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\section{The line search optimisation algorithm |
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\label{sectionoptim}} |
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|
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\subsection{General features} |
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|
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The line search algorithm is based on a quasi-Newton |
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variable storage method which was implemented by |
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\cite{gil-mar:89}. |
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|
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TO BE CONTINUED... |
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|
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\subsection{The online vs. offline version} |
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|
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\begin{itemize} |
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% |
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\item {\bf Online version} \\ |
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Every call to {\it simul} refers to an execution of the |
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forward and adjoint model. |
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Several iterations of optimization may thus be performed within |
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a single run of the main program (lsopt\_top). |
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The following cases may occur: |
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% |
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\begin{itemize} |
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\item |
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cold start only (no optimization) |
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\item |
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cold start, followed by one or several iterations of optimization |
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\item |
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warm start from previous cold start with one or several iterations |
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\item |
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warm start from previous warm start with one or several iterations |
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\end{itemize} |
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% |
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\item {\bf Offline version} \\ |
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Every call to simul refers to a read procedure which |
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reads the result of a forward and adjoint run |
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Therefore, only one call to simul is allowed, |
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{\tt itmax = 0}, for cold start |
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{\tt itmax = 1}, for warm start |
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Also, at the end, {\bf x(i+1)} needs to be computed and saved |
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to be available for the offline model and adjoint run |
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\end{itemize} |
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|
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In order to achieve minimum difference between the online and offline code |
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{\bf xdiff(i+1)} is stored to file at the end of an (offline) iteration, |
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but recomputed identically at the beginning of the next iteration. |
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|
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\subsection{Number of iterations vs. number of simulations} |
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|
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{\tt - itmax:} controls the max. number of iterations \\ |
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{\tt - nfunc:} controls the max. number of simulations |
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within one iteration |
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|
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\paragraph{Summary} ~ \\ |
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From one iteration to the next the descent direction changes. |
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Within one iteration more than one forward and adjoint |
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run may be performed. |
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The updated control used as input for these simulations uses the same |
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descent direction, but different step sizes. |
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|
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\paragraph{Description} ~ \\ |
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From one iteration to the next the descent direction dd changes using |
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the result for the adjoint vector gg of the previous iteration. |
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In lsline the updated control |
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\[ |
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\tt |
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xdiff(i,1) = xx(i-1) + tact(i-1,1)*dd(i-1) |
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\] |
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serves as input for |
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a forward and adjoint model run yielding a new {\tt gg(i,1)}. |
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In general, the new solution passes the 1st and 2nd Wolfe tests |
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so {\tt xdiff(i,1)} represents the solution sought: |
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\[ |
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{\tt xx(i) = xdiff(i,1)} |
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\] |
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If one of the two tests fails, |
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an inter- or extrapolation is invoked to determine |
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a new step size {\tt tact(i-1,2)}. |
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If more than one function call is permitted, |
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the new step size is used together |
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with the "old" descent direction {\tt dd(i-1)} |
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(i.e. dd is not updated using the new {\tt gg(i)}), |
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to compute a new |
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\[ |
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{\tt xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1)} |
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\] |
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that serves as input |
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in a new forward and adjoint run, yielding {\tt gg(i,2)}. |
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If now, both Wolfe tests are successful, |
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the updated solution is given by |
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\[ |
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\tt |
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xx(i) = xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1) |
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\] |
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|
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In order to save memory both the fields dd and xdiff |
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have a double usage. |
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% |
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\begin{itemize} |
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% |
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\item [{\tt xdiff}] ~\\ |
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- in {\it lsopt\_top}: used as {\tt x(i) - x(i-1)} for Hessian update \\ |
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- in {\it lsline}: intermediate result for control update |
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{\tt x = x + tact*dd} |
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% |
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\item [{\tt dd}] ~\\ |
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- in {\it lsopt\_top, lsline}: descent vector, {\tt dd = -gg} |
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and {\tt hessupd} \\ |
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- in {\it dgscale}: intermediate result to compute new preconditioner |
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% |
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\end{itemize} |
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|
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\paragraph{The parameter file lsopt.par} |
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|
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% |
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\begin{itemize} |
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% |
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\item {\bf NUPDATE} |
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max. no. of update pairs {\tt (gg(i)-gg(i-1), xx(i)-xx(i-1))} |
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to be stored in {\tt OPWARMD} to estimate Hessian |
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[pair of current iter. is stored in |
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{\tt (2*jmax+2, 2*jmax+3)} |
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jmax must be > 0 to access these entries] |
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Presently {\tt NUPDATE} must be > 0 |
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(i.e. iteration without reference to previous |
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iterations through {\tt OPWARMD} has not been tested) |
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% |
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\item {\bf EPSX} |
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relative precision on xx bellow which xx should not be improved |
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% |
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\item {\bf EPSG} |
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relative precision on gg below which optimization is |
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considered successful |
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% |
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\item {\bf IPRINT} |
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controls verbose (>=1) or non-verbose output |
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% |
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\item {\bf NUMITER} |
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max. number of iterations of optimisation; |
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NUMTER = 0: cold start only, no optimization |
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% |
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\item {\bf ITER\_NUM} |
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index of new restart file to be created (not necessarily = NUMITER!) |
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% |
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\item {\bf NFUNC} |
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max. no. of simulations per iteration |
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(must be > 0); |
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is used if step size {\tt tact} is inter-/extrapolated; |
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in this case, if NFUNC > 1, a new simulation is performed with |
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same gradient but "improved" step size |
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% |
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\item {\bf FMIN} |
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first guess cost function value |
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(only used as long as first iteration not completed, |
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i.e. for jmax <= 0) |
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% |
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\end{itemize} |
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|
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\paragraph{OPWARMI, OPWARMD files} |
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Two files retain values of previous iterations which are |
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used in latest iteration to update Hessian: |
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\begin{itemize} |
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% |
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\item {\bf OPWARMI}: contains index settings and scalar variables |
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|
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{\footnotesize |
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\begin{tabular}{ll} |
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{\tt n = nn} & no. of control variables \\ |
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{\tt fc = ff} & cost value of last iteration \\ |
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{\tt isize} & no. of bytes per record in OPWARMD \\ |
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{\tt m = nupdate} & max. no. of updates for Hessian \\ |
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{\tt jmin, jmax} & pointer indices for OPWARMD file (cf. below) \\ |
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{\tt gnorm0} & norm of first (cold start) gradient gg \\ |
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{\tt iabsiter} & total number of iterations with respect to cold start |
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\end{tabular} |
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} |
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% |
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\item {\bf OPWARMD}: contains vectors (control and gradient) |
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|
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{\scriptsize |
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\begin{tabular}{cll} |
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entry & name & description \\ |
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\hline |
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1 & {\tt xx(i)} & control vector of latest iteration \\ |
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2 & {\tt gg(i)} & gradient of latest iteration \\ |
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3 & {\tt xdiff(i),diag} & preconditioning vector; (1,...,1) |
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for cold start \\ |
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2*jmax+2 & {\tt gold=g(i)-g(i-1)} & for last update (jmax) \\ |
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2*jmax+3 & {\tt xdiff=tact*d=xx(i)-xx(i-1)} & for last update (jmax) |
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\end{tabular} |
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} |
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% |
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\end{itemize} |
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% |
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\begin{figure}[b!] |
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{\footnotesize |
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\begin{verbatim} |
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|
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Example 1: jmin = 1, jmax = 3, mupd = 5 |
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|
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1 2 3 | 4 5 6 7 8 9 empty empty |
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|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___| |
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0 | 1 2 3 |
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|
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Example 2: jmin = 3, jmax = 7, mupd = 5 ---> jmax = 2 |
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|
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1 2 3 | |
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|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___| |
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| 6 7 3 4 5 |
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|
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\end{verbatim} |
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} |
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\caption{Examples of OPWARM file handling} |
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\label{fig:opwarm} |
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\end{figure} |
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|
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\paragraph{Error handling} |
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|
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|
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|
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\newpage |
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|
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\begin{figure} |
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%\input{part8/lsopt_flow_1} |
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{\scriptsize |
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\begin{verbatim} |
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lsopt_top |
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| |
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|---- check arguments |
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|---- CALL INSTORE |
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| | |
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| |---- determine whether OPWARMI available: |
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| * if no: cold start: create OPWARMI |
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| * if yes: warm start: read from OPWARMI |
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| create or open OPWARMD |
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| |
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|---- check consistency between OPWARMI and model parameters |
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| |
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|---- >>> if COLD start: <<< |
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| | first simulation with f.g. xx_0; output: first ff_0, gg_0 |
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| | set first preconditioner value xdiff_0 to 1 |
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| | store xx(0), gg(0), xdiff(0) to OPWARMD (first 3 entries) |
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| | |
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| >>> else: WARM start: <<< |
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| read xx(i), gg(i) from OPWARMD (first 2 entries) |
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| for first warm start after cold start, i=0 |
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| |
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| |
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| |
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|---- /// if ITMAX > 0: perform optimization (increment loop index i) |
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| ( |
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| )---- save current values of gg(i-1) -> gold(i-1), ff -> fold(i-1) |
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| (---- CALL LSUPDXX |
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| ) | |
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| ( |---- >>> if jmax=0 <<< |
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| ) | | first optimization after cold start: |
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| ( | | preconditioner estimated via ff_0 - ff_(first guess) |
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| ) | | dd(i-1) = -gg(i-1)*preco |
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| ( | | |
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| ) | >>> if jmax > 0 <<< |
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| ( | dd(i-1) = -gg(i-1) |
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| ) | CALL HESSUPD |
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| ( | | |
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| ) | |---- dd(i-1) modified via Hessian approx. |
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| ( | |
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| ) |---- >>> if <dd,gg> >= 0 <<< |
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| ( | ifail = 4 |
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| ) | |
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| ( |---- compute step size: tact(i-1) |
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| ) |---- compute update: xdiff(i) = xx(i-1) + tact(i-1)*dd(i-1) |
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| ( |
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| )---- >>> if ifail = 4 <<< |
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| ( goto 1000 |
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| ) |
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| (---- CALL OPTLINE / LSLINE |
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| ) | |
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... ... ... |
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\end{verbatim} |
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} |
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\caption{Flow chart (part 1 of 3)} |
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\label{fig:lsoptflow1} |
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\end{figure} |
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|
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\begin{figure} |
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%\input{part8/lsopt_flow_2} |
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{\scriptsize |
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\begin{verbatim} |
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... ... |
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| ) |
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| (---- CALL OPTLINE / LSLINE |
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| ) | |
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| ( |---- /// loop over simulations |
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| ) ( |
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| ( )---- CALL SIMUL |
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| ) ( | |
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| ( ) |---- input: xdiff(i) |
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| ) ( |---- output: ff(i), gg(i) |
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| ( ) |---- >>> if ONLINE <<< |
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| ) ( runs model and adjoint |
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| ( ) >>> if OFFLINE <<< |
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| ) ( reads those values from file |
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| ( ) |
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| ) (---- 1st Wolfe test: |
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| ( ) ff(i) <= tact*xpara1*<gg(i-1),dd(i-1)> |
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| ) ( |
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| ( )---- 2nd Wolfe test: |
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| ) ( <gg(i),dd(i-1)> >= xpara2*<gg(i-1),dd(i-1)> |
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| ( ) |
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| ) (---- >>> if 1st and 2nd Wolfe tests ok <<< |
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| ( ) | 320: update xx: xx(i) = xdiff(i) |
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| ) ( | |
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| ( ) >>> else if 1st Wolfe test not ok <<< |
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| ) ( | 500: INTERpolate new tact: |
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| ( ) | barr*tact < tact < (1-barr)*tact |
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| ) ( | CALL CUBIC |
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| ( ) | |
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| ) ( >>> else if 2nd Wolfe test not ok <<< |
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| ( ) 350: EXTRApolate new tact: |
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| ) ( (1+barmin)*tact < tact < 10*tact |
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| ( ) CALL CUBIC |
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| ) ( |
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| ( )---- >>> if new tact > tmax <<< |
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| ) ( | ifail = 7 |
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| ( ) | |
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| ) (---- >>> if new tact < tmin OR tact*dd < machine precision <<< |
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| ( ) | ifail = 8 |
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| ) ( | |
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| ( )---- >>> else <<< |
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| ) ( update xdiff for new simulation |
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| ( ) |
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| ) \\\ if nfunc > 1: use inter-/extrapolated tact and xdiff |
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| ( for new simulation |
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| ) N.B.: new xx is thus not based on new gg, but |
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| ( rather on new step size tact |
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| ) |
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| (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries) |
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| )---- >>> if ifail = 7,8,9 <<< |
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| ( goto 1000 |
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| ) |
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... ... |
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\end{verbatim} |
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} |
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\caption{Flow chart (part 2 of 3)} |
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\label{fig:lsoptflow2} |
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\end{figure} |
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|
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\begin{figure} |
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%\input{part8/lsopt_flow_3} |
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{\scriptsize |
| 350 |
\begin{verbatim} |
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... ... |
| 352 |
| ) |
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| (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries) |
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| )---- >>> if ifail = 7,8,9 <<< |
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| ( goto 1000 |
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| ) |
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| (---- compute new pointers jmin, jmax to include latest values |
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| ) gg(i)-gg(i-1), xx(i)-xx(i-1) to Hessian matrix estimate |
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| (---- store gg(i)-gg(i-1), xx(i)-xx(i-1) to OPWARMD |
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| ) (entries 2*jmax+2, 2*jmax+3) |
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| ( |
| 362 |
| )---- CALL DGSCALE |
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| ( | |
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| ) |---- call dostore |
| 365 |
| ( | | |
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| ) | |---- read preconditioner of previous iteration diag(i-1) |
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| ( | from OPWARMD (3rd entry) |
| 368 |
| ) | |
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| ( |---- compute new preconditioner diag(i), based upon diag(i-1), |
| 370 |
| ) | gg(i)-gg(i-1), xx(i)-xx(i-1) |
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| ( | |
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| ) |---- call dostore |
| 373 |
| ( | |
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| ) |---- write new preconditioner diag(i) to OPWARMD (3rd entry) |
| 375 |
| ( |
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|---- \\\ end of optimization iteration loop |
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| |
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| |
| 379 |
| |
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|---- CALL OUTSTORE |
| 381 |
| | |
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| |---- store gnorm0, ff(i), current pointers jmin, jmax, iterabs to OPWARMI |
| 383 |
| |
| 384 |
|---- >>> if OFFLINE version <<< |
| 385 |
| xx(i+1) needs to be computed as input for offline optimization |
| 386 |
| | |
| 387 |
| |---- CALL LSUPDXX |
| 388 |
| | | |
| 389 |
| | |---- compute dd(i), tact(i) -> xdiff(i+1) = x(i) + tact(i)*dd(i) |
| 390 |
| | |
| 391 |
| |---- CALL WRITE_CONTROL |
| 392 |
| | | |
| 393 |
| | |---- write xdiff(i+1) to special file for offline optim. |
| 394 |
| |
| 395 |
|---- print final information |
| 396 |
| |
| 397 |
O |
| 398 |
\end{verbatim} |
| 399 |
} |
| 400 |
\caption{Flow chart (part 3 of 3)} |
| 401 |
\label{fig:lsoptflow3} |
| 402 |
\end{figure} |
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