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516
The SPLASH2 programs: Characterization and methodological considerations
 INTERNATIONAL SYMPOSIUM ON COMPUTER ARCHITECTURE
, 1995
"... The SPLASH2 suite of parallel applications has recently been released to facilitate the study of centralized and distributed sharedaddressspace multiprocessors. In this context, this paper has two goals. One is to quantitatively characterize the SPLASH2 programs in terms of fundamental propertie ..."
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Cited by 1399 (12 self)
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The SPLASH2 suite of parallel applications has recently been released to facilitate the study of centralized and distributed sharedaddressspace multiprocessors. In this context, this paper has two goals. One is to quantitatively characterize the SPLASH2 programs in terms of fundamental properties and architectural interactions that are important to understand them well. The properties we study include the computational load balance, communication to computation ratio and traffic needs, important working set sizes, and issues related to spatial locality, as well as how these properties scale with problem size and the number of processors. The other, related goal is methodological: to assist people who will use the programs in architectural evaluations to prune the space of application and machine parameters in an informed and meaningful way. For example, by characterizing the working sets of the applications, we describe which operating points in terms of cache size and problem size are representative of realistic situations, which are not, and which re redundant. Using SPLASH2 as an example, we hope to convey the importance of understanding the interplay of problem size, number of processors, and working sets in designing experiments and interpreting their results.
A Theory of Networks for Approximation and Learning
 Laboratory, Massachusetts Institute of Technology
, 1989
"... Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, t ..."
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Cited by 237 (25 self)
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Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of threelayer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the wellknown Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 189 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
Scattered Data Interpolation with Multilevel Splines
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1997
"... This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequen ..."
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Cited by 158 (10 self)
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This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequence of bicubic Bspline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using Bspline refinement to reduce the sum of these functions into one equivalent Bspline function. Experimental results demonstrate that highfidelity reconstruction is possible from a selected set of sparse and irregular samples.
Height and gradient from shading
 International Journal of Computer Vision
, 1990
"... Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented (A gradient field is integrable if it is the gradient of some surface ..."
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Cited by 136 (1 self)
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Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented (A gradient field is integrable if it is the gradient of some surface height function). The robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell (The reflectance map gives the dependence of scene radiance on surface orientation). The new scheme can find an exact solution of a given shapefromshading problem even though a regularizing term is included. The reason is that the penalty term is needed only to stabilize the iterative scheme when it is far from the correct solution; it can be turned off as the solution is approached. This is a reflection of the fact that shapefromshading problems are not illposed when boundary conditions are available, or when the image contains singular points. This paper includes a review of previous work on shape from shading and photoclinometry. Novel features of the new scheme are introduced one at a time to make it easier to see what each contributes. Included is a discussion of implementation details that are important if exact algebraic solutions of synthetic shapefromshading problems are to be obtained. The hope is that better performance on synthetic data will lead to better performance on real data.
A Multilevel Relaxation Algorithm for Simultaneous Localisation and Mapping
, 2004
"... This paper addresses the problem of simultaneous localisation and mapping (SLAM) by a mobile robot. An incremental SLAM algorithm is introduced that is derived from multigrid methods used for solving partial differential equations. The approach improves on the performance of previous relaxation meth ..."
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Cited by 112 (5 self)
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This paper addresses the problem of simultaneous localisation and mapping (SLAM) by a mobile robot. An incremental SLAM algorithm is introduced that is derived from multigrid methods used for solving partial differential equations. The approach improves on the performance of previous relaxation methods for robot mapping because it optimizes the map at multiple levels of resolution. The resulting algorithm has an update time that is linear in the number of estimated features for typical indoor environments, even when closing very large loops, and offers advantages in handling nonlinearities compared to other SLAM algorithms. Experimental comparisons with alternative algorithms using two wellknown data sets and mapping results on a real robot are also presented.
Analysis of the heterogeneous multiscale method for ordinary differential equations
 Commun. Math. Sci
"... Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution. ..."
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Cited by 104 (10 self)
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Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.
Exactly sparse delayedstate filters for viewbased SLAM
 IEEE Transactions on Robotics
, 2006
"... Abstract—This paper reports the novel insight that the simultaneous localization and mapping (SLAM) information matrix is exactly sparse in a delayedstate framework. Such a framework is used in viewbased representations of the environment that rely upon scanmatching raw sensor data to obtain virt ..."
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Cited by 102 (21 self)
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Abstract—This paper reports the novel insight that the simultaneous localization and mapping (SLAM) information matrix is exactly sparse in a delayedstate framework. Such a framework is used in viewbased representations of the environment that rely upon scanmatching raw sensor data to obtain virtual observations of robot motion with respect to a place it has previously been. The exact sparseness of the delayedstate information matrix is in contrast to other recent featurebased SLAM information algorithms, such as sparse extended information filter or thin junctiontree filter, since these methods have to make approximations in order to force the featurebased SLAM information matrix to be sparse. The benefit of the exact sparsity of the delayedstate framework is that it allows one to take advantage of the information space parameterization without incurring any sparse approximation error. Therefore, it can produce equivalent results to the fullcovariance solution. The approach is validated experimentally using monocular imagery for two datasets: a testtank experiment with ground truth, and a remotely operated vehicle survey of the RMS Titanic. Index Terms—Information filters, Kalman filtering, machine vision, mobile robot motion planning, mobile robots, recursive estimation, robot vision systems, simultaneous localization and mapping (SLAM), underwater vehicles. I.
MatrixDependent Prolongations and Restrictions in a BlackBox Multigrid Solver
, 1990
"... Multigrid methods are studied for the solution of linear systems resulting from the 9point discretization of a general linear secondorder elliptic partial differential equation in two dimensions. The rate of convergence of standard multigrid methods often deteriorates when the coefficients in the ..."
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Cited by 76 (8 self)
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Multigrid methods are studied for the solution of linear systems resulting from the 9point discretization of a general linear secondorder elliptic partial differential equation in two dimensions. The rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous, or when dominating firstorder terms are present. These difficulties may be overcome by choosing the prolongation and restriction operators in a special way. A novel way to do this is proposed. As a result, a blackbox solver (written in standard FORTRAN 77) has been developed. Numerical experiments for several hard test problems are described and comparison is made with other algorithms: the standard MG method and a method introduced by Kettler. A significant improvement of robustness and efficiency is found. Note: This chapter has been published in J. Comput. Appl. Math. 33 (1990) 127. 3.1 Introduction Consider the partial differential equation ...
A novel multigrid based preconditioner for heterogeneous Helmholtz problems
 SIAM J. Sci. Comput
, 2006
"... Abstract. An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtztype differential operator with a complex term. A multigrid iteration is used for approximately inverting the prec ..."
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Cited by 65 (6 self)
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Abstract. An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtztype differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner. Key words. Helmholtz equation, nonconstant high wavenumber, complex multigrid preconditioner, Fourier analysis