krotov.second_order module

Support functions for the second-order update equation

Summary

Classes:

Sigma

Function σ(t) for the second order update equation.

Functions:

numerical_estimate_A

Update the second-order parameter \(A\).

__all__: Sigma, numerical_estimate_A

Reference

class krotov.second_order.Sigma[source]

Bases: abc.ABC

Function σ(t) for the second order update equation.

This is an abstract bases class. For any optimization that requires the second-order update equation, an appropriate problem-specific subclass of Sigma must be implemented that defines

  • the evaluation of σ(t) in __call__()

  • the update of any values that σ(t) depends on parametrically (typically: any of the parameters A, B, C), in refresh().

An instantiation of that subclass is then passed as sigma to optimize_pulses().

abstract __call__(t)[source]

Evaluate σ(t)

abstract refresh(forward_states, forward_states0, chi_states, chi_norms, optimized_pulses, guess_pulses, objectives, result)[source]

Recalculate the parametric dependencies of σ(t)

This is called at the end of each control iteration, and may be used to estimate the internal parameters in σ(t)

Parameters

  • forward_states (list) – For each objective, an array-like container (cf. storage in optimize_pulses()) of the initial state forward-propagated under optimized controls from the current iteration.

  • forward_states0 (list) – The forward-propagated states under the guess controls of the current iteration.

  • chi_states (list) – The (normalized) boundary condition for the backward-propagation in the current iteration, as returned by the chi_constructor argument to optimize_pulses().

  • chi_norms (list) – The norms of the un-normalized chi_states.

  • optimized_pulses (list[numpy.ndarray]) – from the current iteration

  • guess_pulses (list[numpy.ndarray]) – current iteration

  • objectives (list[Objective]) – The control objectives

  • result (Result) – The result object, up-to-date for the current iteration

krotov.second_order.numerical_estimate_A(forward_states, forward_states0, chi_states, chi_norms, Delta_J_T)[source]

Update the second-order parameter \(A\).

Calculate the new value of \(A\) according to the equation

\[A^{(i+1)} = \frac{ \sum_k 2 \Re \Braket{\chi_k(T)}{\Delta\phi_k(T)} + \Delta J_T }{ \sum_k \Braket{\Delta \phi_k(T)}{\Delta\phi_k(T)} },\]

where \(\Delta\phi_k\) is the difference of the forward_states \(\ket{\phi_k^{(i)}}\) propagated under the optimized pulse of iteration \((i)\), and the forward_states0 \(\ket{\phi_k^{(i-1)}}\) propagated under the guess pulse of iteration \((i)\) – that is, the guess pulse of iteration \((i-1)\); and \(\Delta J_T\) is the difference of the final time functional,

\[\Delta J_T = J_T(\{\ket{\phi_k^{(i)}(T)}\} - J_T(\{\ket{\phi_k^{(i-1)}(T)}\}.\]
Parameters

  • forward_states (list) – For each objective, the result of a forward-propagation with the optimized pulses of the current iteration.

  • forward_states0 (list) – For each objective, the result of a forward-propagation with the guess pulses of the current iteration

  • chi_states (list) – For each objective, the normalized boundary state \(\ket{\chi_k(T)}/\Abs{\ket{\chi_k(T)}}\) for the backward-propagation with the guess pulse of the current iteration.

  • chi_norms (list) – The norms of the chi_states

  • Delta_J_T (float) – The value by which the final time functional improved in the current iteration.