504 research outputs found

    WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction

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    WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm

    The automatic solution of partial differential equations using a global spectral method

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    A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 22, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree (nx,ny)(n_x,n_y) is computed in O((nxny)3/2)\mathcal{O}((n_x n_y)^{3/2}) operations. Partial differential operators of splitting rank 3\geq 3 are solved via a linear system involving a block-banded matrix in O(min(nx3ny,nxny3))\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3})) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under 6060 seconds. An experimental implementation in the Julia language can currently perform the same solve in 1010 seconds.Comment: 22 page

    Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations

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    Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples

    A gradient-based iterative algorithm for solving coupled Lyapunov equations of continuous-time Markovian jump systems

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    In this paper, a new gradient-based iterative algorithm is proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. A necessary and sufficient condition is established for the proposed gradient-based iterative algorithm to be convergent. In addition, the optimal value of the tunable parameter achieving the fastest convergence rate of the proposed algorithm is given explicitly. Finally, some numerical simulations are given to validate the obtained theoretical results

    On the numerical solution of a class of systems of linear matrix equations

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    We consider the solution of systems of linear matrix equations in two or three unknown matrices. For dense problems we derive algorithms that determine the numerical solution by only involving matrices of the same size as those in the original problem, thus requiring low computational resources. For large and structured systems we show how the problem properties can be exploited to design effective algorithms with low memory and operation requirements. Numerical experiments illustrate the performance of the new methods

    Multiparameter spectral analysis for aeroelastic instability problems

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    This paper presents a novel application of multiparameter spectral theory to the study of structural stability, with particular emphasis on aeroelastic flutter. Methods of multiparameter analysis allow the development of new solution algorithms for aeroelastic flutter problems; most significantly, a direct solver for polynomial problems of arbitrary order and size, something which has not before been achieved. Two major variants of this direct solver are presented, and their computational characteristics are compared. Both are effective for smaller problems arising in reduced-order modelling and preliminary design optimization. Extensions and improvements to this new conceptual framework and solution method are then discussed.Comment: 20 pages, 8 figure

    Toward Solution of Matrix Equation X=Af(X)B+C

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    This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation X=Af(X)B+CX=Af(X) B+C with f(X)=XT,f(X) =X^{\mathrm{T}}, f(X)=Xˉf(X) =\bar{X} and f(X)=XH,f(X) =X^{\mathrm{H}}, where XX is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of W=AWB+CW=\mathcal{A}W\mathcal{B}+\mathcal{C} where the dimensions of the coefficient matrices A,B\mathcal{A},\mathcal{B} and C\mathcal{C} are the same as those of the original equation. Closed-form solutions of equation X=Af(X)B+CX=Af(X) B+C can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation equation X=Af(X)B+CX=Af(X) B+C. Necessary and sufficient conditions are established to guarantee the convergence of the iterations

    Computing singularities of perturbation series

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    Many properties of current \emph{ab initio} approaches to the quantum many-body problem, both perturbational or otherwise, are related to the singularity structure of Rayleigh--Schr\"odinger perturbation theory. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the perturbed Hamiltonian on a vector, and does not rely on the terms in the perturbation series. Some illustrative model problems are studied, including a Helium-like model with δ\delta-function interactions for which M{\o}ller--Plesset perturbation theory is considered and the radius of convergence found.Comment: 11 figures, submitte
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