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46
A Framework for the Construction of Level Set Methods for Shape Optimization and Reconstruction
 Interfaces and Free Boundaries
, 2002
"... The aim of this paper is to develop a functionalanalytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric ..."
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Cited by 46 (6 self)
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The aim of this paper is to develop a functionalanalytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case.
LevenbergMarquardt Level Set Methods for Inverse Obstacle Problems
 Inverse Problems
, 2003
"... The aim of this paper is to construct LevenbergMarquardt level set methods for inverse obstacle problems, and to discuss their numerical realization. Based on a recently developed framework for the construction of level set methods, we can define LevenbergMarquardt level set methods in a general ..."
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Cited by 44 (1 self)
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The aim of this paper is to construct LevenbergMarquardt level set methods for inverse obstacle problems, and to discuss their numerical realization. Based on a recently developed framework for the construction of level set methods, we can define LevenbergMarquardt level set methods in a general way by varying the function space used for the normal velocity. In the typical case of a PDEconstraint, the iterative method yields an indefinite linear system to be solved in each iteration step, which can be reduced to a positive definite problem for the normal velocity. We discuss the structure of this systems and possibilities for its iterative solution.
Image restoration and classification by topological asymptotic expansion. In: Variational formulations in mechanics: theory and applications
, 2007
"... We present in this paper a new way for modeling and solving image restoration and classification problems, the topological gradient method. This method is considered in the frame of variational approaches and the minimization of potential energy with respect to conductivity. The numerical experiment ..."
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Cited by 15 (6 self)
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We present in this paper a new way for modeling and solving image restoration and classification problems, the topological gradient method. This method is considered in the frame of variational approaches and the minimization of potential energy with respect to conductivity. The numerical experiments show the efficiency of the topological gradient approach. The image is most of the time restored or classified at the first iteration of the optimization process. Moreover, the computational cost of this iteration is reduced drastically using spectral methods. We also propose an algorithm which provides the optimal classes (number and values) for the unsupervised regularized classification problem. 1
Efficient Generation of LargeScale ParetoOptimal Topologies
 Structural and Multidisciplinary Optimization
, 2013
"... Abstract The objective of this paper is to introduce an efficient algorithm and implementation for largescale 3D topology optimization. The proposed algorithm is an extension of a recently proposed 2D topologicalsensitivity based method that can generate numerous paretooptimal topologies up to ..."
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Cited by 7 (7 self)
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Abstract The objective of this paper is to introduce an efficient algorithm and implementation for largescale 3D topology optimization. The proposed algorithm is an extension of a recently proposed 2D topologicalsensitivity based method that can generate numerous paretooptimal topologies up to a desired volume fraction, in a single pass. In this paper, we show how the computational challenges in 3D can be overcome. In particular, we consider an arbitrary 3D domainspace that is discretized via hexahedral/brick finite elements. Exploiting congruence between elements, we propose a matrixfree implementation of the finite element method. The latter exploits modern multicore architectures to efficiently solve topology optimization problems involving millions of degrees of freedom. The proposed methodology is illustrated through numerical experiments; comparisons are made against previously published results.
Global optimization, level set dynamics, incomplete sensitivity and regularity control
, 2007
"... We aim to optimize aerodynamic shapes using an incomplete sensitivity concept and regularization (projection) when control parameters are characteristic functions as in level set and immersed boundary approaches. The projection operator is also used to define a rotation operator over the unit sphere ..."
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Cited by 2 (2 self)
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We aim to optimize aerodynamic shapes using an incomplete sensitivity concept and regularization (projection) when control parameters are characteristic functions as in level set and immersed boundary approaches. The projection operator is also used to define a rotation operator over the unit sphere in the admissible space to improve sensitivity definition from the incomplete sensitivity. Hence, the direction of descent is found as a solution of a onedimensional minimization problem. This is suitable for large dimension control spaces, avoids an adjoint formulation and is particularly interesting for sensitivity evaluation for blackbox solvers. The approach is applied to various shape design in supersonic regime with level sets.
Features Detection Based on a Variational Model in Sensornets
 JDCTA
"... With the advent of CMOS cameras, it is now possible to make compact, cheap and lowpower image sensors capable of onboard image processing. These embedded vision sensors provide a rich new sensing modality enabling new classes of wireless sensor networking applications. In order to build these appl ..."
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Cited by 2 (0 self)
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With the advent of CMOS cameras, it is now possible to make compact, cheap and lowpower image sensors capable of onboard image processing. These embedded vision sensors provide a rich new sensing modality enabling new classes of wireless sensor networking applications. In order to build these applications, system designers need to overcome challenges associated with limited bandwith, limited power, group coordination and fusing of multiple camera views with various other sensory inputs. Realtime properties must be upheld if multiple vision sensors are to process data, communicate with each other and make a group decision before the measured environmental feature changes. The appearance of Wireless Multimedia Sensor Networks (WMSNs) requires a new generation technology of image processing for many applications. This paper presents a new approach to License Plate Localization(LPL) in WMSNs. We detect the license plate by a variational model that is embedded in several scalar functions. The motion of the dynamic interface is governed by nonlinear Partial Differential Equations(PDEs). Such variational models are flexible in handling complex environment and are concise in extracting plates boundaries despite of serious pollution. The cost of this method is moderate. The accuracy and efficiency of the proposed algorithm are illustrated by several numerical examples.
Multiobjective Topology Optimization Using a Genetic Algorithm and a Morphological Representation of Geometry
, 2005
"... In this paper, topology optimization of continuum structures is solved as a discrete optimization problem using a genetic algorithm (GA). Early e#orts applying this approach have not been very encouraging due to the lack of an appropriate structural geometry representation, and a good representation ..."
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Cited by 2 (1 self)
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In this paper, topology optimization of continuum structures is solved as a discrete optimization problem using a genetic algorithm (GA). Early e#orts applying this approach have not been very encouraging due to the lack of an appropriate structural geometry representation, and a good representation scheme is essential to the e#ectiveness of the genetic operations and hence the success of any GA procedure. In this work, a recently developed morphological geometric representation scheme is coupled with a GA to perform topology and shape optimization. The scheme uses arrangements of skeleton and surrounding material to define structural geometry in a way that facilitates the transmission of topological/shape characteristics across generations in the evolutionary process, and will not render any undesirable design features such as disconnected segments, checkerboard patterns or singlenode hinge connections.
A level set based shape optimization method for an elliptic obstacle problem
 Math. Models Methods Appl. Sci
"... Abstract. In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the sh ..."
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Abstract. In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.