The numerical solution of Newton’s problem of least resistance

Wachsmuth, Gerd

doi:10.1007/s10107-014-0756-2

Cited by 24 publications

(66 citation statements)

References 11 publications

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“…(c) The ratio of an elementary pair is invariant under a homothety; therefore the ratios of all elementary pairs of the family satisfy (21). Thus, we have 3 Proofs of Theorems 2 and 3…”

Section: Proof Of Lemmamentioning

confidence: 87%

“…Note in passing that te height of the generating triangle of each copy obtained this way is 1, and the base is smaller than d. Therefore the ratios of all obtained elementary pairs satisfy (21).…”

Section: Proof Of Lemmamentioning

confidence: 96%

“…2 (a)-(c). Figure 2 (c) is borrowed from [21]. Several trajectories of flow particles are also shown there.…”

Section: 3mentioning

confidence: 99%

“…Note that the values for Problem (P C ) were calculates with more precision in the recent paper [21]. I could not find in the literature numerical values of minimal resistance concerning Problem (P S ).…”

Section: 3mentioning

confidence: 99%

“…The ratios κ i govern the resistance of mirrors in the family. Roughly speaking, formula (21) indicates that as m → ∞, the resistances of the mirrors can be made arbitrarily close to their smallest value.…”

Section: Proof Of Lemmamentioning

confidence: 99%

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Newton’s problem of minimal resistance under the single-impact assumption

Plakhov

2016

Nonlinearity

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A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the body's surface, the particles are reflected elastically. Assume that each particle hits the body at most once (SIC condition); then the force of resistance of the body along the flow direction can be written down in a simple analytical form.The problem of minimal resistance within this model was first considered by Newton [13] in the class of bodies with a fixed length M along the flow direction and with a fixed maximum orthogonal cross section Ω, under the additional conditions that the body is convex and rotationally symmetric. Here we solve the problem (first stated in [5]) for the wider class of bodies satisfying SIC and with the additional conditions removed. The scheme of solution is inspired by Besicovitch's method of solving the Kakeya problem [1]. If Ω is a disc, the decrease of resistance as compared with the original Newton problem is more than twofold; the ratio tends to 2 as M → 0 and to 20.25 as M → ∞. We also prove that the infimum of resistance is 0 for a wider class of bodies with both single and double impacts allowed. Mathematics subject classifications: 49Q10, 49K30

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“…(c) The ratio of an elementary pair is invariant under a homothety; therefore the ratios of all elementary pairs of the family satisfy (21). Thus, we have 3 Proofs of Theorems 2 and 3…”

Section: Proof Of Lemmamentioning

confidence: 87%

Section: Proof Of Lemmamentioning

confidence: 96%

“…2 (a)-(c). Figure 2 (c) is borrowed from [21]. Several trajectories of flow particles are also shown there.…”

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Proof Of Lemmamentioning

confidence: 99%

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Newton’s problem of minimal resistance under the single-impact assumption

Plakhov

2016

Nonlinearity

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Adaptive, anisotropic and hierarchical cones of discrete convex functions

Mirebeau

2015

Numer. Math.

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We introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions. Among the various mathematical problems which posses such a formulation, the Monopolist problem [24, 10] arising in economics is our main motivation.Consider a two dimensional domain Ω, sampled on a grid X of N points. We show that the cone Conv(X) of restrictions to X of convex functions on Ω is typically characterized by ≈ N 2 linear inequalities; a direct computational use of this description therefore has a prohibitive complexity. We thus introduce a hierarchy of sub-cones Conv(V) of Conv(X), associated to stencils V which can be adaptively, locally, and anisotropically refined. We show, using the arithmetic structure of the grid, that the trace U |X of any convex function U on Ω is contained in a cone Conv(V) defined by only O(N ln 2 N ) linear constraints, in average over grid orientations.Numerical experiments for the Monopolist problem, based on adaptive stencil refinement strategies, show that the proposed method offers an unrivaled accuracy/complexity trade-off in comparison with existing methods. We also obtain, as a side product of our theory, a new average complexity result on edge flipping based mesh generation.A number of mathematical problems can be formulated as the optimization of a convex functional over the cone of convex functions on a domain Ω (here compact and two dimensional):This includes optimal transport, as well as various geometrical conjectures such as Newton's problem [16,18]. We choose for concreteness to emphasize an economic application: the Monopolist (or Principal Agent) problem [24], in which the objective is to design an optimal product line, and an optimal pricing catalog, so as to maximize profit in a captive market. The following minimal instance is numerically studied in [1, 10, 21] and on Figure 1.We refer to the numerical section §6, and to [24] for the economic model details; let us only say here that the Monopolist's optimal product line is {∇U (z); z ∈ Ω}, and that the optimal prices are given by the Legendre-Fenchel dual of U . Consider the following three regions, defined for k ∈ {0, 1, 2} (implicitly excluding points z ∈ Ω close to which U is not smooth) Ω k := {z ∈ Ω; rank(Hessian U (z)) = k}.(2) *

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Conforming approximation of convex functions with the finite element method

Wachsmuth

2017

Numer. Math.

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The numerical solution of Newton’s problem of least resistance

Cited by 24 publications

References 11 publications

Newton’s problem of minimal resistance under the single-impact assumption

Newton’s problem of minimal resistance under the single-impact assumption

Adaptive, anisotropic and hierarchical cones of discrete convex functions

Conforming approximation of convex functions with the finite element method

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