164 research outputs found
Analysis of measurement errors for a superconducting phase qubit
We analyze several mechanisms leading to errors in a course of measurement of
a superconducting flux-biased phase qubit. Insufficiently long measurement
pulse may lead to nonadiabatic transitions between qubit states and
, before tunneling through a reduced barrier is supposed to distinguish
the qubit states. Finite (though large) ratio of tunneling rates for these
states leads to incomplete discrimination between and .
Insufficiently fast energy relaxation after the tunneling of state may
cause the repopulation of the quantum well in which only the state is
supposed to remain. We analyze these types of measurement errors using
analytical approaches as well as numerical solution of the time-dependent
Schr\"{o}dinger equation.Comment: 14 pages, 14 figure
Robust multi-fidelity design of a micro re-entry unmanned space vehicle
This article addresses the preliminary robust design of a small-scale re-entry unmanned space vehicle by means of a hybrid optimization technique. The approach, developed in this article, closely couples an evolutionary multi-objective algorithm with a direct transcription method for optimal control problems. The evolutionary part handles the shape parameters of the vehicle and the uncertain objective functions, while the direct transcription method generates an optimal control profile for the re-entry trajectory. Uncertainties on the aerodynamic forces and characteristics of the thermal protection material are incorporated into the vehicle model, and a Monte-Carlo sampling procedure is used to compute relevant statistical characteristics of the maximum heat flux and internal temperature. Then, the hybrid algorithm searches for geometries that minimize the mean value of the maximum heat flux, the mean value of the maximum internal temperature, and the weighted sum of their variance: the evolutionary part handles the shape parameters of the vehicle and the uncertain functions, while the direct transcription method generates the optimal control profile for the re-entry trajectory of each individual of the population. During the optimization process, artificial neural networks are utilized to approximate the aerodynamic forces required by the optimal control solver. The artificial neural networks are trained and updated by means of a multi-fidelity approach: initially a low-fidelity analytical model, fitted on a waverider type of vehicle, is used to train the neural networks, and through the evolution a mix of analytical and computational fluid dynamic, high-fidelity computations are used to update it. The data obtained by the high-fidelity model progressively become the main source of updates for the neural networks till, near the end of the optimization process, the influence of the data obtained by the analytical model is practically nullified. On the basis of preliminary results, the adopted technique is able to predict achievable performance of the small spacecraft and the requirements in terms of thermal protection materials
An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems
We study Bayesian inference methods for solving linear inverse problems,
focusing on hierarchical formulations where the prior or the likelihood
function depend on unspecified hyperparameters. In practice, these
hyperparameters are often determined via an empirical Bayesian method that
maximizes the marginal likelihood function, i.e., the probability density of
the data conditional on the hyperparameters. Evaluating the marginal
likelihood, however, is computationally challenging for large-scale problems.
In this work, we present a method to approximately evaluate marginal likelihood
functions, based on a low-rank approximation of the update from the prior
covariance to the posterior covariance. We show that this approximation is
optimal in a minimax sense. Moreover, we provide an efficient algorithm to
implement the proposed method, based on a combination of the randomized SVD and
a spectral approximation method to compute square roots of the prior covariance
matrix. Several numerical examples demonstrate good performance of the proposed
method
A Scheme to Numerically Evolve Data for the Conformal Einstein Equation
This is the second paper in a series describing a numerical implementation of
the conformal Einstein equation. This paper deals with the technical details of
the numerical code used to perform numerical time evolutions from a "minimal"
set of data.
We outline the numerical construction of a complete set of data for our
equations from a minimal set of data. The second and the fourth order
discretisations, which are used for the construction of the complete data set
and for the numerical integration of the time evolution equations, are
described and their efficiencies are compared. By using the fourth order scheme
we reduce our computer resource requirements --- with respect to memory as well
as computation time --- by at least two orders of magnitude as compared to the
second order scheme.Comment: 20 pages, 12 figure
Quantum Collapse of a Small Dust Shell
The full quantum mechanical collapse of a small relativistic dust shell is
studied analytically, asymptotically and numerically starting from the exact
finite dimensional classical reduced Hamiltonian recently derived by
H\'aj{\'\i}\v{c}ek and Kucha\v{r}. The formulation of the quantum mechanics
encounters two problems. The first is the multivalued nature of the Hamiltonian
and the second is the construction of an appropriate self adjoint momentum
operator in the space of the shell motion which is confined to a half line. The
first problem is solved by identifying and neglecting orbits of small action in
order to obtain a single valued Hamiltonian. The second problem is solved by
introducing an appropriate lapse function. The resulting quantum mechanics is
then studied by means of analytical and numerical techniques. We find that the
region of total collapse has very small probability. We also find that the
solution concentrates around the classical Schwarzschild radius. The present
work obtains from first principles a quantum mechanics for the shell and
provides numerical solutions, whose behavior is explained by a detailed WKB
analysis for a wide class of collapsing shells.Comment: 23 pages, 8 figures, Revtex4 fil
A unified approach for the solution of the Fokker-Planck equation
This paper explores the use of a discrete singular convolution algorithm as a
unified approach for numerical integration of the Fokker-Planck equation. The
unified features of the discrete singular convolution algorithm are discussed.
It is demonstrated that different implementations of the present algorithm,
such as global, local, Galerkin, collocation, and finite difference, can be
deduced from a single starting point. Three benchmark stochastic systems, the
repulsive Wong process, the Black-Scholes equation and a genuine nonlinear
model, are employed to illustrate the robustness and to test accuracy of the
present approach for the solution of the Fokker-Planck equation via a
time-dependent method. An additional example, the incompressible Euler
equation, is used to further validate the present approach for more difficult
problems. Numerical results indicate that the present unified approach is
robust and accurate for solving the Fokker-Planck equation.Comment: 19 page
Numerical Approximations Using Chebyshev Polynomial Expansions
We present numerical solutions for differential equations by expanding the
unknown function in terms of Chebyshev polynomials and solving a system of
linear equations directly for the values of the function at the extrema (or
zeros) of the Chebyshev polynomial of order N (El-gendi's method). The
solutions are exact at these points, apart from round-off computer errors and
the convergence of other numerical methods used in connection to solving the
linear system of equations. Applications to initial value problems in
time-dependent quantum field theory, and second order boundary value problems
in fluid dynamics are presented.Comment: minor wording changes, some typos have been eliminate
A multidomain approach of the Fourier pseudospectral method using discontinuous grid for elastic wave modeling
Nonlinear electrochemical relaxation around conductors
We analyze the simplest problem of electrochemical relaxation in more than
one dimension - the response of an uncharged, ideally polarizable metallic
sphere (or cylinder) in a symmetric, binary electrolyte to a uniform electric
field. In order to go beyond the circuit approximation for thin double layers,
our analysis is based on the Poisson-Nernst-Planck (PNP) equations of dilute
solution theory. Unlike most previous studies, however, we focus on the
nonlinear regime, where the applied voltage across the conductor is larger than
the thermal voltage. In such strong electric fields, the classical model
predicts that the double layer adsorbs enough ions to produce bulk
concentration gradients and surface conduction. Our analysis begins with a
general derivation of surface conservation laws in the thin double-layer limit,
which provide effective boundary conditions on the quasi-neutral bulk. We solve
the resulting nonlinear partial differential equations numerically for strong
fields and also perform a time-dependent asymptotic analysis for weaker fields,
where bulk diffusion and surface conduction arise as first-order corrections.
We also derive various dimensionless parameters comparing surface to bulk
transport processes, which generalize the Bikerman-Dukhin number. Our results
have basic relevance for double-layer charging dynamics and nonlinear
electrokinetics in the ubiquitous PNP approximation.Comment: 25 pages, 17 figures, 4 table
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