The Role of Exact Conditions in TDDFT

This chapter is devoted to exact conditions in time-dependent density functional theory. Many conditions have been derived for the exact ground-state density functional, and several have played crucial roles in the construction of popular approximations. We believe that the reliability of the most fundamental approximation of any density functional theory, the local density approximation (LDA), is due to the exact conditions that it satisfies. Improved approximations should satisfy at least those conditions that LDA satisfies, plus others. (Which others is part of the art of functional approximation). In the time-dependent case, as we shall see, the adiabatic LDA (ALDA) plays the same role as LDA in the ground-state case, as it satisfies many exact conditions. But we do not have a generally applicable improvement beyond ALDA that includes nonlocality in time. For TDDFT, we have a surfeit of exact conditions, but that only makes finding those that are useful to impose an even more demanding task.Comment: 13 pages, 3 figure

Kernel Controllers: A Systems-Theoretic Approach for Data-Driven Modeling and Control of Spatiotemporally Evolving Processes

We consider the problem of modeling, estimating, and controlling the latent state of a spatiotemporally evolving continuous function using very few sensor measurements and actuator locations. Our solution to the problem consists of two parts: a predictive model of functional evolution, and feedback based estimator and controllers that can robustly recover the state of the model and drive it to a desired function. We show that layering a dynamical systems prior over temporal evolution of weights of a kernel model is a valid approach to spatiotemporal modeling that leads to systems theoretic, control-usable, predictive models. We provide sufficient conditions on the number of sensors and actuators required to guarantee observability and controllability. The approach is validated on a large real dataset, and in simulation for the control of spatiotemporally evolving function

Kinetic energy density functionals for non-periodic systems

Kinetic energy functionals of the electronic density are used to model large systems in the context of density functional theory, without the need to obtain electronic wavefunctions. We discuss the problems associated with the application of widely used kinetic energy functionals to non-periodic systems. We develop a method that circumvents this difficulty and allows the kinetic energy to be evaluated entirely in real space. We demonstrate that the method is efficient [O(N)] and accurate by comparing the results of our real-space formulation to calculations performed in reciprocal space, and to calculations using traditional approaches based on electronic states.Comment: 12 pages, 4 figure

Learning Output Kernels for Multi-Task Problems

Simultaneously solving multiple related learning tasks is beneficial under a variety of circumstances, but the prior knowledge necessary to correctly model task relationships is rarely available in practice. In this paper, we develop a novel kernel-based multi-task learning technique that automatically reveals structural inter-task relationships. Building over the framework of output kernel learning (OKL), we introduce a method that jointly learns multiple functions and a low-rank multi-task kernel by solving a non-convex regularization problem. Optimization is carried out via a block coordinate descent strategy, where each subproblem is solved using suitable conjugate gradient (CG) type iterative methods for linear operator equations. The effectiveness of the proposed approach is demonstrated on pharmacological and collaborative filtering data

Excited states from time-dependent density functional theory

Time-dependent density functional theory (TDDFT) is presently enjoying enormous popularity in quantum chemistry, as a useful tool for extracting electronic excited state energies. This article explains what TDDFT is, and how it differs from ground-state DFT. We show the basic formalism, and illustrate with simple examples. We discuss its implementation and possible sources of error. We discuss many of the major successes and challenges of the theory, including weak fields, strong fields, continuum states, double excitations, charge transfer, high harmonic generation, multiphoton ionization, electronic quantum control, van der Waals interactions, transport through single molecules, currents, quantum defects, and, elastic electron-atom scattering.Comment: 38 pages, 17 figures and 11 tables. Submitted to Reviews of Computational Chemistry. Caution: Large Fil

Next generation extended Lagrangian first principles molecular dynamics

Extended Lagrangian Born-Oppenheimer molecular dynamics [Phys. Rev. Lett., ${\bf 100}$, 123004 (2008)] is formulated for general Hohenberg-Kohn density functional theory and compared to the extended Lagrangian framework of first principles molecular dynamics by Car and Parrinello [Phys. Rev. Lett. ${\bf 55}$, 2471 (1985)]. It is shown how extended Lagrangian Born-Oppenheimer molecular dynamics overcomes several shortcomings of regular, direct Born-Oppenheimer molecular dynamics, while improving or maintaining important features of Car-Parrinello simulations. The accuracy of the electronic degrees of freedom in extended Lagrangian Born-Oppenheimer molecular dynamics, with respect to the exact Born-Oppenheimer solution, is of second order in the size of the integration time step and of fourth order in the potential energy surface. Improved stability over recent formulations of extended Lagrangian Born-Oppenheimer molecular dynamics is achieved by generalizing the theory to finite temperature ensembles, using fractional occupation numbers in the calculation of the inner-product kernel of the extended harmonic oscillator that appears as a preconditioner in the electronic equations of motion. Materials systems that normally exhibit slow self-consistent field convergence can be simulated using integration time steps of the same order as in direct Born-Oppenheimer molecular dynamics, but without the requirement of an iterative, non-linear electronic ground state optimization prior to the force evaluations and without a systematic drift in the total energy. In combination with proposed low-rank and on-the-fly updates of the kernel, this formulation provides an efficient and general framework for quantum based Born-Oppenheimer molecular dynamics simulations.Comment: 18 pages, 10 figure

The Random Feature Model for Input-Output Maps between Banach Spaces

Well known to the machine learning community, the random feature model, originally introduced by Rahimi and Recht in 2008, is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional, function space viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation

The Generalized Fractional Calculus of Variations

We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental and isoperimetric problems, natural boundary conditions, and Noether type theorems.Comment: This is a preprint of a paper whose final and definite form will appear in Southeast Asian Bulletin of Mathematics (2014

A Unified SVM Framework for Signal Estimation

This paper presents a unified framework to tackle estimation problems in Digital Signal Processing (DSP) using Support Vector Machines (SVMs). The use of SVMs in estimation problems has been traditionally limited to its mere use as a black-box model. Noting such limitations in the literature, we take advantage of several properties of Mercer's kernels and functional analysis to develop a family of SVM methods for estimation in DSP. Three types of signal model equations are analyzed. First, when a specific time-signal structure is assumed to model the underlying system that generated the data, the linear signal model (so called Primal Signal Model formulation) is first stated and analyzed. Then, non-linear versions of the signal structure can be readily developed by following two different approaches. On the one hand, the signal model equation is written in reproducing kernel Hilbert spaces (RKHS) using the well-known RKHS Signal Model formulation, and Mercer's kernels are readily used in SVM non-linear algorithms. On the other hand, in the alternative and not so common Dual Signal Model formulation, a signal expansion is made by using an auxiliary signal model equation given by a non-linear regression of each time instant in the observed time series. These building blocks can be used to generate different novel SVM-based methods for problems of signal estimation, and we deal with several of the most important ones in DSP. We illustrate the usefulness of this methodology by defining SVM algorithms for linear and non-linear system identification, spectral analysis, nonuniform interpolation, sparse deconvolution, and array processing. The performance of the developed SVM methods is compared to standard approaches in all these settings. The experimental results illustrate the generality, simplicity, and capabilities of the proposed SVM framework for DSP.Comment: 22 pages, 13 figures. Digital Signal Processing, 201

Output-Feedback Stabilization of the Korteweg-de Vries Equation

The present paper develops boundary output-feedback stabilization of the Korteweg-de Vries (KdV) equation with sensors and an actuator located at different boundaries (anti collocated set-up) using backstepping method. The feedback control law and output injection gains are found using the backstepping method for linear KdV equation. The proof of stability is based on construction of a strict Lyapunov functional which includes the observer states. A numerical simulation is presented to validate the result