Reference.Programs.qm_truncate

qm_truncate: Truncate states versus averaging

Upon balancing transformation, the obtained Hankel singular values (HSVs) simultaneously(!) indicate controllability and observability, and the resulting modes are ordered accordingly. To reduce dimensionality, there are two methods currently implemented:

(simple) Truncatation:
In simple tuncation one simply eliminates the low HSV modes because they are weakly controllable and weakly observable at the same time.

Singular perturbation theory:
The idea of confinement truncation is based on the analogy between large HSV-modes with slow dof's and low HSV-modes with fast dof's. Then an averaging principle ( based on singular perturbation theory) can be used to derive equations of motion for the former one, where the latter one is confined to its average, or rather, the t → ∞ limit.

A = A11 - A12 A22 \ A21
N = A11 - N12 A22 \ A21
C = C1 - C2 A22 \ A21

see work by Carsten Hartmann et al. ... add references ...

Source code

The MATLAB function qm_truncate.m can be found here

Note that qm_truncate.m works only with objects of the main classes given in the following table:

class name	description
ket	state vecors, in eigen/energy representation
rho	density matrices, in eigen/energy representation

Input parameters

The following input parameters are required when calling the function qm_truncate.m

parameter	description
method	't': truncation 's': singular perturbation theory
dim	dimensionality of truncated / reduced model equations

File I/O

As input this function requires matrices A,B,N,C,D along with vectors x_i, x_e etc in balanced representation which are read from input data file ket_bal_0.mat or rho_bal_0.mat, for TDSE or LvNE simulations, respectively, typically generated using our function qm_balance.

As output this function produces truncated matrices A,B,N,C,D along with vectors x_i, x_e etc which is written to output data file tdse_xn_0.mat or lvne_xn_0.mat for TDSE or LvNE simulations, respectively, where n indicates the reduced dimension after truncation and where x will be t or s for truncation or singular perturbation theory, respectively. This can subsequently be used as input for the bilinear control problem in reduced dimensionality by running qm_propa and/or its optimal control variant implemented in qm_optimal .

Algorithms

For more details on the balancing transformation and subsequent truncation, see this page of our WavePacket central wiki. In particular, there are two options for performing the dimension reduction:

• Simple truncation of low HSV modes

• Averaging principle (based on singular perturbation theory)

Variables

Note that function qm_init should be run previously in order to initialize all variables in use. Of particular interest are the following structures:

• state

References

1. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: J. Chem. Phys. 135, 014112 (2011)