. The algorithm uses Oðlog nÞ applications of the matrix on structured random test vectors and Oðn log nÞ extra computational cost, where n is the dimension of the unknown matrix. Numerical examples on constructing Green's functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm.