Pseudospectral Approximation Research Articles

Optimal pulse design in quantum control: A unified computational method

Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.

Frictionless atom cooling in harmonic traps: A time-optimal approach

In this article we formulate frictionless atom cooling in harmonic traps as a time-optimal control problem, permitting imaginary values of the trap frequency for trasient time intervals during which the trap becomes an expulsive parabolic potential. We show that the minimum time solution has "bang-bang" form, where the frequency jumps suddenly at certain instants and then remains constant, and calculate estimates of the minimum cooling time for various numbers of such jumps. A numerical optimization method based on pseudospectral approximations is used to obtain suboptimal realistic solutions without discontinuities, which may be implemented experimentally.

Control of a Network of Spiking Neurons

Systems consisting of a large number of coupled or decoupled oscillators with different natural oscillation frequencies have been a subject of scientific interest. In this article, we study the problem of simultaneous control of a network of spiking neurons characterized by different natural dynamics. The evolution of such neuron systems is described by phase-reduced models. We investigate controllability properties of such neuron systems for various phase models and present a robust computational method for optimal control of neurons based on pseudospectral approximations.

An hp‐adaptive pseudospectral method for solving optimal control problems

AbstractAn hp‐adaptive pseudospectral method is presented for numerically solving optimal control problems. The method presented in this paper iteratively determines the number of segments, the width of each segment, and the polynomial degree required in each segment in order to obtain a solution to a user‐specified accuracy. Starting with a global pseudospectral approximation for the state, on each iteration the method determines locations for the segment breaks and the polynomial degree in each segment for use on the next iteration. The number of segments and the degree of the polynomial on each segment continue to be updated until a user‐specified tolerance is met. The terminology ‘ hp’ is used because the segment widths (denoted h) and the polynomial degree (denoted p) in each segment are determined simultaneously. It is found that the method developed in this paper leads to higher accuracy solutions with less computational effort and memory than is required in a global pseudospectral method. Consequently, the method makes it possible to solve complex optimal control problems using pseudospectral methods in cases where a global pseudospectral method would be computationally intractable. Finally, the utility of the method is demonstrated on a variety of problems of varying complexity. Copyright © 2010 John Wiley & Sons, Ltd.

A class Hermite pseudospectral approximate with ω(x) ≡ 1 and application to reaction-diffusion equation

In this paper, we continue the discussion of [12] to establish the Hermite pseudospectral method with weight ω(x) ≡ 1. As an application, we consider the pseudospectral approximation of the reaction-diffusion equation on the whole line, we prove the existence of the approximate attractor and give the error estimate for the approximate solution.

Flow Dynamics and Bed Resistance of Wave Propagation over Bed Ripples

The viscous, two-dimensional, free-surface flow induced by the propagation of nonlinear water waves over a rigid rippled bed was simulated numerically. The simulations were based on the numerical solution of the Navier-Stokes equations subject to fully nonlinear free-surface boundary conditions and appropriate bottom, inflow, and outflow boundary conditions. The equations were properly transformed so that the computational domain became time-independent. A hybrid scheme was used for the spatial discretization with finite differences in the streamwise direction and a pseudospectral approximation with Chebyshev polynomials in the vertical direction. A fractional time-step scheme was used for the temporal discretization. Over the rippled bed, the wave boundary layer thickness increased significantly, while vortex shedding at the ripple crest generated alternating circulation regions over the ripple trough. The velocity of the Eulerian drift profile was opposite to the direction of wave propagation far above the ripples, whereas close to the bed, its magnitude was influenced by the ripples up to a height of about six times the ripple height above the ripple crest. The amplitude of the wall shear stress on the ripples increased with increasing ripple steepness, whereas the amplitude of the corresponding friction drag force on a ripple was insensitive to this increase. The amplitude of the form drag force attributable to the dynamic pressure increased with increasing ripple steepness; therefore, the percentage of friction in the total drag force decreased with increasing ripple steepness. The period-averaged drag forces on a ripple were very weak, while the influence of form drag increased with increasing ripple steepness.

A pseudospectral method for optimal control of open quantum systems

In this paper, we present a unified computational method based on pseudospectral approximations for the design of optimal pulse sequences in open quantum systems. The proposed method transforms the problem of optimal pulse design, which is formulated as a continuous-time optimal control problem, to a finite-dimensional constrained nonlinear programming problem. This resulting optimization problem can then be solved using existing numerical optimization suites. We apply the Legendre pseudospectral method to a series of optimal control problems on open quantum systems that arise in nuclear magnetic resonance spectroscopy in liquids. These problems have been well studied in previous literature and analytical optimal controls have been found. We find an excellent agreement between the maximum transfer efficiency produced by our computational method and the analytical expressions. Moreover, our method permits us to extend the analysis and address practical concerns, including smoothing discontinuous controls as well as deriving minimum-energy and time-optimal controls. The method is not restricted to the systems studied in this article and is applicable to optimal manipulation of both closed and open quantum systems.

Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.

An efficient spectral code for incompressible flows in cylindrical geometries

An efficient and accurate numerical scheme is proposed to solve the incompressible Navier–Stokes equations in a bounded cylinder. The scheme is based on a projection method formulated in primitive variables to maintain the incompressibility constraint, with a second-order semi-implicit scheme for the time integration, and a pseudospectral approximation for the space variables. The Chebyshev-collocation method applied in the radial and axial directions, and the Fourier–Galerkin approximation used in the azimuthal direction lead to a sequence of two-dimensional Helmholtz and Poisson equations for every azimuthal coefficient that are solved by a diagonalization technique. Radial expansions are considered in the diameter of the cell in order to avoid clustering about the axis, and the number of points are selected to ensure that r = 0 is not a collocation point. A minimal number of regularity conditions are imposed implicitly at the origin by forcing the proper parity of the Fourier expansions in the radial direction. The method has been tested on analytical solutions and compared with other reliable three-dimensional results. The improvements introduced in the treatment of the spatial discretization reduce significantly the difficulty of implementation of the code, and facilitate the use of high resolutions. Different boundary conditions can also be easily implemented.

High accuracy representation of the free propagator

A new matrix approximation to the free propagator of quantum mechanics is proposed. It is a variant of the well-known Fourier pseudospectral approximation. Numerical comparisons involving the linear and nonlinear Schrödinger equations are presented.

Comparisons between pseudospectral and radial basis function derivative approximations

Fourier-based pseudospectral (PS) methods have been used since the 1970s for obtaining spectrally accurate solutions to partial differential equations (PDEs) in periodic geometries. Radial basis functions (RBFs) were introduced about the same time for interpolation on scattered nodes in irregular geometries. As was later recognized, they can also be used for accurate numerical solution of PDEs. Although the main strength of RBFs lies in their outstanding geometric flexibility, also offering possibilities of spectral accuracy over irregularly shaped finite domains, it is still of interest to compare them against Fourier-based PS methods in the extremely simple geometries (infinite or periodic domains) where the latter can also be used. Mostly by means of heuristic arguments and graphical illustrations based on Fourier analysis and numerical experiments, we show that there are notable differences (more pronounced in increasing numbers of dimensions) in how the two spectral approaches approximate derivatives.

Computing Gowdy spacetimes via spectral evolution in future and past directions

We consider a system of nonlinear wave equations with constraints that arises from the Einstein equations of general relativity and describes the geometry of the so-called Gowdy symmetric spacetimes on T3. We introduce two numerical methods, which are based on the pseudo-spectral approximation. The first approach relies on marching in the future timelike direction and toward the singularity t = 0 and is based on a novel nonlinear transformation, which allows us to reduce the nonlinear source terms to simple quadratic products of the unknown variables. The second approach is designed from asymptotic formulae that are available near this singularity, and evolves the solutions in the past timelike direction from the ‘final’ data given at t = 0. Numerical experiments are presented in various regimes, including cases where ‘spiky’ structures are observed as the singularity is approached. The proposed backward strategy leads to a robust numerical method which allows us to accurately simulate the long-time behavior of a large class of Gowdy spacetimes.

Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics

Pseudo-spectral approximations are constructed for the model equations describing the population kinetics of human tumor cells in vitro and their responses to radiotherapy or chemotherapy. These approximations are more efficient than finite-difference approximations. The spectral accuracy of the pseudo-spectral method allows us to resolve the model with a much smaller number of spatial grid-points than required for the finite-difference method to achieve comparable accuracy. This is demonstrated by numerical experiments which show a good agreement between predicted and experimental data.

Gauss-Hermite wave packet dynamics: convergence of the spectral and pseudo-spectral approximation

The time dependent linear Schrodinger equation for nuclei on the whole space is semi-discretised using Hermite and Gauss-Hermite basis functions. These are well suited on the one hand for the conservation properties of the numerical solution and, on the other hand, for their remarkable approximation properties. We investigate theoretically and numerically the convergence of the spectral and pseudo- spectral Gauss-Hermite semi-discretisation schemes. Schrodinger equation, Gauss- Hermite approximation, spectral and pseudo-spectral methods

Relative source time functions of seismic events at the Rudna copper mine, Poland: estimation of inversion uncertainties

The relative source time function (RSTF) inversion uncertainty assessment was performed for two small, mining-induced seismic events (MW=2.9 and 3.0) that occurred at Rudna copper mine in Poland. The seismograms of se- lected events were recorded by the seismic net- work composed of over 60, short-period, vertical seismometers, recording ground velocity, located in the distance ranging from 400 m up to 8 km from their hypocenters. The RSTFs were cal- culated for each seismic station independently, using the empirical Green's function technique. The pseudospectral approximation of the sought RSTF by a finite sum of Gaussian kernel func- tions was used and the inverse problem was solved with the adaptive simulated annealing algorithm. Both methods improved the stability of the deconvolution procedure and physical correctness of the final solution in comparison to the classical deconvolution methods. To estimate the inversion uncertainties, classical Markov- chain Monte-Carlo techniques were used. The uncertainty analysis allows for improved selection of a priori data to the following inversion for kinematic rupture process.

Pseudospectral time-dependent density functional theory

Time-dependent density functional theory (TDDFT) is implemented within the Tamm-Dancoff approximation (TDA) using a pseudospectral approach to evaluate two-electron repulsion integrals. The pseudospectral approximation uses a split representation with both spectral basis functions and a physical space grid to achieve a reduction in the scaling behavior of electronic structure methods. We demonstrate here that exceptionally sparse grids may be used in the excitation energy calculation, following earlier work employing the pseudospectral approximation for determining correlation energies in wavefunction-based methods with similar conclusions. The pseudospectral TDA-TDDFT method is shown to be up to ten times faster than a conventional algorithm for hybrid functionals without sacrificing chemical accuracy.

Pseudospectral integration matrix and boundary value problems

This paper presents a new spectral successive integration matrix. This matrix is used to construct a Chebyshev expansion method for the solution of boundary value problems. The method employs the pseudospectral approximation of the highest-order derivative to generate an approximation to the lower-order derivatives. Application to the linear stability problem for plane Poiseuille flow is presented. The present numerical results are in satisfactory agreement with the exact solutions.

Finite-difference preconditioners for superconsistent pseudospectral approximations

The superconsistent collocation method, which is based on a collocation grid different from the one used to represent the solution, has proven to be very accurate in the resolution of various functional equations. Excellent results can be also obtained for what concerns preconditioning. Some analysis and numerous experiments, regarding the use of finite-differences preconditioners, for matrices arising from pseudospectral approximations of advection-diffusion boundary value problems, are presented and discussed, both in the case of Legendre and Chebyshev representation nodes.

Technique for Evaluation of Profiles of a Composite Chiral Slab Through Inversion and Pseudospectral Approximation

In this paper, a technique is developed to simultaneously evaluate the permeability, permittivity and chirality parameter of a composite chiral slab where they are functions of depth of the slab. Additionally, the chirality parameter is a function of frequency. The developed technique is based on the use of the Riccati equation for the reflection coefficient associated with electromagnetic propagation through the slab and the Green functions. The technique also makes use of Nth degree interpolation polynomials to approximate the unknowns using Legendre-Gauss-Lobatto collocation nodes. A numerical example is given through which the application of the technique is utilized.

The large time error estimation of Chebyshev pseudospectral approximation for a class of semilinear parabolic equations

In this paper, we use Chebyshev pseudospectral method to analyze a class of semilinear parabolic equations. We give out the error estimation between spectral approximate solution and exact solution on large time using local continuation method.