The Experts below are selected from a list of 81 Experts worldwide ranked by ideXlab platform
Omri Azencot - One of the best experts on this subject based on the ideXlab platform.
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discrete derivatives of Vector fields on surfaces an operator approach
ACM Transactions on Graphics, 2015Co-Authors: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, Mirela BenchenAbstract:Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to Representing Vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing Vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent Vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a Vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and Vector field design, by posing algebraic constraints on the covariant derivative operator.
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Discrete Derivatives of Vector Fields on Surfaces – An Operator Approach
ACM Transactions on Graphics, 2015Co-Authors: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, M Ben-chenAbstract:Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to Representing Vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing Vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent Vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a Vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear operators which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and Vector field design, by posing algebraic constraints on the covariant derivative operator. ACM Reference Format: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete derivatives of Vector fields on surfaces – An operator approach .
M Ben-chen - One of the best experts on this subject based on the ideXlab platform.
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Discrete Derivatives of Vector Fields on Surfaces – An Operator Approach
ACM Transactions on Graphics, 2015Co-Authors: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, M Ben-chenAbstract:Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to Representing Vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing Vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent Vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a Vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear operators which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and Vector field design, by posing algebraic constraints on the covariant derivative operator. ACM Reference Format: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete derivatives of Vector fields on surfaces – An operator approach .
Uwe Naumann - One of the best experts on this subject based on the ideXlab platform.
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Accumulating Jacobians as chained sparse matrix products
Mathematical Programming, 2003Co-Authors: Andreas Griewank, Uwe NaumannAbstract:The chain rule – fundamental to any kind of analytical differentiation - can be applied in various ways to computational graphs Representing Vector functions. These variants result in different operations counts for the calculation of the corresponding Jacobian matrices. The minimization of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a hard combinatorial optimization problem.
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Efficient Calculation of Jacobians Using Dynamic Programming
1999Co-Authors: Uwe NaumannAbstract:The chain rule - fundamental for Automatic Differentiation (AD) - can be applied to computational graphs Representing Vector functions in arbitrary orders resulting in different operations counts for the calculation of their Jacobian matrices. Very few authors have looked at this interesting subject so far and there is no generally accepted terminology for dealing with these combinations of the forward and reverse modes of AD. The minimizati- on of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a computationally hard combinatorial optimization problem. In this paper we will describe an approach to the solution of the edge elimination problem in computational graphs that builds on the idea of optimizing chained matrix products using dynamic programming techniques. We will discuss the theory and present some test results by regarding this approach in comparison with other methods for computing Jacobians using a minimal number of arithmetic operations.
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Heuristics for Efficiently Calculating Jacobians by Edge Elimination in Computational Graphs
1999Co-Authors: Uwe NaumannAbstract:The chain rule - fundamental for Automatic Differentiation (AD) - can be applied to computational graphs Representing Vector functions in arbitrary orders resulting in different operations counts for the calculation of their Jacobian matrices. Very few authors have looked at this interesting subject so far and there is no generally accepted terminology for dealing with these combinations of the forward and reverse modes of AD. The minimizati- on of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a computationally hard combinatorial optimization problem. After an extensive introduction into the problem of computing Jacobians using a minimal number of arithmetic operations we will describe several heuristics for reducing the size the computational graph as well as for solving the edge elimination problem in computational graphs. We will discuss the ideas behind the heuristics and present some test results. Finally we will give an outlook on the role that edge eliminatio- n could play in the development of new AD tools.
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SAVE - Simulated Annealing Applied to the Vertex Elimination Problem in Computational Graphs
1999Co-Authors: Uwe NaumannAbstract:The chain rule - fundamental for Automatic Differentiation (AD) - can be applied to computational graphs Representing Vector functions in arbitrary orders resulting in different operations counts for the calculation of their Jacobian matrices. Very few authors have looked at this interesting subject so far and there is no generally accepted terminology for dealing with these combinations of the forward and reverse modes of AD. The minimizati- on of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a computationally hard combinatorial optimization problem. In this paper we will describe a new heuristic approach to the solution of the vertex elimination problem in computational graphs which can easily adapted to the more general case of eliminating edges. Simulated annealing is widely regarded as a suitable method for solving combinatorial optimization problems. We will discuss different annealing schedules and present some test results. Finally we will regard this approach in comparison with other methods for computing Jacobians using a minimal number of arithmetic operations.
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Optimizing the Accumulation of Jacobians by Edge Elimination in the Computational Graph
1999Co-Authors: Uwe NaumannAbstract:The chain rule - fundamental for Automatic Differentiation (AD) - can be applied to computational graphs Representing Vector functions in arbitrary orders resulting in different operations counts for the calculation of their Jacobian matrices. Very few authors have dealt with this interesting subject so far and there is no generally accepted terminology for handling these combinations of the forward and reverse modes of AD. The minimization of the number of arithmetic operations required for the calculation of the complete Jacobian leads to a computationally hard combinatorial optimizati- on problem. In this paper we will give a formal description of this problem, which also is sometimes referred to as the cross-country elimination problem in computational graphs, in terms of a shortest path problem in the so-called metagraph. The well-known strategy of eliminating vertices will be refined by introducing the elimination of edges. We will show that edge elimination is in general superior to vertex elimination with respect to the operations count. As an outlook we will present a selection of methods for solving the general edge elimination problem heuristically
Mirela Benchen - One of the best experts on this subject based on the ideXlab platform.
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discrete derivatives of Vector fields on surfaces an operator approach
ACM Transactions on Graphics, 2015Co-Authors: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, Mirela BenchenAbstract:Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to Representing Vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing Vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent Vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a Vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and Vector field design, by posing algebraic constraints on the covariant derivative operator.
Maks Ovsjanikov - One of the best experts on this subject based on the ideXlab platform.
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discrete derivatives of Vector fields on surfaces an operator approach
ACM Transactions on Graphics, 2015Co-Authors: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, Mirela BenchenAbstract:Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to Representing Vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing Vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent Vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a Vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and Vector field design, by posing algebraic constraints on the covariant derivative operator.
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Discrete Derivatives of Vector Fields on Surfaces – An Operator Approach
ACM Transactions on Graphics, 2015Co-Authors: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, M Ben-chenAbstract:Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to Representing Vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing Vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent Vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a Vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear operators which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and Vector field design, by posing algebraic constraints on the covariant derivative operator. ACM Reference Format: Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete derivatives of Vector fields on surfaces – An operator approach .