Multiple shooting

The multiple shooting method ( English multiple shooting method ), and multi-target method is in the mathematics a numerical method for solving boundary value problems in ordinary differential equations . The interval on which the solution of the boundary value problem is to be determined is first divided into smaller sub-intervals, on each of which an initial value problem is then solved. With additional continuity conditions, a solution is then determined for the entire interval. This method is an essential further development of the single shooting process , especially with regard to numerical stability .

Problem

A boundary value problem of the form is given

${\displaystyle y'(t)=f(t,y(t)),\quad t\in [a,b],\quad g(y(a),y(b))=0}$,

where the right-hand side and the two-point boundary condition are given continuous functions and a differentiable function is sought. To solve such a boundary value problem, the single-shot method proceeds as follows: Let be the solution of the initial value problem ${\displaystyle f\colon [a,b]\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle g\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle y\colon [a,b]\to \mathbb {R} ^{n}}$ ${\displaystyle y_{p}(t)}$

${\displaystyle y'(t)=f(t,y(t)),\quad t\in [a,b],\quad y(a)=p}$,

then the free parameter is determined so that the boundary condition ${\displaystyle p\in \mathbb {R} ^{n}}$

${\displaystyle g(p,y_{p}(b))=0}$

is satisfied. An iterative method , such as Newton's method , is usually used to solve this vector equation . With stiff initial value problems , however, small changes in the initial condition can lead to large changes in the solution , making the method numerically unstable . ${\displaystyle p}$ ${\displaystyle y_{p}(b)}$

Procedure

The multi-shot method now uses a subdivision to improve stability

${\displaystyle a=t_{1}<t_{2}<\cdots <t_{N+1}=b}$.

of the interval into sub-intervals and computes the solutions to a number of initial value problems ${\displaystyle [a,b]}$ ${\displaystyle N}$ ${\displaystyle y_{k,p_{k}}(t),k=1,\ldots ,N,}$

${\displaystyle {\begin{aligned}y'(t)&=f(t,y(t)),\quad t\in [t_{1},t_{2}],\quad y(t_{1})=p_{1}\\y'(t)&=f(t,y(t)),\quad t\in [t_{2},t_{3}],\quad y(t_{2})=p_{2}\\&\,\,\,\vdots \\y'(t)&=f(t,y(t)),\quad t\in [t_{N},t_{N+1}],\quad y(t_{N})=p_{N}\end{aligned}}}$

in these sub-intervals. The free parameters are determined in such a way that the continuity conditions ${\displaystyle p_{1},\ldots ,p_{N}\in \mathbb {R} ^{n}}$

${\displaystyle y_{k,p_{k}}(t_{k+1})=p_{k+1}~{\text{für}}~k=1,\ldots ,N-1}$

and the boundary condition

${\displaystyle g(p_{1},y_{N,p_{N}}(t_{N+1}))=0}$

are fulfilled. The composite function is thus defined by ${\displaystyle y\colon [a,b]\to \mathbb {R} ^{n}}$

${\displaystyle y(t)=y_{k,p_{k}}(t)~{\text{für}}~t\in [t_{k},t_{k+1}]}$

not only continuous, but also differentiable, and thus a solution to the initial problem. To determine the parameters , a non-linear vector equation system with equations and unknowns has to be solved, which in turn is carried out with an iterative method. ${\displaystyle p_{k}}$ ${\displaystyle N}$

literature

• Josef Stoer, Roland Bulirsch : Numerical Mathematics 2 . 5th edition. Springer-Verlag, 2005, ISBN 3-540-23777-1 , chapter 7.3.5 ff.
• Hans Georg Bock , Karl J. Plitt: A multiple shooting algorithm for direct solution of optimal control problems . In: Proceedings of the 9th IFAC World Congress . Budapest 1984.
• Morrison, David D. and Riley, James D. and Zancanaro, John F .: Multiple shooting method for two-point boundary value problems . In: Commun. ACM . tape 5 , no. 12 . ACM, New York, NY, USA December 1962, pp. 613-614 .