S. F. McCormick; Multilevel Projection Methods for Partial Differential Equations; SIAM, 1992  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Research Center, Hampton, Virginia, 23681, USA. 1 Introduction The efficiency of multigrid (MG) techniques in solving large scale eigenvalue problems derived from discretizations of partial differential eigenvalue problems, was shown in multiple works, for example in [1] 3] 13] 16] 18] [20] [21] 27] This work is motivated by the need of tools of analyzing, and designing robust and efficient MG eigenvalue solvers. These tools were needed for the algorithms presented in [4] 5] 6] 19] 24] and in the reports [7] 8] 9] 10] The algorithms were applied to electromagnetism and ....

S. F. McCormick; Multilevel Projection Methods for Partial Differential Equations; SIAM, 1992

.... Either multigrid has been applied directly to (1) without stabilization (see [1] as an example) which produces poor quality reconstructions for high noise to signal ratios (due to the ill posedness of the problem) or stabilization has been applied, but multigrid displays slow convergence (see [2]) In this paper it will be demonstrated how to overcome these difficulties with existing multigrid tools, obtaining a fast algorithm to approximate u in (1) To stabilize problem (1) Tikhonov regularization, or penalized least squares, is used: Research was supported in part by a DOE EPSCoR ....

McCormick, S. F.: Multilevel Projection Methods for Partial Differential Equa- tions, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 62 (1992), Section 4.1, pp. 62-70.

....linear problems, non linear problems generally can only be treated by linearization near a solution. In point of fact, the literature is remarkably sparse in the area of founding theory for the FAS method. A new technique, called multilevel projection methods (PML) has recently been introduced, [18] in an effort to provide a unifying, thematic approach to the design of a multilevel solver for a given problem. The main feature of PML methods is that the only basic choices that must be made concern the subspaces that will be used in relaxation and coarsening. All other components of the ....

....PML methods is that the only basic choices that must be made concern the subspaces that will be used in relaxation and coarsening. All other components of the method, such as interlevel transfers, scaling, coarse level problems, etc. are determined by projection between appropriate subspaces. In [18], several prototypical problems are developed to illustrate the principals involved. It now appears that the best hope of obtaining a strong founding theory for multilevel treatment of nonlinear problems may well be through careful and judicious application of PML, and our future research into ....

Stephen F. McCormick. Multilevel projection methods for partial differential equations, volume 62 of CBM$-NSF regional conference series in applied mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.

....that the subspace problem is also a generalized eigenvalue problem which allows to apply the algorithm recursively and formulate a multilevel method of optimal complexity. Solution of eigenvalue problems by multigrid methods using linearization was discussed by Hackbusch ( Hac84] and McCormick ([McC92]) The idea to use coordinate relaxation applied directly for a matrix eigenvalue problem goes back to the book by Fadeev and Fadeeva [FF63] 1963) where they applied a technique similar to Gauss Seidel method for minimizing the Rayleigh quotient. This approach was extended by Kaschiev [Kas88] and ....

McCormick S. (1992) Multilevel Projection Methods for Partial Differential Equations. SIAM, Philadelphia.

....process. Further, projections with respect to former determined eigenproblem solutions are not necessary. In the case of a selfadjoint eigenproblem and computation of a simple eigenvalue, the just described method coincides with the Rayleigh quotient multigrid minimization of Mandel and McCormick [17, 19]. An important feature of this method is its monotonicity. This means, that the sequence of Rayleigh quotients formed in every step and at every stage of the algorithm decreases monotonically. Since the Rayleigh quotient is bounded from below, the sequence is always convergent. This property is ....

S. F. McCormick. Multilevel Projection Methods for Partial Differential Equations. CBMS-NSF 62, SIAM Philadelphia, Pennsylvania, 1992.

....decomposition methods, when applied to the fixed dimension eigenproblem. Among the latter are the linear (multiplicative) multigrid method of Hackbusch [13] the preconditioned linear (additive) multigrid method due to Bramble et al. 2] the nonlinear multigrid method of Mandel and McCormick [19, 20], and the domain decomposition method due to Chan and Sharapov [4] The first of these methods is constructed to apply to both the selfadjoint and the nonselfadjoint case, whereas the other methods are restricted to the selfadjoint case. For approach (II) the above mentioned multigrid methods ....

....nested grids is based on a posteriori discretization error estimators or, at least, on error indicators, then the term adaptive multigrid method is justified. Examples of this kind have been worked out by Deuflhard et al. 6] as a modification and extension of the nonlinear multigrid method [19, 20] or by Leinen et al. 17] as a hierarchical basis implementation of the preconditioned linear multigrid method [2] In [6] the nonlinear multigrid method has been shown to be more robust for relatively coarse grids than the linear multigrid method a feature, which is essential within adaptive ....

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S. F. McCormick. Multilevel Projection Methods for Partial Differential Equations. CBMS-NSF 62, SIAM Philadelphia, Pennsylvania, 1992.

....of the function v h . The use of all nodal basis functions results in the nonlinear Gau Seidel minimization, the use of global, mutually orthogonal basis functions leads to the nonlinear cg minimization (Bradwick, Fletcher 1966 [4] Polak 1971 [3] 10 A multigrid formulation (McCormick 1992, [2]) is obtained, if additional nodal basis functions vH 2 VH of a coarser space VH ae V h as search directions are added R(u h tv H ) min t2R a(u h tv H ; u h tv H ) u h tv H ; u h tv H ) min t2R a(u h ; u h ) 2t Delta a(u h ; vH ) t 2 a(v H ; vH ) u h ; u h ) 2t Delta ....

S. McCormick, Multilevel Projection Methods for Partial Differential Equations, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992

..... In the special case of an elliptic selfadjoint problem (i.e. OE j 0) one step of a classical multigrid V cycle with Gau Seidel smoother can be regarded as the successive minimization of the energy functional J in the direction of the multilevel nodal basis functions l 2 S (cf. e.g. McCormick [21], Xu [23] or Yserentant [24] We will use a straightforward extension of this multilevel relaxation as the starting point for the construction of monotone multigrid methods for the non smooth optimization problem (1.6) For this reason, we introduce the splitting S j = m X l=1 V l ; 2.1) ....

S.F. McCormick. Multilevel Projection Methods for Partial Differential Equations. SIAM, Philadalphia, 1992.

....whose elements are in S h . The advantage of this approach is that a least squares discretization converts the SN equations, which are a coupled system of first order equations, into a self adjoint variational formulation. Based on this variational formulation MultiLevel Projection Methods [7] can be applied in order to guide the development of a multigrid solver for the resulting discrete system. Unfortunately, this discretization does not behave correctly in the diffusion limit. In order to explain this fact, we use the moment equations, since OE 0 = in the diffusion limit. ....

.... Scalar Flux: meshsize h = 1.25 ) Figure 4.1: Scalar flux solution of problem (4. 1) Therefore, by performing one full multigrid V cycle, a solution with an error on the order of the truncation error is obtained (cf. [7]) As test problem we used the same problem that was used by Larsen, Morel and Miller in [4] which is shown below: 8 : j j x 100 j Gamma 100 N X =1 = 0:01 j (0) 0 for j 0 j (10) 0 for j 0 9 = 4:1) 11 0 5 10 15 20 ....

S.F. McCormick, Multilevel Projection Methods for Partial Differential Equations, SIAM, (1992). 14

....whose elements are in S h . The advantage of this approach is that a least squares discretization converts the SN equations, which are a coupled system of first order equations, into a self adjoint variational formulation. Based on this variational formulation Multi Level Projection Methods [7] can be applied in order to guide the development of a multigrid solver for the resulting discrete system. Unfortunately, this discretization does not behave correctly in the diffusion limit. In order to explain this fact, we use the moment equations, since OE 0 = in the diffusion limit. ....

....coarse level solution as a starting guess and performing a single V cycle on the next finer mesh. This algorithm yields V cycle convergence rates that are below 0:09. Therefore, by performing one full multigrid V cycle, a solution with an error on the order of the truncation error is obtained (cf. [7]) As test problem we used the same problem that was used by Larsen, Morel and Miller in [4] which is shown below: 8 : j j x 100 j Gamma 100 N X =1 = 0:01 j (0) 0 for j 0 j (10) 0 for j 0 9 = 4:1) In our ....

S.F. McCormick, Multilevel Projection Methods for Partial Differential Equations, SIAM, (1992).

....recursive splitting of the data is equivalent to a multilevel windowing operation. The window width is the same as the length of the data at the top level and is halved at every successive level. This multilevel windowing operation can be related to multigrid methods as described in Section 3. 2 of McCormick (1992). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 4 3 2 1 0 1 Time Successive Partitioning of a Dyadic Tree ffl The spectrum in each block is estimated. ffl Then an optimal pruning algorithm is used to recombine adjacent segments for which the spectra are the same. This rule of spliiting the ....

McCormick, S. (1992). Multilevel Projection Methods For Partial Differential Equations.

....a Gauss Seidel relaxation on each level. The restrictions of these V cycles are a semicoarsening. Thus, the multilevel algorithm is similar to the multilevel algorithm in [4] and [5] The Gauss Seidel relaxation and the restriction and prolongation is made like the multilevel projection method in [6]. The multilevel cycle of the sparse grid multilevel algorithm is called Q cycle. The problem of this Q cycle is the calculation of the right hand side during the restriction. In case of general second order elliptic differential the exact stiffness matrix is so complicated that it is not possible ....

McCormick, S., Multilevel Projection Methods for Partial Differential Equations, volume 62 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.

....V such that a(un ; vn ) f(vn ) for every v n 2 V: 2.1) There are two types of iterative solvers for this problem. These are the additive and multiplicative subspace correction methods (see [64] For adaptive composite grids, AFAC and FAC are a very efficient realization of these methods (see [39] and [40] The two correction methods can be used as preconditioners for other iterative solvers (e.g. cg iteration) In most cases, however, one prefers to use the additive correction method as a preconditioner and the multiplicative correction method as a Gauss Seidel iteration. This leads to ....

S. McCormick. Multilevel Projection Methods for Partial Differential Equations, volume 62 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1992.

....factor of O i 1 1 Gammafl j of the complexity of finding the optimal solution of an MDP (Chow and Tsitsiklis, 1989) where fl is the discount factor. They also show that when an ergodicity condition holds, their algorithm is within a constant factor of optimal. Multigrid methods (Briggs, 1987; McCormick, 1992; Rude 1993) have been applied to a variety of physical boundary value problems where the problem solution can be formulated as an iteration to convergence over a lattice. These problems include linear systems, finite element analysis networks, algebraic and differential eigenvalue problems, and ....

McCormick, S. F. (1992) Multilevel Projection Methods for Partial Differential Equations, Society for Industrial and Applied Mathematics, Capital City Press.

....objective function f(x) It is shown that all the limit points of the generated sequence are in the solution set of (1.11) The recursive application of this techniques results in a multilevel algorithm. Application of the multilevel and multigrid techniques was considered in the book by McCormick [24] who mostly focuses on linear problems but also discusses multilevel techniques for solving the eigenvalue problem and the variational form of the Riccati equation. In his book on multigrid methods Hackbusch [15] also discusses nonlinear applications including applications to the eigenvalue ....

S.F. McCormick. Multilevel Projection Methods for Partial Differential Equations. SIAM, Philadelphia, 1992.

....[19] 25] and in [20] In this section we restrict the presentation of numerical results to a full multigrid solver for problems in slab geometry. We refer the reader, who is not familiar with multigrid methods to (Briggs [5] for an introduction and to (Hackbusch [11] and (McCormick [21] 22] [23]) for more advanced topics. The proper choice of the components, namely, the inter grid transfer operators, coarse grid problems, and relaxation schemes, is essential for the efficiency of a multigrid solver. The choice of the first two components is naturally given by the leastsquares variational ....

.... operator, which is a mapping from a coarse grid to the next finer grid in the grid sequence, is formed directly by composing the isomorphisms between the discrete spaces and their corresponding coordinate spaces with the injection mapping between V k Gamma1 and V k (Bramble [4] McCormick [23]) The restriction operators, which are mappings from a finer grid to the next coarser grid, are just the adjoints of the prolongation operators. Therefore, the only multigrid components that need to be chosen here are the sequence of discrete spaces and the relaxation. For the discrete subspaces, ....

S.F. McCormick, Multilevel Projection Methods for Partial Differential Equations, SIAM, Philadelphia, 1992.

.... from linear algebra introduced in 1937 by Kaczmarz [35] A proof of the linear convergence for general Hilbert space projection methods coupled with Kaczmarz iteration can be found in [51] We mention here that in the multigrid literature Kaczmarz method is frequently used as a smoother (see e.g. [41]) and will come back to this fact again later. 3.3. Other Methods We would like to briefly mention some of the other approaches taken to solve various differential equations problems by RBFs. In the dual reciprocity method [6, 7, 24, 25, 26, 27] one considers an elliptic PDE of the form Lu = ....

McCormick, S. F., Multilevel Projection Methods for Partial Differential Equations, CBMS-NSF Regional Conference Series in Applied Mathematics 62, SIAM, Philadelphia, 1992.

....linear problems, non linear problems generally can only be treated by linearization near a solution. In point of fact, the literature is remarkably sparse in the area of founding theory for the FAS method. A new technique, called multilevel projection methods (PML) has recently been introduced, [18] in an effort to provide a unifying, thematic approach to the design of a multilevel solver for a given problem. The main feature of PML methods is that the only basic choices that must be made concern the subspaces that will be used in relaxation and coarsening. All other components of the ....

....PML methods is that the only basic choices that must be made concern the subspaces that will be used in relaxation and coarsening. All other components of the method, such as interlevel transfers, scaling, coarse level problems, etc. are determined by projection between appropriate subspaces. In [18], several prototypical problems are developed to illustrate the principals involved. It now appears that the best hope of obtaining a strong founding theory for multilevel treatment of nonlinear problems may well be through careful and judicious application of PML, and our future research into ....

Stephen F. McCormick. Multilevel projection methods for partial differential equations, volume 62 of CBMS-NSF regional conference series in applied mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.

....steps are needed to propagate the effect of the boundary conditions to interior points of the domain over which the problem is defined. Multigrid methods have been developed for the solution of boundary value problems as a way to decrease the number of iterations needed in a relaxation approach (McCormick, 1992; Rude, 1993) The problem is transformed to equivalent problems defined over the domain discretized at different resolutions. Here we apply the multigrid approach to the Q Learning algorithm (Watkins, 1989) In the remaining sections, we recast Q Learning as a multigrid method and describe ....

McCormick, S. F. (1992). Multilevel Projection Methods for Partial Differential Equations. SIAM, Philadelphia, Pennsylvania.

..... Alternatively, we can interpret the GS relaxation on the semidefinite system as a subspace correction method [Xu, 1990] where we relax a function u ( 2 V k with respect to a OE 2 E k by u ( 1) u ( Delta OE with a(u ( 1) OE) f(OE) 43) compare also the PMG method in [McCormick, 1992]. The BGS relaxation corresponds to the simultaneous relaxation of a set Phi ae E k by u ( 1) u ( X OE2 Phi OE Delta OE with 8OE 2 Phi : a(u ( 1) OE) f(OE) 44) which is equivalent to the solution of the variational problem for the error e ( u Gamma u ( 8OE 2 ....

McCormick, S. F. (1992). Multilevel projection methods for partial differential equations. In CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM.

....engineering applications involves structured meshes. These meshes may be nested (as in multigrid codes) or may be irregularly coupled (called Multiblock or Irregularly Coupled Regular Mesh Problems) Multigrid is a common technique for accelerating the solution of partial differential equations [14]. Multigrid codes employ a number of meshes at different levels of resolution. The restriction and prolongation operations for shifting between different multigrid levels require moving regular array sections with non unit strides. In multiblock problems, the data is divided into several ....

S. McCormick. Multilevel Projection Methods for Partial Differential Equations. SIAM, 1992.

....linear problems, non linear problems generally can only be treated by linearization near a solution. In point of fact, the literature is remarkably sparse in the area of founding theory for the FAS method. A new technique, called multilevel projection methods (PML) has recently been introduced, [18] in an effort to provide a unifying, thematic approach to the design of a multilevel solver for a given problem. The main feature of PML methods is that the only basic choices that must be made concern the subspaces that will be used in relaxation and coarsening. All other components of the ....

....PML methods is that the only basic choices that must be made concern the subspaces that will be used in relaxation and coarsening. All other components of the method, such as interlevel transfers, scaling, coarse level problems, etc. are determined by projection between appropriate subspaces. In [18], several prototypical problems are developed to illustrate the principals involved. It now appears that the best hope of obtaining a strong founding theory for multilevel treatment of nonlinear problems may well be through careful and judicious application of PML, and our future research into ....

Stephen F. McCormick. Multilevel projection methods for partial differential equations, volume 62 of CBMS-NSF regional conference series in applied mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.

....a Gau Seidel relaxations on each level. The restrictions of these V cycles are a semicoarsening. Thus, the multilevel algorithm is similar to the multilevel algorithm in [4] and [5] The Gau Seidel relaxation and the restriction and prolongation is made like the multilevel projection method in [6]. The multilevel cycle of the sparse grid multilevel algorithm is called Q cycle. The problem of this Q cycle is the calculation of the right hand side during the restriction. In case of general second order elliptic differential the exact stiffness matrix is so complicated that it is not possible ....

McCormick, S., Multilevel Projection Methods for Partial Differential Equations, volume 62 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.

....applications involves structured meshes. These meshes may be nested (as in multigrid codes) or may be irregularly coupled (called Multiblock or Irregularly Coupled Regular Mesh Problems) 9] Multigrid is a common technique for accelerating the solution of partial differential equations [5, 30]. Multigrid codes employ a number of meshes at different levels of resolution. The restriction and prolongation operations for shifting between different multigrid levels require moving regular array sections [19] with non unit strides. In multiblock problems, the data is divided into several ....

S. McCormick. Multilevel Projection Methods for Partial Differential Equations. SIAM, 1992.

....solution process that is faithful to the least squares principles, and therefore avoids the disadvantages of earlier attempts. Multilevel Projection Methods 3 This short note gives an overview of the basic FOSLS approach and its multigrid treatment by the multilevel projection method (PML, [5]) in the next two sections, respectively. The fourth section describes extensions to other applications of the methods developed in [1] and [2] for second order elliptic type problems. The final section contains a few brief remarks. 2. FOSLS In this section, we illustrate the basic concepts ....

....such that G(u; p; f) min (v;q)2W ThetaV G(v; q; f) 3.1) The Rayleigh Ritz version of PML is ideally suited to this numerical task because it designs the relaxation and coarsening processes in concert with the Rayleigh Ritz discretization of (3. 1) To explain this formally and in brief (see [5] for more detail, and for other incarnations of PML, together with several applications) consider a discrete subspace W h Theta V h ae W Theta V indexed by the mesh parameter h on which (3.1) is to be solved. To implement the PML methodology, we need relaxation subspaces W h k Theta V ....

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McCormick,S. Multilevel Projection Methods for Partial Differential Equations, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 112 pages.

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