Discontinuous Galerkin method for Navier-Stokes equations using kinetic flux vector splitting

Kinetic schemes for compressible flow of gases are constructed by exploiting the connection between Boltzmann equation and the Navier-Stokes equations. This connection allows us to construct a flux splitting for the Navier-Stokes equations based on the direction of molecular motion from which a numerical flux can be obtained. The naive use of such a numerical flux function in a discontinuous Galerkin (DG) discretization leads to an unstable scheme in the viscous dominated case. Stable schemes are constructed by adding additional terms either in a symmetric or non-symmetric manner which are motivated by the DG schemes for elliptic equations. The novelty of the present scheme is the use of kinetic fluxes to construct the stabilization terms. In the symmetric case, interior penalty terms have to be added for stability and the resulting schemes give optimal convergence rates in numerical experiments. The non-symmetric schemes lead to a cell energy/entropy inequality but exhibit sub-optimal convergence rates. These properties are studied by applying the schemes to a scalar convection-diffusion equation and the 1-D compressible Navier-Stokes equations. In the case of Navier-Stokes equations, entropy variables are used to construct stable schemes

Two-state quantum walk on two- and three-dimensional lattices

We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.Comment: 7 pages, 5 figures, revised versio

Disorder induced localization and enhancement of entanglement in one- and two-dimensional quantum walks

The time evolution of one- and two-dimensional discrete-time quantum walk with increase in disorder is studied. We use spatial, temporal and spatio-temporal broken periodicity of the unitary evolution as disorder to mimic the effect of disordered/random medium in our study. Disorder induces a dramatic change in the interference pattern leading to localization of the quantum walks in one- and two-dimensions. Spatial disorder results in the decreases of the particle and position entanglement in one-dimension and counter intuitively, an enhancement in entanglement with temporal and spatio-temporal disorder is seen. The study signifies that the Anderson localization of quantum state without compromising on the degree of entanglement could be implement in a large variety of physical settings where quantum walks has been realized. The study presented here could make it feasible to explore, theoretically and experimentally the interplay between disorder and entanglement. This also brings up a variety of intriguing questions relating to the negative and positive implications on algorithmic and other applications.Comment: 13 Pages, 6 Figures, appendix included in the revised versio

Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices

From the unitary operator used for implementing two-state discrete-time quantum walk on one-, two- and three- dimensional lattice we obtain a two-component Dirac-like Hamiltonian. In particular, using different pairs of Pauli basis as position translation states we obtain three different form of Hamiltonians for evolution on one-dimensional lattice. We extend this to two- and three-dimensional lattices using different Pauli basis states as position translation states for each dimension and show that the external coin operation, which is necessary for one-dimensional walk is not a necessary requirement for a walk on higher dimensions but can serve as an additional resource to control the dynamics. The two-component Hamiltonian we present here for quantum walk on different lattices can serve as a general framework to simulate, control, and study the dynamics of quantum systems governed by Dirac-like Hamiltonian.Comment: 14 pages, 5 figures ; Published version ; It includes some parts of arxiv:1103.270

Dipole trap for 87 Rb atoms using lasers of different wavelength

The parity of atomic wave functions prevents neutral atoms from having permanent electric-dipole moment. Electric-dipole moment is induced in an atom when exposed to strong light, the electric field of the light. Hence the optical trapping of neutral atoms relies on the induced dipole moment. Here we present the calculated numerical values of the detuning, potential depth, minimum laser power required to trap $^{87}$Rb ($D_2$ line) atoms using lasers of wavelength 1064 nm, 850 nm, 820 nm and 800 nm and beam waists 50$\mu$m, 100$\mu$m and 200$\mu$m.Comment: 2 pages, 4 table

Discrete time quantum walk model for single and entangled particles to retain entanglement in coin space

In most widely discussed discrete time quantum walk model, after every unitary shift operator, the particle evolves into the superposition of position space and settles down in one of its basis states, loosing entanglement in the coin space in the new position. The Hadamard operation is applied to let the particle to evolve into the superposition in the coin space and the walk is iterated. We present a model with a additional degree of freedom for the unitary shift operator $U^{\prime}$. The unitary operator with additional degree of freedom will evolve the quantum particle into superposition of position space retaining the entanglement in coin space. This eliminates the need for quantum coin toss (Hadamard operation) after every unitary displacement operation as used in most widely studied version of the discrete time quantum walk model. This construction is easily extended to a multiple particle quantum walk and in this article we extend it for a pair of particles in pure state entangled in coin degree of freedom by simultaneously subjecting it to a pair of unitary displacement operators which were constructed for single particle. We point out that unlike for single particle quantum walk, upon measurement of its position after $N$ steps, the entangled particles are found together with 1/2 probability and at different positions with 1/2 probability. This can act as an advantage in applications of the quantum walk. A special case is also treated using a complex physical system such as, inter species two-particle entangled Bose-Einstein condensate, as an example.Comment: 9 pages, 3 figure

Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution

On the Self-Dual Geometry of N=2 Strings

We discuss the precise relation of the open N=2 string to a self-dual Yang-Mills (SDYM) system in 2+2 dimensions. In particular, we review the description of the string target space action in terms of SDYM in a ``picture hyperspace'' parametrised by the standard vectorial R^{2,2} coordinate together with a commuting spinor of SO(2,2). The component form contains an infinite tower of prepotentials coupled to the one representing the SDYM degree of freedom. The truncation to five fields yields a novel one-loop exact lagrangean field theory.Comment: 6 pages, LaTeX; contribution to the 31st Ahrenshoop International Symposium on the Theory of Elementary Particles, Buckow, Germany, 2-6 September 199

Approximate dynamic programming with (min⁡,+)(\min,+) linear function approximation for Markov decision processes

Markov Decision Processes (MDP) is an useful framework to cast optimal sequential decision making problems. Given any MDP the aim is to find the optimal action selection mechanism i.e., the optimal policy. Typically, the optimal policy ($u^*$) is obtained by substituting the optimal value-function ($J^*$) in the Bellman equation. Alternately $u^*$ is also obtained by learning the optimal state-action value function $Q^*$ known as the $Q$ value-function. However, it is difficult to compute the exact values of $J^*$ or $Q^*$ for MDPs with large number of states. Approximate Dynamic Programming (ADP) methods address this difficulty by computing lower dimensional approximations of $J^*$/$Q^*$. Most ADP methods employ linear function approximation (LFA), i.e., the approximate solution lies in a subspace spanned by a family of pre-selected basis functions. The approximation is obtain via a linear least squares projection of higher dimensional quantities and the $L_2$ norm plays an important role in convergence and error analysis. In this paper, we discuss ADP methods for MDPs based on LFAs in $(\min,+)$ algebra. Here the approximate solution is a $(\min,+)$ linear combination of a set of basis functions whose span constitutes a subsemimodule. Approximation is obtained via a projection operator onto the subsemimodule which is different from linear least squares projection used in ADP methods based on conventional LFAs. MDPs are not $(\min,+)$ linear systems, nevertheless, we show that the monotonicity property of the projection operator helps us to establish the convergence of our ADP schemes. We also discuss future directions in ADP methods for MDPs based on the $(\min,+)$ LFAs.Comment: 16 pages, 2 figure

Linear Stochastic Approximation: Constant Step-Size and Iterate Averaging

We consider $d$-dimensional linear stochastic approximation algorithms (LSAs) with a constant step-size and the so called Polyak-Ruppert (PR) averaging of iterates. LSAs are widely applied in machine learning and reinforcement learning (RL), where the aim is to compute an appropriate $\theta_{*} \in \mathbb{R}^d$ (that is an optimum or a fixed point) using noisy data and $O(d)$ updates per iteration. In this paper, we are motivated by the problem (in RL) of policy evaluation from experience replay using the \emph{temporal difference} (TD) class of learning algorithms that are also LSAs. For LSAs with a constant step-size, and PR averaging, we provide bounds for the mean squared error (MSE) after $t$ iterations. We assume that data is \iid with finite variance (underlying distribution being $P$) and that the expected dynamics is Hurwitz. For a given LSA with PR averaging, and data distribution $P$ satisfying the said assumptions, we show that there exists a range of constant step-sizes such that its MSE decays as $O(\frac{1}{t})$. We examine the conditions under which a constant step-size can be chosen uniformly for a class of data distributions $\mathcal{P}$, and show that not all data distributions `admit' such a uniform constant step-size. We also suggest a heuristic step-size tuning algorithm to choose a constant step-size of a given LSA for a given data distribution $P$. We compare our results with related work and also discuss the implication of our results in the context of TD algorithms that are LSAs.Comment: 16 pages, 2 figures, was submitted to NIPS 201