## Multiscale scientific computation: Review 2001 (2001)

Venue: | Multiscale and Multiresolution Methods |

Citations: | 40 - 7 self |

### BibTeX

@INPROCEEDINGS{Brandt01multiscalescientific,

author = {Achi Brandt},

title = {Multiscale scientific computation: Review 2001},

booktitle = {Multiscale and Multiresolution Methods},

year = {2001},

pages = {1--96},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

8530 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...[110] for an early multiscale approach), vertex ordering (see, e.g., in [4]), two-dimensional embedding or the problem of drawing graphs nicely (successfully multiscaled in [99]), min-cut or max-flow =-=[72]-=-, sparse spanners [93], dense subgraphs [81], and others. The general approach in these graph problems is that of coarsening: recursive transition to increasingly coarser graphs, each having only a fr... |

720 |
Linear and Nonlin ear Waves
- Whitham
(Show Context)
Citation Context ...relative change per wavelength. Although in the latter case the rays are curved, they can still be followed by Snell's law, or more generally by marching solutions of the eikonal equation (see, e.g., =-=[160]-=-). Discontinuities in k(x) can also be accommodated by geometrical optics, employing the usual rules of reflection and refraction, as long as the surfaces of discontinuity have curvature radii large c... |

247 | A review of algebraic multigrid
- Stüben
(Show Context)
Citation Context ...braic multigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [23, x13.1], [56], [58], [25], =-=[134]-=-, [146]. The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discretized partial diff... |

212 |
Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes
- Jameson, Schmidt, et al.
- 1981
(Show Context)
Citation Context ...e [22]), the recommended cures seemed complicated, and code developers opted for partial efficiency. The leading multigrid method has been based on multi-stage pseudo-time-stepping relaxation schemes =-=[105]-=-, [104]. Although such schemes can be optimized to damp high-frequency errors [157], the resulting algorithms are still relatively slow, because some intermediate (neither highfrequency nor very smoot... |

200 |
The Radon transform and some of its applications
- Deans
- 1983
(Show Context)
Citation Context ...rom its Radon transform has proven vital in X-ray computed tomography (CT), emission computed tomography, nuclear magnetic resonance (NMR) imaging, astronomy, geophysics, and a number of other fields =-=[76]-=-. One of the best known reconstruction al65sgorithms is the convolution backprojection method (CB), which is widely used in commercial medical CT devices [76] (with "rebinning" for divergent-beam proj... |

178 | Inferring global perceptual contours from local features
- Guy, Medioni
- 1996
(Show Context)
Citation Context ...spectively, 3 and 1 times the pixel size). The completion field value vj of edge j can be built from the set of all elementary responses ui in a variety of ways (see different approaches in [161] and =-=[98]-=-). As a representative example for our discussion here, we can take vj = N1mX i=1 aijui ; (j = 1; : : : ; N1m) ; (15:1) where aij expresses the "affinity" of edgels i and j: it is large if edgel j is ... |

176 |
A global uniqueness theorem for an inverse boundary value problem
- Sylvester, Uhlmann
- 1987
(Show Context)
Citation Context ...ven by Calderon [69]. Kohn and Vogelius [115] showed that under certain assumptions the conductivity of a medium is uniquely determined by the Neumann-to-Dirichlet mapping. Then Sylvester and Uhlmann =-=[149]-=- provided a general framework for proving uniqueness of the solution of the inverse problem. Alessandrini [1] gave a mathematical explanation for the blurriness of conductivity images and proved that ... |

176 | Stochastic completion fields: A neural model of illusory contour shape and salience,” ICCV
- Williams, Jacobs
- 1995
(Show Context)
Citation Context ... being, respectively, 3 and 1 times the pixel size). The completion field value vj of edge j can be built from the set of all elementary responses ui in a variety of ways (see different approaches in =-=[161]-=- and [98]). As a representative example for our discussion here, we can take vj = N1mX i=1 aijui ; (j = 1; : : : ; N1m) ; (15:1) where aij expresses the "affinity" of edgels i and j: it is large if ed... |

173 |
1977] - Multi level adaptive solutions to boundary-value problems
- Brandt
(Show Context)
Citation Context ...he problems, which usually requires many iterative applications of a linear solver. FAS multigrid procedures can directly solve nonlinear problems, as efficiently as solving their linear counterparts =-=[19]-=-, [23], [24]. The development of such one-shot solvers for the Schr"odinger operator depends on the ability to update the selfconsistent potential as the solution changes on the coarse grids. This is ... |

151 | Multigrid techniques: 1984 guide with applications to fluid dynamics, The Weizmann Institute of Science, Rehovot - Brandt - 1984 |

127 |
Algebraic multigrid theory: The symmetric case
- Brandt
- 1986
(Show Context)
Citation Context ..., then the propagationdirection meshsize for that level is O(J 2k\Gamma 1), while the perpendicular-direction one is O(Jk\Gamma 1). Incidentally, such oriented grids should have also been employed in =-=[25]-=-, reducing the order of complexity stated there to the same one as in the non-oscillatory case (with an additional O(log n) factor in the case of integral transforms or integral equations defined on a... |

121 | R.: Fast multiscale image segmentation
- Sharon, Brandt, et al.
- 2000
(Show Context)
Citation Context ... inexpensive processing, so that the entire algorithm costs only several dozen computer operations per pixel. A detailed account of our current multiscale algorithm for image segmentation is given in =-=[140]-=-, demonstrating its properties on several line drawings and real images. The future strategy is to enhance the algorithm in several ways indicated above: adding more coarse-level observables, introduc... |

120 |
A geometrical theory of Diffraction
- Keller
(Show Context)
Citation Context ...n the small special regions (around an aperture, corners, edges, a radar target, etc.), yielding a general numerical tool for computing diffraction (the rays produced by small-scale disturbances; cf. =-=[111]-=-). 8 Rigorous Quantitative Analysis of Multigrid Since the early days of multigrid development, the "local mode analysis" (LMA), based on heuristic local Fourier decomposition of the error function, h... |

109 |
Statistical Mechanics of Chain Molecules
- Flory
- 1969
(Show Context)
Citation Context ...are reported in [7] and briefly reviewed below. For alternative coarsening schemes, see [8]. The simple polymer is a non-branching long chain of n repeat units (monomers; see the comprehensive survey =-=[82]-=-). In the united-atom model each monomer is considered as a single unit and details of its internal structure and interactions are ignored, so mathematically the polymer is represented as a chain of p... |

93 |
Guide to multigrid development
- Brandt
- 1982
(Show Context)
Citation Context ...blems, which usually requires many iterative applications of a linear solver. FAS multigrid procedures can directly solve nonlinear problems, as efficiently as solving their linear counterparts [19], =-=[23]-=-, [24]. The development of such one-shot solvers for the Schr"odinger operator depends on the ability to update the selfconsistent potential as the solution changes on the coarse grids. This is also r... |

86 |
On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro
- Calderón
- 1980
(Show Context)
Citation Context ...) boundary condition resulting from any Neumann (@u=@n) condition. The inverse EIT problem is to calculate oe from this mapping. The first description of the inverse EIT problem was given by Calderon =-=[69]-=-. Kohn and Vogelius [115] showed that under certain assumptions the conductivity of a medium is uniquely determined by the Neumann-to-Dirichlet mapping. Then Sylvester and Uhlmann [149] provided a gen... |

85 |
Algebraic multigrid (AMG) for sparse matrix equations. Sparsity and its Applications
- Brandt, McCormick, et al.
- 1984
(Show Context)
Citation Context ...roaches Algebraic multigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [23, x13.1], [56], =-=[58]-=-, [25], [134], [146]. The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discretized... |

81 |
Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations
- Brandt, McCormick, et al.
- 1982
(Show Context)
Citation Context ...ew Approaches Algebraic multigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [23, x13.1], =-=[56]-=-, [58], [25], [134], [146]. The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discr... |

67 |
Hamiltonian formulation of Wilson’s lattice gauge theories”, Phys
- Kogut, Susskind
- 1975
(Show Context)
Citation Context ...(Nc) model the gauge group elements are unitary complex Nc \ThetasNc matrices whose determinant is 1, and OEh(z) and f h(z) are complex Nc-vectors. See more about these models in [162], [112], [113], =-=[114]-=-, [147], and about a multigrid approach to related, simplified models in [108] and [109]. These systems, despite their linearity and good ellipticity measures, are very challenging, due to their topol... |

66 |
Stable determination of conductivity by boundary measurements, Applicable Anal
- Alessandrini
- 1988
(Show Context)
Citation Context ...eled by the eigenvalue of the equation\Gammas\Gammas\Deltas+ V (x; z)\Delta (x) = *(x) ; x = (x1; x2) 2 D : (9:5) We chose V (x; z) that models the Coloumbic potential at x of the two-ion system, D = =-=[0; 1]-=- \Thetas[0; 1], andswas required to satisfy periodic boundary conditions on D (having chosen V and E also with this periodicity). The Euler equations for minimizing (9.4) under the constraint (9.5) ca... |

56 |
Multilevel matrix multiplication and fast solution of integral equations
- Brandt, Lubrecht
- 1990
(Show Context)
Citation Context ...ace\GammasG(x; y) = exp(\Gamma xy)\Deltastransforms, for example. More generally, the algorithms yield fast matrix multiplication rules for dense matrices that include large lowrank submatrices. (See =-=[53]-=-, [29], [158] and references in [158]). The same algorithms can also be used for the fast \Gamma O(N log 1ffl )\Deltassummation of all the forces that N particles exerts on each other (see Sec. 14.3 b... |

56 |
Determining the conductivity by boundary measurements
- Kohn, Vogelius
- 1984
(Show Context)
Citation Context ...ulting from any Neumann (@u=@n) condition. The inverse EIT problem is to calculate oe from this mapping. The first description of the inverse EIT problem was given by Calderon [69]. Kohn and Vogelius =-=[115]-=- showed that under certain assumptions the conductivity of a medium is uniquely determined by the Neumann-to-Dirichlet mapping. Then Sylvester and Uhlmann [149] provided a general framework for provin... |

55 |
Updating the singular value decomposition
- Bunch, Nielsen
- 1978
(Show Context)
Citation Context ...s applications in numerical linear algebra, such as subspace computations [80], [84], solving constrained least-squares type problems [83], [92], updating the singular value decomposition of matrices =-=[68]-=-, and modifying the symmetric eigenvalue problem [75]; see survey of literature and solution methods in [125], [126]. The equations should often be solved many times as a subproblem of a larger one. A... |

54 | The partition of unity method
- Babuška, Melenk
- 1997
(Show Context)
Citation Context ...nal O(log n) factor in the case of integral transforms or integral equations defined on a curved manifold of codimension 1, e.g., a boundary). A finite-element representation akin to (7.2) appears in =-=[5]-=- and [124], but only on one level, and without the above-mentioned grid alignment. Unlike that representation, the present one can be used to bridge the entire gap between the wave discretization need... |

54 |
Multi-level adaptive technique (MLAT) for fast numerical solutions to boundary value problems
- Brandt
(Show Context)
Citation Context ...). 2 Steady-State Fluid Dynamics 2.1 Objective: textbook multigrid efficiency An efficient multigrid algorithm for steady-state incompressible viscous flows in two dimensions appeared already in 1972 =-=[18]-=-, a relatively efficient multigrid solver for a compressible inviscid transonic flow was demonstrated in 1975 [145], and a fully efficient solver for a system of several coupled differential equations... |

54 |
U.: Quadratically constrained least squares and quadratic problems
- Golub, Matt
- 1991
(Show Context)
Citation Context ...e real, oe ? 0 and uk ? 0 for all k. This problem has various applications in numerical linear algebra, such as subspace computations [80], [84], solving constrained least-squares type problems [83], =-=[92]-=-, updating the singular value decomposition of matrices [68], and modifying the symmetric eigenvalue problem [75]; see survey of literature and solution methods in [125], [126]. The equations should o... |

51 | General highly accurate algebraic coarsening, Elect
- Brandt
(Show Context)
Citation Context ...ese difficulties. So, as already suggested in [31], we have returned to the development of AMG Dirac solvers. Our first AMG Dirac solver has been based on the highly-accurate coarsening techniques of =-=[37]-=- (see Sec. 17.1 below). This solver, briefly described in [37, App. C] and detailed in [132], has been tested for the Schwinger model (two-dimension, with U (1) gauge). We use Kacmarz or least square ... |

49 | Completion energies and scale
- Sharon, Brandt, et al.
(Show Context)
Citation Context ... are themselves multiscaled. A detailed study of multiscale completion fields, their parameterization and fast implementation has been conducted with Eitan Sharon and Ronen Basri. It is summarized in =-=[139]-=-. Intriguing possibilities of combining the developed algorithms in a variety of ways should be investigated: 1. Iterating a multiscale algorithm, with the output of the first iteration (e.g., the set... |

48 |
Accelerated multigrid convergence and high Reynolds recirculating flows
- Brandt, Yavneh
- 1993
(Show Context)
Citation Context ...on (or convection-diffusion) operators. The former's multigrid solver is classical; the latter's can be based on downstream relaxation [64], with additional special procedures for recirculation flows =-=[65]-=-, 9s[166]. Indeed, incorporating such procedures into the relaxation schemes for the appropriate ghost functions yields very efficient solvers for incompressible flows even at high Reynolds numbers an... |

47 |
Multilevel computations of integral transforms and particle interactions with oscillatory kernels
- Brandt
- 1991
(Show Context)
Citation Context ...ic noise; electronic waves in condensed matter; etc. We have developed two basic approaches relevant to the problem. One is a general multiscale solver for integral equations with oscillatory kernels =-=[29]-=-, which is a very efficient way to solve wave propagation in homogeneous (and some piecewise homogeneous) media by replacing the differential equations with boundary integral equations. Multiscale ray... |

46 |
A multigrid tutorial (2nd ed
- Briggs, Henson, et al.
- 2000
(Show Context)
Citation Context ...ters summarize important recent techniques, and some less known older ones, in the field of multigrid PDE solvers, assuming a general familiarity with its basic elements. (For introductory books, see =-=[67]-=- and [153]; or at least read the "Elementary acquaintance with multigrid" in [24]; see also the basic insight described in Sec. 17 below.) Some algorithms and concepts are explained in more details th... |

43 |
Algebraic Multigrid (AMG): An Introduction with Applications
- Stüben
- 1999
(Show Context)
Citation Context ...ultigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [23, x13.1], [56], [58], [25], [134], =-=[146]-=-. The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discretized partial differentia... |

42 |
On multigrid solution of high-Reynolds incompressible entering flows
- Brandt, Yavneh
- 1992
(Show Context)
Citation Context ...e resulting algorithms are still relatively slow, because some intermediate (neither highfrequency nor very smooth) "characteristic components" cannot adequately be reduced by coarse grids (cf. [22], =-=[64]-=-). Other multigrid solvers were based on incomplete LU decomposition (ILU) and related relaxation schemes [159], [155], [144]. While such schemes give excellent results in some cases, they cannot 8scu... |

39 |
An introduction to lattice gauge theory and spin systems
- Kogut
- 1979
(Show Context)
Citation Context ...m.) In the SU (Nc) model the gauge group elements are unitary complex Nc \ThetasNc matrices whose determinant is 1, and OEh(z) and f h(z) are complex Nc-vectors. See more about these models in [162], =-=[112]-=-, [113], [114], [147], and about a multigrid approach to related, simplified models in [108] and [109]. These systems, despite their linearity and good ellipticity measures, are very challenging, due ... |

39 |
On Generalized Finite Element Methods
- Melenk
- 1995
(Show Context)
Citation Context ...g n) factor in the case of integral transforms or integral equations defined on a curved manifold of codimension 1, e.g., a boundary). A finite-element representation akin to (7.2) appears in [5] and =-=[124]-=-, but only on one level, and without the above-mentioned grid alignment. Unlike that representation, the present one can be used to bridge the entire gap between the wave discretization needed at smal... |

38 |
Multigrid solutions to elliptic flow problems. Numerical Methods for Partial Differential Equations
- Brandt, Dinar
- 1979
(Show Context)
Citation Context ...as demonstrated in 1975 [145], and a fully efficient solver for a system of several coupled differential equations, characteristic to computational fluid dynamics (CFD), was presented already in 1978 =-=[40]-=-. However, in the decades that followed, the development in this area has not been fully satisfactory. In particular, the efficiency of solvers for nonelliptic steady-state systems (such as Euler and ... |

37 |
Solution of the Euler equations by a multigrid method
- Jameson
- 1983
(Show Context)
Citation Context ..., the recommended cures seemed complicated, and code developers opted for partial efficiency. The leading multigrid method has been based on multi-stage pseudo-time-stepping relaxation schemes [105], =-=[104]-=-. Although such schemes can be optimized to damp high-frequency errors [157], the resulting algorithms are still relatively slow, because some intermediate (neither highfrequency nor very smooth) "cha... |

35 |
The dense k-subgraph problem. Algorithmica
- Feige, Peleg, et al.
- 2001
(Show Context)
Citation Context ...tex ordering (see, e.g., in [4]), two-dimensional embedding or the problem of drawing graphs nicely (successfully multiscaled in [99]), min-cut or max-flow [72], sparse spanners [93], dense subgraphs =-=[81]-=-, and others. The general approach in these graph problems is that of coarsening: recursive transition to increasingly coarser graphs, each having only a fraction of the number of nodes and edges at t... |

32 | Rigorous quantitative analysis of multigrid, I. Constant coefficients two-level cycle with L2-norm
- Brandt
- 1994
(Show Context)
Citation Context ...domains with reentrant corners are a notorious example. A general rigorous framework for the local mode analysis on one hand, and for the treatment of boundaries on the other hand, has appeared [28], =-=[32]-=-. 24sFor general linear elliptic PDE systems with piecewise smooth coefficients in general domains discretized by uniform grids, it has been proved that, in the limit of small meshsizes, the convergen... |

31 |
Local mesh refinement multilevel techniques
- Bai, Brandt
- 1987
(Show Context)
Citation Context ...particularly flexible and efficient framework for that purpose, with 20ssome additional benefits. The general description of this framework has been given elsewhere [19, xx7-9], [23, x9], [24, x9] or =-=[6]-=-, so we only summarize the main points. Local refinements are created by adding local patches of finer uniform grids over desired parts of a domain covered by a "parent" coarse grid. (The system is re... |

30 |
Statistical Mechanics
- Hill
(Show Context)
Citation Context ...e multilevel process, the configurations produced at the coarsest level correspond to the canonical ensemble; at finer levels they yield direct accurate representation of the grand canonical ensemble =-=[103]-=-. At sufficiently coarse levels, this entire algorithm effectively produces macroscopic "equations" for the simulated system, in the form of numerical CP tables. This can yield a macroscopic numerical... |

29 |
On the stationary values of a second-degree polynomial on the unit sphere
- Forsythe, Golub
- 1965
(Show Context)
Citation Context ... dN are real, oe ? 0 and uk ? 0 for all k. This problem has various applications in numerical linear algebra, such as subspace computations [80], [84], solving constrained least-squares type problems =-=[83]-=-, [92], updating the singular value decomposition of matrices [68], and modifying the symmetric eigenvalue problem [75]; see survey of literature and solution methods in [125], [126]. The equations sh... |

26 | Multilevel computations: Review and recent developments - Brandt - 1988 |

26 | Wave-ray multigrid method for standing wave equations
- Brandt, Livshits
- 1997
(Show Context)
Citation Context ...sional solver and a preliminary version of the twodimensional solver were given in [122]. The current version of the two-dimensional solver, together with numerical results, is described in detail in =-=[51]-=-; its accuracy is analyzed in [52]. An important feature of the solver is the alignment of the grid on which Aj(x) is discretized with the propagation direction of the corresponding eikonal (the direc... |

24 |
Recombining iterants in multigrid algorithms and problems with small islands
- Brandt, Mikulinsky
- 1995
(Show Context)
Citation Context ... iterants (taking linear combinations of results of the latest multigrid cycles so as to minimize the residual L2 norm; which can also be done on coarser levels of the multigrid hierarchy; see [130], =-=[59]-=-) together with local relaxation passes added around the local discontinuities. With these devices, and with the localgauge intergrid transfers and transport gauge averaging mentioned above, the multi... |

20 |
Multigrid solvers for non-elliptic and singular-perturbation steady-state problems. (unpublished). The Weizmann
- Brandt
- 1981
(Show Context)
Citation Context ...f magnitude behind the ideal efficiency that had been attained for general elliptic systems. Although the main reasons for this inefficiency have also been understood for a long time (see for example =-=[22]-=-), the recommended cures seemed complicated, and code developers opted for partial efficiency. The leading multigrid method has been based on multi-stage pseudo-time-stepping relaxation schemes [105],... |

20 |
Multilevel methods for differential eigenproblems
- Brandt, McCormick, et al.
- 1983
(Show Context)
Citation Context ... the theory of the electron gas. In the local density approximation, Vxc(r) depends only on ae(r) and possibly on the gradient (rae)(r). Fast multigrid eigenproblem solvers have been developed before =-=[57]-=-, but the ab-initio problem includes new traits and difficulties that call for new multiscale techniques, such as in the following list: (1) Singularities. The nuclear potential energy harbors a singu... |

20 |
Multi-level approaches to discrete-state and stochastic problems
- Brandt, Ron, et al.
- 1986
(Show Context)
Citation Context ... orchestrated large-scale moves are constructed. This is done by what we will generally call "multilevel annealing", whose main features are described below. Its first, incomplete version appeared in =-=[61]-=-. 18.2.1 Identifying multiscale variables or moves In multilevel optimization, the main role of annealing is to identify increasingly larger-scale degrees of freedom that are acceptable to simulation ... |

20 |
Design of optimally-smoothing multi-stage schemes for the Euler equations. AIAA Paper 89-1933
- Leer, Tai, et al.
- 1989
(Show Context)
Citation Context ...rtial efficiency. The leading multigrid method has been based on multi-stage pseudo-time-stepping relaxation schemes [105], [104]. Although such schemes can be optimized to damp high-frequency errors =-=[157]-=-, the resulting algorithms are still relatively slow, because some intermediate (neither highfrequency nor very smooth) "characteristic components" cannot adequately be reduced by coarse grids (cf. [2... |

19 |
Multigrid methods in lattice field computations
- Brandt
- 1992
(Show Context)
Citation Context ...c system are first order), on pre-coarsening gauge smoothing and on multiscale iterate recombination, had previously been applied to the twodimensional (d = 2) U (1) model (see general description in =-=[31]-=-, and full account in [130]). More recently we have been working on the U (1) and SU (2) gauge models in 4D [131], [132]. For the 4D-U (1) gauge model, general conditions have been formulated under wh... |