The numerical solution of Newton’s problem of least resistance

In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R 2 . We propose two different methods for solving it numerically. First, we discretize this problem by writing the c...

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Vydáno v:Mathematical programming Ročník 147; číslo 1-2; s. 331 - 350
Hlavní autor: Wachsmuth, Gerd
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2014
Springer Nature B.V
Témata:
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computer programs
Equivalence
Full Length Paper
Infimum
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Studies
Texts
Theoretical
Topological manifolds
52B55 (Computational aspects related to convexity)
65D15 (Algorithms for functional approximation)
65K10 (Optimization and variational techniques)
ISSN:0025-5610, 1436-4646
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Shrnutí:In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R 2 . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in R 1 . Deriving its Euler–Lagrange equation yields a program with two unknowns, which can be solved quickly.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-014-0756-2