504 research outputs found
WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction
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
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 , 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 is computed in
operations. Partial differential operators of
splitting rank are solved via a linear system involving a block-banded
matrix in 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 seconds. An experimental implementation in the Julia
language can currently perform the same solve in seconds.Comment: 22 page
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
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
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
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
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
This paper studies the solvability, existence of unique solution, closed-form
solution and numerical solution of matrix equation with and where 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
where the dimensions of the coefficient
matrices and are the same as those of
the original equation. Closed-form solutions of equation 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
. Necessary and sufficient conditions are established to guarantee
the convergence of the iterations
Computing singularities of perturbation series
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
-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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