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HighOrder Collocation Methods for Differential Equations with Random Inputs
 SIAM Journal on Scientific Computing
"... Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling met ..."
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Cited by 188 (13 self)
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Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a highorder stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods. Key words. collocation methods, stochastic inputs, differential equations, uncertainty quantification
Electrical Impedance Tomography
 SIAM REVIEW
, 1999
"... This paper surveys some of the work our group has done in electrical impedance tomography. ..."
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Cited by 164 (2 self)
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This paper surveys some of the work our group has done in electrical impedance tomography.
Development of Parallel Methods for a 1024Processor Hypercube
 SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING
, 1988
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Estimating differential quantities using polynomial fitting of osculating jets
"... This paper addresses the pointwise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation ..."
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Cited by 117 (7 self)
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This paper addresses the pointwise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities —such as normal, curvatures, extrema of curvature. On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation / approximation, a wellstudied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of R 3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.
Is Gauss Quadrature Better Than Clenshaw–Curtis?
, 2008
"... We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to exp ..."
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Cited by 109 (4 self)
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We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O’Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z +1)/(z − 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw–Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [−1, 1].
Pattern clustering by multivariate mixture analysis
 Multivariate Behavioral Research
, 1970
"... Cluster analysis is reformulated as a problem of estimating the parameters of a mixture of multivariate distributions. The maximumlikelihood theory and numerical solution techniques are developed for a fairly general class of distributions. The theory is applied to mixtures of multivariate normal ..."
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Cited by 101 (0 self)
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Cluster analysis is reformulated as a problem of estimating the parameters of a mixture of multivariate distributions. The maximumlikelihood theory and numerical solution techniques are developed for a fairly general class of distributions. The theory is applied to mixtures of multivariate normals (“NORMIX”) and mixtures of multivariate Bernoulli distributions (“Latent Classes”). The feasibility of the procedures is demonstrated by two examples of computer solutions for normal mixture models of the Fisher Iris data and of artificially generated clusters with unequal covariance matrices. This paper is addressed to the problem which has been variously called cluster analysis, Qanalysis, typology, grouping, clumping, classif ication, numerical taxonomy, and unsupervised pattern recognition. The variety of nomenclature may be due to the importance of the subject in such diverse fields as psychology, biology, signal detection, artificial intelligence, and information retrieval.
RungeKutta Methods in Optimal Control and the Transformed Adjoint System
 Numerische Mathematik
, 1999
"... The convergence rate is determined for RungeKutta discretizations of nonlinear control problems. The analysis utilizes a connection between the KuhnTucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle. This connection can also be exp ..."
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Cited by 84 (5 self)
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The convergence rate is determined for RungeKutta discretizations of nonlinear control problems. The analysis utilizes a connection between the KuhnTucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle. This connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function.
Convergence analysis of pseudotransient continuation
 SIAM J. Num. Anal
, 1998
"... Abstract. Pseudotransient continuation (Ψtc) is a wellknown and physically motivated technique for computation of steady state solutions of timedependent partial differential equations. Standard globalization strategies such as line search or trust region methods often stagnate at local minima. Ψ ..."
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Cited by 84 (30 self)
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Abstract. Pseudotransient continuation (Ψtc) is a wellknown and physically motivated technique for computation of steady state solutions of timedependent partial differential equations. Standard globalization strategies such as line search or trust region methods often stagnate at local minima. Ψtc succeeds in many of these cases by taking advantage of the underlying PDE structure of the problem. Though widely employed, the convergence of Ψtc is rarely discussed. In this paper we prove convergence for a generic form of Ψtc and illustrate it with two practical strategies.
A Practical Approach to Dynamic Load Balancing
 IEEE Transactions on Parallel and Distributed Systems
, 1998
"... Abstract—This paper presents a cohesive, practical load balancing framework that improves upon existing strategies. These techniques are portable to a broad range of prevalent architectures, including massively parallel machines, such as the Cray T3D/E and Intel Paragon, shared memory systems, such ..."
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Cited by 82 (8 self)
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Abstract—This paper presents a cohesive, practical load balancing framework that improves upon existing strategies. These techniques are portable to a broad range of prevalent architectures, including massively parallel machines, such as the Cray T3D/E and Intel Paragon, shared memory systems, such as the Silicon Graphics PowerChallenge, and networks of workstations. As part of the work, an adaptive heat diffusion scheme is presented, as well as a task selection mechanism that can preserve or improve communication locality. Unlike many previous efforts in this arena, the techniques have been applied to two largescale industrial applications on a variety of multicomputers. In the process, this work exposes a serious deficiency in current load balancing strategies, motivating further work in this area. Index Terms—Dynamic load balancing, diffusion, massively parallel computing, irregular problems. ————————— — F —————————— 1