11,945 research outputs found
Linear Scaling Density Matrix Real Time TDDFT: Propagator Unitarity \& Matrix Truncation
Real time, density matrix based, time dependent density functional theory
proceeds through the propagation of the density matrix, as opposed to the
Kohn-Sham orbitals. It is possible to reduce the computational workload by
imposing spatial cut-off radii on sparse matrices, and the propagation of the
density matrix in this manner provides direct access to the optical response of
very large systems, which would be otherwise impractical to obtain using the
standard formulations of TDDFT. Following a brief summary of our
implementation, along with several benchmark tests illustrating the validity of
the method, we present an exploration of the factors affecting the accuracy of
the approach. In particular we investigate the effect of basis set size and
matrix truncation, the key approximation used in achieving linear scaling, on
the propagator unitarity and optical spectra. Finally we illustrate that, with
an appropriate density matrix truncation range applied, the computational load
scales linearly with the system size and discuss the limitations of the
approach.Comment: Accepted for publication in J. Chem. Phy
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
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