The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Ludovic Noels - One of the best experts on this subject based on the ideXlab platform.
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an incremental Secant mean field homogenization method with second statistical moments for elasto visco plastic composite materials
Mechanics of Materials, 2017Co-Authors: Ling Wu, Laurent Adam, Issam Doghri, Ludovic NoelsAbstract:Abstract This paper presents an extension of the recently developed incremental-Secant mean-field homogenization (MFH) procedure in the context of elasto-plasticity to elasto-visco-plastic composite materials while accounting for second statistical moments. In the incremental-Secant formulation, a virtual elastic unloading is performed at the composite level in order to evaluate the residual stress and strain states in the different phases, from which a Secant MFH formulation is applied. When applying the Secant MFH process, the linear-comparison-composite (LCC) is built from the piece-wise heterogeneous residual strain-stress state using naturally isotropic Secant tensors defined using either first or second statistical moment values. As a result non-proportional and non-radial loading conditions can be considered because of the incremental-Secant formulation, and accurate predictions can be obtained as no isotropization step is required. The limitation of the incremental-Secant formulation previously developed was the requirement in case of hard inclusions to cancel the residual stress in the matrix phase, resulting from the composite material unloading, to avoid over-stiff predictions. It is shown in this paper that in the case of hard inclusions by defining a proper second statistical moment estimate of the von Mises stress, the residual stress can be kept in the different composite phases. Moreover it is shown that the method can be extended to visco-plastic behaviors without modifying the homogenization process as the incremental-Secant formulation only requires the definition of the Secant operator of the different phase material models. Finally, it is shown that although it is also possible to define a proper second statistical moment estimate of the von Mises stress in the case of soft inclusions, this does not improve the accuracy as compared to the increment-Secant method with first order statistical moment estimates.
Chris Peterson - One of the best experts on this subject based on the ideXlab platform.
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A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction
2018 17th International Symposium on Parallel and Distributed Computing (ISPDC), 2018Co-Authors: Henry Kvinge, Elin Farnell, Michael Kirby, Chris PetersonAbstract:Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because Secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids Secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology. Computing all (unit) Secants for a set of points is by nature computationally expensive, thus opening the door for exploitation of GPU architecture for achieving fast versions of these algorithms. We present a polynomial-time data-reduction algorithm that produces a meaningful low-dimensional representation of a data set by iteratively constructing improved projections within the framework described above. Key to our algorithm design and implementation is the use of GPUs which, among other things, minimizes the computational time required for the calculation of all Secant lines. One goal of this report is to share ideas with GPU experts and to discuss a class of mathematical algorithms that may be of interest to the broader GPU community.
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Too many Secants: a hierarchical approach to Secant-based dimensionality reduction on large data sets
2018 IEEE High Performance extreme Computing Conference (HPEC), 2018Co-Authors: Henry Kvinge, Elin Farnell, Michael Kirby, Chris PetersonAbstract:A fundamental question in many data analysis settings is the problem of discerning the “natural” dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all Secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various dimensions, one may infer the dimension of the manifold of origination. That is, one may learn the dimension at which it is possible to construct a diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using Whitney's embedding theorem, we can relate this information to the natural dimension of the data. A drawback of the SAP algorithm is that a data set with T points has O(T2) Secants, making the computation and storage of all Secants infeasible for very large data sets. In this paper, we propose a novel algorithm that generalizes the SAP algorithm with an emphasis on addressing this issue. That is, we propose a hierarchical Secant-based dimensionality-reduction method, which can be employed for data sets where explicitly calculating all Secants is not feasible.
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HPEC - Too many Secants: a hierarchical approach to Secant-based dimensionality reduction on large data sets
2018 IEEE High Performance extreme Computing Conference (HPEC), 2018Co-Authors: Henry Kvinge, Elin Farnell, Michael Kirby, Chris PetersonAbstract:A fundamental question in many data analysis settings is the problem of discerning the “natural” dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all Secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various dimensions, one may infer the dimension of the manifold of origination. That is, one may learn the dimension at which it is possible to construct a diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using Whitney's embedding theorem, we can relate this information to the natural dimension of the data. A drawback of the SAP algorithm is that a data set with $T$ points has $O(T^{2})$ Secants, making the computation and storage of all Secants infeasible for very large data sets. In this paper, we propose a novel algorithm that generalizes the SAP algorithm with an emphasis on addressing this issue. That is, we propose a hierarchical Secant-based dimensionality-reduction method, which can be employed for data sets where explicitly calculating all Secants is not feasible.
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Induction for Secant varieties of Segre varieties
Transactions of the American Mathematical Society, 2008Co-Authors: Hirotachi Abo, Giorgio Ottaviani, Chris PetersonAbstract:This paper studies the dimension of Secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the Secant variety in a high dimensional case to a sequence of calculations of partial Secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective p-Secant varieties to Segre varieties for p < 6. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of Pn. We determine the set of p for which unbalanced Segre varieties have defective p-Secant varieties. In addition, we completely describe the dimensions of the Secant varieties to the deficient Segre varieties P1 x P1 x Pn x Pn and P2 x P3 x P3. In the final section we propose a series of conjectures about defective Segre varieties.
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Induction for Secant varieties of Segre varieties
arXiv: Algebraic Geometry, 2006Co-Authors: Hirotachi Abo, Giorgio Ottaviani, Chris PetersonAbstract:This paper studies the dimension of Secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the Secant variety in a high dimensional case to a sequence of calculations of partial Secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective $t$-Secant varieties to Segre varieties for t < 7. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of P^n. We determine the set of p for which unbalanced Segre varieties have defective p-Secant varieties. In addition, we show that the Segre varieties P^1 x P^1 x P^n x P^n and P^2 x P ^3 x P^3 are deficient and completely describe the dimensions of their Secant varieties. In the final section we propose a series of conjectures about defective Segre varieties.
Ling Wu - One of the best experts on this subject based on the ideXlab platform.
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an incremental Secant mean field homogenization method with second statistical moments for elasto visco plastic composite materials
Mechanics of Materials, 2017Co-Authors: Ling Wu, Laurent Adam, Issam Doghri, Ludovic NoelsAbstract:Abstract This paper presents an extension of the recently developed incremental-Secant mean-field homogenization (MFH) procedure in the context of elasto-plasticity to elasto-visco-plastic composite materials while accounting for second statistical moments. In the incremental-Secant formulation, a virtual elastic unloading is performed at the composite level in order to evaluate the residual stress and strain states in the different phases, from which a Secant MFH formulation is applied. When applying the Secant MFH process, the linear-comparison-composite (LCC) is built from the piece-wise heterogeneous residual strain-stress state using naturally isotropic Secant tensors defined using either first or second statistical moment values. As a result non-proportional and non-radial loading conditions can be considered because of the incremental-Secant formulation, and accurate predictions can be obtained as no isotropization step is required. The limitation of the incremental-Secant formulation previously developed was the requirement in case of hard inclusions to cancel the residual stress in the matrix phase, resulting from the composite material unloading, to avoid over-stiff predictions. It is shown in this paper that in the case of hard inclusions by defining a proper second statistical moment estimate of the von Mises stress, the residual stress can be kept in the different composite phases. Moreover it is shown that the method can be extended to visco-plastic behaviors without modifying the homogenization process as the incremental-Secant formulation only requires the definition of the Secant operator of the different phase material models. Finally, it is shown that although it is also possible to define a proper second statistical moment estimate of the von Mises stress in the case of soft inclusions, this does not improve the accuracy as compared to the increment-Secant method with first order statistical moment estimates.
Issam Doghri - One of the best experts on this subject based on the ideXlab platform.
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an incremental Secant mean field homogenization method with second statistical moments for elasto visco plastic composite materials
Mechanics of Materials, 2017Co-Authors: Ling Wu, Laurent Adam, Issam Doghri, Ludovic NoelsAbstract:Abstract This paper presents an extension of the recently developed incremental-Secant mean-field homogenization (MFH) procedure in the context of elasto-plasticity to elasto-visco-plastic composite materials while accounting for second statistical moments. In the incremental-Secant formulation, a virtual elastic unloading is performed at the composite level in order to evaluate the residual stress and strain states in the different phases, from which a Secant MFH formulation is applied. When applying the Secant MFH process, the linear-comparison-composite (LCC) is built from the piece-wise heterogeneous residual strain-stress state using naturally isotropic Secant tensors defined using either first or second statistical moment values. As a result non-proportional and non-radial loading conditions can be considered because of the incremental-Secant formulation, and accurate predictions can be obtained as no isotropization step is required. The limitation of the incremental-Secant formulation previously developed was the requirement in case of hard inclusions to cancel the residual stress in the matrix phase, resulting from the composite material unloading, to avoid over-stiff predictions. It is shown in this paper that in the case of hard inclusions by defining a proper second statistical moment estimate of the von Mises stress, the residual stress can be kept in the different composite phases. Moreover it is shown that the method can be extended to visco-plastic behaviors without modifying the homogenization process as the incremental-Secant formulation only requires the definition of the Secant operator of the different phase material models. Finally, it is shown that although it is also possible to define a proper second statistical moment estimate of the von Mises stress in the case of soft inclusions, this does not improve the accuracy as compared to the increment-Secant method with first order statistical moment estimates.
Laurent Adam - One of the best experts on this subject based on the ideXlab platform.
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an incremental Secant mean field homogenization method with second statistical moments for elasto visco plastic composite materials
Mechanics of Materials, 2017Co-Authors: Ling Wu, Laurent Adam, Issam Doghri, Ludovic NoelsAbstract:Abstract This paper presents an extension of the recently developed incremental-Secant mean-field homogenization (MFH) procedure in the context of elasto-plasticity to elasto-visco-plastic composite materials while accounting for second statistical moments. In the incremental-Secant formulation, a virtual elastic unloading is performed at the composite level in order to evaluate the residual stress and strain states in the different phases, from which a Secant MFH formulation is applied. When applying the Secant MFH process, the linear-comparison-composite (LCC) is built from the piece-wise heterogeneous residual strain-stress state using naturally isotropic Secant tensors defined using either first or second statistical moment values. As a result non-proportional and non-radial loading conditions can be considered because of the incremental-Secant formulation, and accurate predictions can be obtained as no isotropization step is required. The limitation of the incremental-Secant formulation previously developed was the requirement in case of hard inclusions to cancel the residual stress in the matrix phase, resulting from the composite material unloading, to avoid over-stiff predictions. It is shown in this paper that in the case of hard inclusions by defining a proper second statistical moment estimate of the von Mises stress, the residual stress can be kept in the different composite phases. Moreover it is shown that the method can be extended to visco-plastic behaviors without modifying the homogenization process as the incremental-Secant formulation only requires the definition of the Secant operator of the different phase material models. Finally, it is shown that although it is also possible to define a proper second statistical moment estimate of the von Mises stress in the case of soft inclusions, this does not improve the accuracy as compared to the increment-Secant method with first order statistical moment estimates.
