File:NewtonHook.f90: Difference between revisions

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The Newton-Krylov method

The code above implements the Newton-Raphson-Krylov method for solving \[\vec{F}(\vec{x})=\vec{0},\] where $\vec{x}$ and $\vec{F}$ are $n$-vectors, using a Hookstep-Trust-region approach.

This is a powerful method that can solve for $\vec{x}$ for complex $\vec{F}(\vec{x})$. E.g., for a periodic orbit, let $\vec{F}(\vec{x}) = \vec{X}(\vec{x})-\vec{x}$, where $\vec{X}(\vec{x})$ is the result of time integration of an initial condition $\vec{x}$.

For each iteration, $\vec{x}_{i+1} = \vec{x}_i + \vec{\delta x}$, the linear problem \[\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}\,\vec{\delta x}=-\vec{F}(\vec{x}_i) \] needs to be solved for the step $\vec{\delta x}$ subject to the condition that it be smaller than some magnitude $\delta$, where $\delta$ is the size of the trust region. This problem is in the form $A\vec{x}=\vec{b}$, where $A$ is an $n\times n$ matrix, and is solved using the Krylov-subspace method, GMRES(m). See File:GMRESm.f90.

To perform the GMRES(m) operations, for a given $\vec{\delta x}$, the product $\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}\,\vec{\delta x}$ may be approximated, by e.g. $\frac{1}{\epsilon}(\vec{F}(\vec{x}_i+\epsilon\,\vec{\delta x})-\vec{F}(\vec{x}_i))$ for some small value $\epsilon$. Thus, the method does need to know the matrix $\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}$ itself, only a routine for calculating $\vec{F}(\vec{x})$ is required.

Note that provided that each step $\vec{\delta x}$ of the Newton method is essentially in the correct direction, the method is expected to converge. Thus the tolerance specified in the accuracy of the solution for $\vec{\delta x}$, calculated via the GMRES method, typically need not be so stringent as the tolerance placed on the Newton method for the solution $\vec{x}$.

For further details of the method, see e.g. Newton-Krylov-Hookstep (channelflow.org).

The code

To download, click the link above. The code also requires File:GMRESm.f90 and Lapack.

• Prior to calling the subroutine newtonhook(...), the arrays new_x(:)$\equiv\vec{x}_i$ and new_fx(:)$\equiv\vec{F}(\vec{x}_i)$ need to be allocated the dimension $n$.
• Also prior to calling newtonhook(...), new_x(:) must be assigned the initial guess $\vec{x}_0$.
• In addition to scalar and array variables, newtonhook(...) needs to be passed
  • a function that calculates the dot product of two vectors,
  • a function that calculates $\vec{F}(\vec{x})$ for a given $\vec{\delta x}$,
  • a function that calculates $\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}\,\vec{\delta x}$ for a given $\vec{\delta x}$. See possible approximation above.
  • a subroutine that replaces a vector $\vec{x}$ with the solution of $M\vec{x}'=\vec{x}$ for $\vec{x}'$, where $M$ is a preconditioner matrix. This may simply be an empty subroutine if no preconditioner is required, i.e. $M=I$.
  • a subroutine that is called at the end of each iteration. This may be used to save the current state after each iteration, if desired.
• The functions above may require auxiliary data to $\vec{x}$. Place this data in a module and access via 'use mymodule' in the function/subroutine.

• As the calculation proceeds, the vector new_x(:) is updated with improved vectors, having smaller corresponding new_f(:).

Parallel use

No special adaptations are required for parallel (MPI) use -- let each thread pass its subsection for the vector sv, and let the dotprod function allreduce the result of the dot product.