PMC:1570349 / 15121-16535 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/1570349","sourcedb":"PMC","sourceid":"1570349","source_url":"https://www.ncbi.nlm.nih.gov/pmc/1570349","text":"Here we give an outline of the algorithm for general inverse bifurcation problems. The main ingredients are applications of the projection operator F(p), as well as the adjoint, F'*(p). The computation of the former is denoted by the routine APPLYF (see Figure 4). Each time APPLYF is called, corrector steps (using, for instance, Newton's method) have to be carried out on the previously computed xinit to find the initial solution for the current value of ps. Once the corrected solution xcorr is computed, the iterative procedure LOCMINDIST (see Figure 5) is called to compute the nearest point on the bifurcation manifold of interest. This procedure is based on a series of one-parameter continuations and gradient calculations according to (6). The inputs include the following: v the initial search direction in parameter space; εproj the relative tolerance on the iterative proceure. In general, several search directions (denoted by V = {v1,⋯, vmax}) have to be used to approximate the globally closest point. However, for the examples shown in the paper the initial search space V is only 1 dimensional. Once F(p) and the corresponding solution are obtained, F'*(p) is then computed. The derivative information, together with constraints (plow, pupp, c(p)) and Hessian approximation (HessA) are then used to compute an SQP step and update the Hessian. Figure 6 describes the inverse bifurcation algorithm.","tracks":[]}