Differential Equations Solutions 41

Differential Equations Solutions 41 - I chose pattern...

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51 Quasi-Newton: often converges superlinearly when started close enough to a solution, but requires Frst derivatives (or good approximations of them) and storage of a matrix of size 2000 × 2000, unless the matrix is accumulated implicitly by saving the vectors s and y . Pattern search: converges only linearly, but has good global behavior and requires only function values, no derivatives. If Frst derivatives (or approximations) were available, I would use quasi-Newton, with updating of the matrix decomposition (or a limited memory version). Other- wise, I would use pattern search. CHALLENGE 9.12. I would use pattern search to minimize F ( x )= y (1) as a function of x . When a function value F ( x ) is needed, I would call one of MATLAB’s sti± ode solvers, since I don’t know whether the problem is sti± or not, and return the value computed as y (1). The value of x would need to be passed to the function that evaluates f for the ode solver.
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Unformatted text preview: I chose pattern search because it has proven convergence and does not require derivatives of F with respect to x . Note that these derivatives are not available for this problem: we can compute derivatives of y with respect to t but not with respect to x . And since our value of y (1) is only an approximation, the use of Fnite di±erences to estimate derivatives with respect to x would yield values too noisy to be useful. CHALLENGE 9.13. Method conv. rate Storage f evals/itn g evals/itn H evals/itn Truncated Newton > 1 O ( n ) ≤ n + 1 Newton 2 O ( n 2 ) 1 1 1 Quasi-Newton > 1 2 O ( n 2 ) 1 1 steepest descent 1 O ( n ) 1 1 Conjugate gradients 1 O ( n ) 1 1 Notes on the table: 1. Once the counts for the linesearch are omitted, no function evaluations are needed. 2. ²or a single step, Quasi-Newton is superlinear; it is n-step quadratic....
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