A numerical tour of wave propagation |

The gradient-like method can be summarized as

The conjugate gradient (CG) algorithm decreases the misfit function along the conjugate gradient direction:

(81) |

There are many ways to compute :

To achieve best convergence rate, in practice we suggest to use a hybrid scheme combing Hestenes-Stiefel and Dai-Yuan:

(83) |

Iterating with Eq. (80) needs to find an appropriate . Here we provide two approaches to calculate .

Approach 1: Currently, the objective function is

(84) |

Setting gives

Approach 2: Recall that

Using the 1st-order approximation, we have

(87) |

Setting gives

In fact, Eq. (88) can also be obtained from Eq. (85) in terms of Eq. (72): .

In terms of Eq. (86), the term is computed conventionally using a 1st-order-accurate finite difference approximation of the partial derivative of :

(89) |

with a small parameter . In practice, we chose an such that

(90) |

A numerical tour of wave propagation |

2021-08-31