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55
Spatially adaptive techniques for level set methods and incompressible flow
 Comput. Fluids
"... Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes ..."
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Cited by 73 (15 self)
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Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as HamiltonJacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56].
Sobolev active contours
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2007
"... All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. Ho ..."
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Cited by 66 (9 self)
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All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolevtype inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edgebased and regionbased active contour models as Sobolev active contours and show the substantial improvements gained in segmentation.
On level set regularization for highly illposed distributed parameter estimation problems
, 2005
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Phase–field relaxation of topology optimization with local stress constraints
 Local Stress Constraints, SFBReport 0435 (SFB F013, University Linz, 2004), and submitted
, 2005
"... We introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phasefield method. The starting point of the relaxation is a reformulation of the material problem involving linear and 0–1 constraints only. The 0–1 constraints are then rela ..."
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Cited by 17 (1 self)
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We introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phasefield method. The starting point of the relaxation is a reformulation of the material problem involving linear and 0–1 constraints only. The 0–1 constraints are then relaxed and approximated by a CahnHilliard type penalty in the objective functional, which yields convergence of minimizers to 0–1 designs as the penalty parameter decreases to zero. A major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter. The relaxation scheme yields a largescale optimization problem with a high number of linear inequality constraints. We discretize the problem by finite elements and solve the arising finitedimensional programming problems by a primaldual interior point method. Numerical experiments for problems with stress constraints based on different criteria indicate the success and robustness of the new approach.
Adaptive finite volume methods for distributed nonsmooth parameter identification
, 2007
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Photonic Design: From Fundamental Solar Cell Physics to Computational Inverse Design
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OPTIMAL DESIGN OF THE SUPPORT OF THE CONTROL FOR THE 2D WAVE EQUATION: A NUMERICAL METHOD
, 2007
"... We consider in this paper the homogeneous 2D wave equation defined on Ω ⊂ R2. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset ω ⊂ Ω, the control vω of minimal L2(ω × (0, T))norm which drives to rest the system at a time T> 0 large enough. We address the questi ..."
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Cited by 7 (2 self)
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We consider in this paper the homogeneous 2D wave equation defined on Ω ⊂ R2. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset ω ⊂ Ω, the control vω of minimal L2(ω × (0, T))norm which drives to rest the system at a time T> 0 large enough. We address the question of the optimal position of ω which minimize the functional J: ω → vω L2(ω×(0,T)). Assuming ω ∈ C1(Ω), we express the shape derivative of J as a curvilinear integral on ∂ω × (0, T) independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to J. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the wellposedness of the problem by considering its relaxation.
Parametric level set methods for inverse problems, (anticipated
, 2010
"... In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of para ..."
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Cited by 7 (7 self)
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In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, reinitialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasiNewton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a “narrowbanding ” advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, Xray computed tomography and diffuse optical tomography.
Artificial time integration
, 2006
"... Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now timedependent differential system, usually employing for ..."
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Cited by 7 (2 self)
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Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now timedependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be “close enough” to the dynamics of the continuous system (which is typically easier to analyze) provided that small – hence many – time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought. In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.
Generalized polarization tensors for shape description
 Numer. Math
, 2014
"... Abstract With each domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper [9], a recursive optimal control scheme to recover fine shape ..."
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Cited by 7 (3 self)
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Abstract With each domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper [9], a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this paper, we show that the GPTs can be used for shape description. We also show that highfrequency oscillations of the boundary of a domain are only contained in its highorder GPTs. Indeed, we provide a stability and resolution analysis for the reconstruction of small shape changes from the GPTs. By developing a level set version of the recursive optimization scheme, we make the change of topology possible and show that the GPTs can capture the topology of the domain. We provide numerical evidence that GPTs can capture topology and highfrequency shape oscillations. Both the analytical and numerical results of this paper clearly show that the concept of GPTs is a very promising new tool for shape description.