The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Marc Boullé - One of the best experts on this subject based on the ideXlab platform.
-
A Bayesian Approach for Supervised
2014Co-Authors: Marc Boullé, France Telecom RAbstract:In supervised machine learning, some algorithms are restricted to discrete data and thus need to discretize continuous attributes. In this paper, we present a new Discretization method called MODL, based on a Bayesian approach. The MODL method relies on a model space of Discretizations and on a prior distribution defined on this model space. This allows setting up an evaluation criterion of Discretization, which is minimal for the most probable Discretization given the data, i.e. the Bayes optimal Discretization. We compare this approach with the MDL approach and statistical approaches used in other Discretization methods, from a theoretical and experimental point of view. Extensive experiments show that the MODL method builds high quality Discretizations
-
modl a bayes optimal Discretization method for continuous attributes
Machine Learning, 2006Co-Authors: Marc BoulléAbstract:While real data often comes in mixed format, discrete and continuous, many supervised induction algorithms require discrete data. Efficient Discretization of continuous attributes is an important problem that has effects on speed, accuracy and understandability of the induction models. In this paper, we propose a new Discretization method MODL1, founded on a Bayesian approach. We introduce a space of Discretization models and a prior distribution defined on this model space. This results in the definition of a Bayes optimal evaluation criterion of Discretizations. We then propose a new super-linear optimization algorithm that manages to find near-optimal Discretizations. Extensive comparative experiments both on real and synthetic data demonstrate the high inductive performances obtained by the new Discretization method.
-
modl a bayes optimal Discretization method for continuous attributes
Machine Learning, 2006Co-Authors: Marc BoulléAbstract:While real data often comes in mixed format, discrete and continuous, many supervised induction algorithms require discrete data. Efficient Discretization of continuous attributes is an important problem that has effects on speed, accuracy and understandability of the induction models. In this paper, we propose a new Discretization method MODL1, founded on a Bayesian approach. We introduce a space of Discretization models and a prior distribution defined on this model space. This results in the definition of a Bayes optimal evaluation criterion of Discretizations. We then propose a new super-linear optimization algorithm that manages to find near-optimal Discretizations. Extensive comparative experiments both on real and synthetic data demonstrate the high inductive performances obtained by the new Discretization method.
-
Optimal bin number for equal frequency Discretizations in supervized learning
Intelligent Data Analysis, 2005Co-Authors: Marc BoulléAbstract:While real data often comes in mixed format, discrete and continuous, many supervised induction algorithms require discrete data. Although efficient supervised Discretization methods are available, the unsupervised Equal Frequency Discretization method is still widely used by the statistician both for data exploration and data preparation. In this paper, we propose an automatic method, based on a Bayesian approach, to optimize the number of bins for Equal Frequency Discretizations in the context of supervised learning. We introduce a space of Equal Frequency Discretization models and a prior distribution defined on this model space. This results in the definition of a Bayes optimal evaluation criterion for Equal Frequency Discretizations. We then propose an optimal search algorithm whose run-time is super-linear in the sample size. Extensive comparative experiments demonstrate that the method works quite well in many cases.
Vikram Gavini - One of the best experts on this subject based on the ideXlab platform.
-
higher order adaptive finite element methods for kohn sham density functional theory
Journal of Computational Physics, 2013Co-Authors: Phani Motamarri, M R Nowak, Kenneth W Leiter, Jaroslaw Knap, Vikram GaviniAbstract:We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element Discretization of Kohn-Sham density-functional theory (DFT). To this end, we develop an a priori mesh-adaption technique to construct a close to optimal finite-element Discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100-200-fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element Discretizations of the Kohn-Sham DFT problem. Our studies suggest that staggering computational savings-of the order of 1000-fold-relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations, by using higher-order finite-element Discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element Discretization of the Kohn-Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element Discretizations suggests that the performance of finite-element basis is competing with the plane-wave Discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations to within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.
-
higher order adaptive finite element methods for orbital free density functional theory
Journal of Computational Physics, 2012Co-Authors: Phani Motamarri, Jaroslaw Knap, Mrinal Iyer, Vikram GaviniAbstract:In the present work, we study various numerical aspects of higher-order finite-element Discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element Discretizations. We next study the convergence properties of higher-order finite-element Discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element Discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element Discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient Discretization of electronic structure calculations using the finite-element basis.
-
a numerical analysis of the finite element Discretization of orbital free density functional theory
2011Co-Authors: Phani Motamarri, Jaroslaw Knap, Mrinal Iyer, Vikram GaviniAbstract:In the present work, we investigate the computational efficiency afforded by higherorder finite-element Discretization of the saddle-point formulation of orbital-free densityfunctional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electrondensity is a robust solution procedure for higher-order finite-element Discretizations. We next study the numerical convergence rate for various orders of finite-element approximations on benchmark problems. We obtain close to optimal convergence rates in our studies, although orbital-free density-functional theory is nonlinear in nature and some benchmark problems have Coulomb singular potential fields. We finally investigate the computational efficiency of various higher-order finite-element Discretizations by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use optimal mesh coarse-graining rates that are derived from error estimates and a priori knowledge of the asymptotic solution of electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element Discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient Discretization of electronic structure calculations using the finite-element basis.
Phani Motamarri - One of the best experts on this subject based on the ideXlab platform.
-
higher order adaptive finite element methods for kohn sham density functional theory
Journal of Computational Physics, 2013Co-Authors: Phani Motamarri, M R Nowak, Kenneth W Leiter, Jaroslaw Knap, Vikram GaviniAbstract:We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element Discretization of Kohn-Sham density-functional theory (DFT). To this end, we develop an a priori mesh-adaption technique to construct a close to optimal finite-element Discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100-200-fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element Discretizations of the Kohn-Sham DFT problem. Our studies suggest that staggering computational savings-of the order of 1000-fold-relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations, by using higher-order finite-element Discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element Discretization of the Kohn-Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element Discretizations suggests that the performance of finite-element basis is competing with the plane-wave Discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations to within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.
-
higher order adaptive finite element methods for orbital free density functional theory
Journal of Computational Physics, 2012Co-Authors: Phani Motamarri, Jaroslaw Knap, Mrinal Iyer, Vikram GaviniAbstract:In the present work, we study various numerical aspects of higher-order finite-element Discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element Discretizations. We next study the convergence properties of higher-order finite-element Discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element Discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element Discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient Discretization of electronic structure calculations using the finite-element basis.
-
a numerical analysis of the finite element Discretization of orbital free density functional theory
2011Co-Authors: Phani Motamarri, Jaroslaw Knap, Mrinal Iyer, Vikram GaviniAbstract:In the present work, we investigate the computational efficiency afforded by higherorder finite-element Discretization of the saddle-point formulation of orbital-free densityfunctional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electrondensity is a robust solution procedure for higher-order finite-element Discretizations. We next study the numerical convergence rate for various orders of finite-element approximations on benchmark problems. We obtain close to optimal convergence rates in our studies, although orbital-free density-functional theory is nonlinear in nature and some benchmark problems have Coulomb singular potential fields. We finally investigate the computational efficiency of various higher-order finite-element Discretizations by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use optimal mesh coarse-graining rates that are derived from error estimates and a priori knowledge of the asymptotic solution of electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element Discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient Discretization of electronic structure calculations using the finite-element basis.
Ludmil T Zikatanov - One of the best experts on this subject based on the ideXlab platform.
-
stability and monotonicity for some Discretizations of the biot s consolidation model
Computer Methods in Applied Mechanics and Engineering, 2016Co-Authors: Carmen Rodrigo, F J Gaspar, Ludmil T ZikatanovAbstract:Abstract We consider finite element Discretizations of the Biot’s consolidation model in poroelasticity with MINI and stabilized P1–P1 elements. We analyze the convergence of the fully discrete model based on spatial Discretization with these types of finite elements and implicit Euler method in time. We also address the issue related to the presence of non-physical oscillations in the pressure approximation for low permeabilities and/or small time steps. We show that even in 1D a Stokes-stable finite element pair fails to provide a monotone Discretization for the pressure in such regimes. We then introduce a stabilization term which removes the oscillations. We present numerical results confirming the monotone behavior of the stabilized schemes.
Eitan Tadmor - One of the best experts on this subject based on the ideXlab platform.
-
Strong Stability-Preserving High-Order Time Discretization Methods
SIAM Review, 2001Co-Authors: Sigal Gottlieb, Chi-wang Shu, Eitan TadmorAbstract:In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time Discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time Discretizations, these high-order time Discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.
