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Constructing reparametrization invariant metrics on spaces of plane curves
, 2012
"... on spaces of plane curves ..."
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A Riemannian view on shape optimization
 Foundations of Computational Mathematics
"... Abstract. Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shapeNewton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined possessing o ..."
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Abstract. Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shapeNewton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined possessing often sought properties like symmetry and quadratic convergence for Newton optimization methods. AMS subject classifications. 49Q10, 49M15, 53B20 Key words. Shape optimization, Riemannian manifold, Newton method
RTRANSFORMS FOR SOBOLEV H 2METRICS ON SPACES OF PLANE CURVES
"... Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 metric with ..."
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Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 metric without zero order terms. We find isometries (called Rtransforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We demonstrate the value of using Rtransforms by some numerical experiments.
GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES
, 2013
"... We study properties of Sobolevtype metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolevtype metrics with constant coefficients of order 2 and higher is globally wellposed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus t ..."
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We study properties of Sobolevtype metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolevtype metrics with constant coefficients of order 2 and higher is globally wellposed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives. 2010 Mathematics Subject Classification: 58D15 (primary); 35G55, 53A04, 58B20 (secondary) 1.
CONSTRUCTING REPARAMETERIZATION INVARIANT METRICS ON SPACES OF PLANE CURVES
"... Abstract. Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into ..."
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Abstract. Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolevtype Riemannian metrics of order one on the space Imm(S1, R2) of parameterized plane curves and the quotient space Imm(S1, R2) / Diff(S 1) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are nonnegative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes. 1.
TOWARDS A LAGRANGENEWTON APPROACH FOR PDE CONSTRAINED SHAPE OPTIMIZATION
"... Abstract. The novel Riemannian view on shape optimization developed in [22] is extended to a LagrangeNewton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example. 1. Introduc ..."
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Abstract. The novel Riemannian view on shape optimization developed in [22] is extended to a LagrangeNewton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example. 1. Introduction. Shape