S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: \Multiscale Wavelet Methods for PDEs", (W. Dahmen, A. Kurdila, and P. Oswald, Eds.), Academic Press, San Diego, 1997, 237-284.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... problem with right hand side f 2 B 1 L2( 1 and a stable wavelet basis with at least vanishing moments, we obtain for the procedure choosing the n largest weighted coecients d of P 1 n=1 n s=d n;1 (u) s 1 d 1 2 1 for all 0 s min( d 2(d 1) 1 3 ) see [2]. This is the optimal degrees of freedom to error rate n 1=d of uniform grid re nement and H 2 regular problems, as long as d 2 2d 2 . The best n term approximation gives an explicit characterisation of (quasi ) optimal grids. We can study singularities which may occur for the solution ....

S. Dahlke, S., Dahmen, W., DeVore, R. Nonlinear approximation and adaptive techniques for solving elliptic operator equations. in: Multiscale Techniques for PDEs, Dahmen, W., Kurdilla, A., Oswald, P., Academic Press (1997), 237-284.

....to 2 Gammaj d= 2(d Gamma1) 5.3 Adaptivity A formal advantage of multilevel Riesz basis and frame decompositions is that they provide isomorphisms to coefficient spaces for a number of BesovSobolev spaces on domains. This can be used for optimal approximation resp. compression purposes (see [53, 32, 33, 20] for some information in this direction) Roughly speaking, adaptivity strategies based on coefficient information are successful (and can be justified theoretically) if a suitable decomposition of the function under consideration is available at low cost. This is the case for many applications in ....

....are successful (and can be justified theoretically) if a suitable decomposition of the function under consideration is available at low cost. This is the case for many applications in signal and image processing but much harder to implement for the solution of operator equations (see, however, [21, 20] for a possible strategy) As a compromise, we outline here an approach which is partly based on heuristic arguments. It is implemented in several adaptive finite element codes [3, 6] see also [62] for a somewhat different variant) and has led to satisfactory results in standard applications. ....

Dahlke, S., W. Dahmen and R. A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, this volume.

....in a fixed cone have occured earlier in Chang, Krantz, and Stein [5] but in a form not sufficient for our purposes. Also, the extension operator of the type (3) has something in common with wavelet constructions proposed to extend Besov spaces from Lipschitz domains in Dahlke, Dahmen, and DeVore [7]. We discuss these similarities in a more detail in Section 2. To formulate our solution of Problem B, we introduce Peetre type maximal functions. For an f 2 D 0 ( Omega Gamma and x 2 Omega these are given by Omega j;N f(x) sup y2 Omega j j f(y)j (1 2 j jx Gamma yj) N (N 0) ....

....after this work has been finished, Prof. R. A. DeVore informed us about interesting similarities between (2. 7) and extension operators that has recently arisen as a by product of wavelet analysis on Lipschitz domains developed in Cohen, Dahmen, and DeVore [6] and Dahlke, Dahmen, and DeVore [7]. The idea can be roughly described as follows (see [7, p. 18] Let Omega be an s.L.d. and let f( j;m ; j;m ) j 2 N 0 ; m 2 Z n g be compactly supported, sufficiently smooth, biorthogonal wavelet basis on R n , so that the identity f X j;m hf; j;m i j;m holds in a variety of ....

S. Dahlke, W. Dahmen, and R. A. DeVore, `Nonlinear approximation and adaptive techniques for solving elliptic operator equations', IGPM-Bericht Nr. 132 (RWTH Aachen, 1996).

....approximation of operator equations can be a challenging field of application. The motivation for using wavelets here is at least twofold: they provide optimal preconditioning of the arising ill conditioned linear systems [J, DK, DPS] and they allow the definition of efficient adaptive schemes, [LTc, MPR, B, DDHS, CC, DDV]. In addition, the flexibility in the construction of biorthogonal wavelets leaves some room which can be used to adapt these systems to special problems at hand, see [DW, U, DKU1] for example. Biorthogonal wavelet systems on the unitary interval, which can be required to satisfy certain boundary ....

S. Dahlke, W. Dahmen, and R. deVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, RWTH Aachen, Preprint IGPM No. 132, 1996; in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdilla, and P. Oswald, eds., Academic Press, to appear.

....approximation of operator equations can be a challenging field of application. The motivation for using wavelets here is at least twofold: they provide optimal preconditioning of the arising ill conditioned linear systems [24, 18, 21] and they allow the definition of efficient adaptive schemes, [28, 30, 3, 13, 5, 14]. In addition, the flexibility in the construction of biorthogonal wavelets leaves some room which can be used to adapt these systems to special problems at hand, see [15, 34, 19] for example. Biorthogonal wavelet systems on the unitary interval, which can be required to satisfy certain boundary ....

S. Dahlke, W. Dahmen, and R. deVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, RWTH Aachen, Preprint IGPM No. 132, 1996; in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdilla, and P. Oswald, eds., Academic Press, to appear.

....BR78, BW85, V94] and integral equations [St96, Dahl96] and have proven to be very powerful especially for problems with low global Sobolev smoothness due to edge or corner singularities or singular perturbed operators or not sufficiently smooth right hand sides. We refer to the recent papers [Dahm97, DDD97, DeV98] where excellent results were obtained using N term best approximation. However, these techniques pose huge technical problems connected with mesh refinement strategies especially in higher dimensions. Therefore a combination of adaptivity to capture the non smooth parts of the solution and a ....

....of adaptivity without complicated mesh refinement strategies especially for problems in higher dimensions. Nevertheless, for singularly perturbed problems with large ellipticity constants and problems that exhibit boundary singularities, a posteriori adaptivity is still necessary. We refer to [Dahm97, DDD97] and [DeV98] for very promising results on nonlinear approximation and adaptivity. ffl The constructions of the approximation spaces presented in this paper are not restricted to biorthogonal wavelets as basis functions, but can be carried over to other multiscale basis functions as well. ....

S. Dahlke, W. Dahmen, R.A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdila, P. Oswald, (eds.), Academic Press, 1997, 237-284.

....strategy based on i) is known as threshold coding in the engineering literature; it will be referred to as absolute cut off in the present paper. The compression problem as stated above is a non linear approximation problem, which has been thoroughly investigated in [12] 13] 14] 15] 17] [8]. The absolute cut off is shown to yield optimal results if the error is measured in a family of Besov spaces, in which the summability index depends upon the regularity index. A basic tool of the analysis is the characterization of Besov seminorms in terms of wavelet coefficients. In the absolute ....

S. Dahlke, W. Dahmen and R.A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, preprint, RWTH Aachen, 1996.

.... 1 N 1 if and only if F 2 B s (L (R d ) s=d 1=2) 1 ; 3) where the B s (L (R d ) are the Besov spaces (see, e.g. 15, 20] for the de nition and the main properties of Besov spaces) Similar results also hold for other norms such as L p and Sobolev norms, see, e.g. [7, 13] for details. Of course, best N term approximation is not directly applicable in our setting for catching the N biggest wavelet coecients requires knowing all coecients of the unknown solution u. Nevertheless, quite recently, an implementable adaptive wavelet scheme has been developed which ....

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: \Multiscale Wavelet Methods for PDEs", (W. Dahmen, A. Kurdila, and P. Oswald, Eds.), Academic Press, San Diego, 1997, 237-284.

....basis in H t satisfying a norm equivalence of the form (2.2.1) with suitable scaling matrix D t . In analogy to (4.1.1) let N;H t (v) inf w;#w N kv w T (D t ) 1 kH t (5.4.1) denote the error of best wavelet N term approximation in H t . The following fact has been shown in [12]. Proposition 5.6. Whenever t s let 1 = r t d 1 2 : 5.4.2) Then (for a suciently regular basis ) one has 1 X N=1 N (r t) d N;H t (v) 1 i v 2 B r (L ( 9 : 5.4.3) Note that B r (L ( 5 is the largest space of smoothnes r in L which is still embedded in H ....

S. Dahlke, W. Dahmen, and R.A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdila, and P. Oswald, eds., Academic Press, San Diego, 1997, 237-284.

....basis in H t satisfying a norm equivalence of the form (2.2.1) with suitable scaling matrix D t . In analogy to (4.1.1) let N;H t (v) inf w;#w N kv w T (D t ) 1 kH t (5.4.1) denote the error of best wavelet N term approximation in H t . The following fact has been shown in [10]. Proposition 5.6. Whenever t s let 1 = r t d 1 2 : 5.4.2) Then (for a suciently regular basis ) one has 1 X N=1 N (r t) d N;H t (v) 1 i v 2 B s (L ( 9 : 5.4.3) Note that B r (L ( 5 is the largest space of smoothnes r in L which is still embedded in ....

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: \Multiscale Wavelet Methods for PDEs", (W. Dahmen, A. Kurdila, and P. Oswald, Eds.), Academic Press, San Diego, 1997, 237-284.

.... w (J ) Using (3.2.12) one can show that for s and related through (4.3.7) u 2 means for D I = 2 tjIj that u 2 B sd t (L ) The exact relation between the rate of best N term approximation (in the energy norm) and a certain Besov regularity can be stated as follows. We follow [42] and suppose again that H = H t . Let 0 denote the sup of all such that H . Then the following holds [42] 13 Proposition 4.5 Assume that t and let for t 1 : t d 1 2 : 4.3.9) Then one has 1 X n=1 N ( t) d N;H t(g) 1 (4.3.10) if ....

....u 2 B sd t (L ) The exact relation between the rate of best N term approximation (in the energy norm) and a certain Besov regularity can be stated as follows. We follow [42] and suppose again that H = H t . Let 0 denote the sup of all such that H . Then the following holds [42]. 13 Proposition 4.5 Assume that t and let for t 1 : t d 1 2 : 4.3.9) Then one has 1 X n=1 N ( t) d N;H t(g) 1 (4.3.10) if and only if g 2 B (L ( 1 . Of course, 4.3.10) implies that the best N term approximation in H t (and ....

[Article contains additional citation context not shown here]

S. Dahlke, W. Dahmen, R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdila, P. Oswald (eds.), Academic Press, London, 237-283, 1997.

No context found.

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic equations, in: Multiscale Techniques for PDEs, W. Dahmen, A. Kurdila, and P. Oswald (eds), Academic Press, p. 237-284, 1997.

....typical facts suited to the present context. To this end, consider oe N;t (g) inf 8 : kg Gamma X 2 d kH t : d 2 IR; 2 ae r; # = N 9 = Employing the norm equivalence (5.3.8) yields oe N;t (v) oe N;0 ( Sigma t v) oe N ( Sigma t v) 11.5. 1) which in turn leads to [55] Remark 11.3 Let v 2 H t : We take N to be a set of N indices for which 2 tjj jhv; ij is largest. Then one has oe N;t (v) kv Gamma Q N vk H t ; N 2 IN : 11.5.2) 131 Thus, picking the N first largest weighted coefficients realizes asymptotically the best N term approximation ....

....to the norm k Delta k H t and hence, in case (11.2.2) also relative to the energy norm k Delta k. Combining (11.5. 2) with analogous results about oe N;t for t = 0, the best N term approximation of a function v relative to k Delta k H t can be characterized in terms of its Besov regularity [55]. Proposition 11.4 Assume that ff Gamma t fl and let for t ff 1 : ff Gamma t n 1 2 : 11.5.3) Then one has 1 X N=1 i N (ff Gammat) n oe N;t (v) j 1; 11.5.4) where n is again the spatial dimension of the underlying domain Omega ) if and only if v 2 B ff (L ....

[Article contains additional citation context not shown here]

S. Dahlke, W. Dahmen, R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdila, P. Oswald, (eds.), Academic Press, to appear.

No context found.

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdila, and P. Oswald, eds., Academic Press, San Diego, 1997, pp. 237-284.

No context found.

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic equations, in: Multiscale Techniques for PDEs, W. Dahmen, A. Kurdila, and P. Oswald (eds), Academic Press, p. 237--284, 1997.

....W s which ensures O(N s ) accuracy for uniform discretization with N parameters. In several instances of the elliptic problems, e.g. when the right hand side f has singularities, or when the boundary of# has corners, the Besov regularity of the solution will exceed its Sobolev regularity (see [16] and [18] So these solutions can be approximated better by best N term approximation than by uniformly refined spaces and the use of adaptive methods is suggested. Another important feature of N term approximation is that a near best approximation is produced by thresholding, i.e. simply ....

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic equations, in: Multiscale Techniques for PDEs, W. Dahmen, A. Kurdila, and P. Oswald (eds), Academic Press, 1997, San Diego, 237--284.

....ensures O(N Gammas ) accuracy for uniform discretization with N parameters. In several instances of the elliptic problems, e.g. when the right hand side f has singularities, or when the boundary of Omega has corners, the Besov regularity of the solution will exceed its Sobolev regularity (see [16] and [18] So these solutions can be approximated better by best N term approximation than by uniformly refined spaces and the use of adaptive methods is suggested. Another important feature of N term approximation is that a near best approximation is produced by thresholding, i.e. simply ....

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic equations, in: Multiscale Techniques for PDEs, W. Dahmen, A. Kurdila, and P. Oswald (eds), Academic Press, 1997, San Diego, 237--284.

No context found.

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic equations, in: Multiscale Techniques for PDEs, W. Dahmen, A. Kurdila, and P. Oswald (eds), Academic Press, p. 237--284, 1997.

....needed to ensure the validity of an estimate like (2. 37) The point here is that best N term approximation can be (nearly) used to characterize spaces from another regularity scale, namely certain Besov spaces, see [37, 38, 36] The type of result needed here can be formulated as follows [23]. Suppose again that H = H t for simplicity and de ne N;t (g) inf 8 : kg X 2 d kH t : d 2 IR; 2 J ; # = N 9 = Let 0 denote the supremum of all such that H . Then the following holds [23] Proposition 2.6 Assume that t and let for t 1 ....

....The type of result needed here can be formulated as follows [23] Suppose again that H = H t for simplicity and de ne N;t (g) inf 8 : kg X 2 d kH t : d 2 IR; 2 J ; # = N 9 = Let 0 denote the supremum of all such that H . Then the following holds [23]. Proposition 2.6 Assume that t and let for t 1 : t d 1 2 : 2.38) Then one has 1 X n=1 N ( t) d N;t (g) 1 (2.39) if and only if g 2 B (L ( 2 . Of course, 2.39) implies that the best N term approximation in H t (and hence the near ....

[Article contains additional citation context not shown here]

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: \Multiscale Wavelet Methods for PDEs", (W. Dahmen, A. Kurdila, and P. Oswald, eds.), Academic Press, San Diego, 1997, 237-284.

No context found.

S. Dahlke, W. Dahmen and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdila, P. Oswald (eds.), Academic Press, London, 237--283.

....problems are: ffl Convergent and efficient adaptive wavelet methods for positive definite problems. ffl Construction of adapted wavelet bases. Let us describe this in more detail. Recently, an adaptive wavelet strategy has been introduced for symmetric positive definite operators, 19] see also [18]. It was proven there that this strategy gives rise to a convergent adaptive algorithm. The original method in [19] was somewhat modified in [14] resulting in a strategy that in addition was proven to be asymptotically optimal efficient. The construction of (biorthogonal) wavelet bases leaves some ....

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Methods for PDEs, W. Dahmen, A. Kurdilla, and P. Oswald, eds., Academic Press, San Diego, 1997, 237--283.

....spaces, the usual Galerkin approach can be interpreted as some kind of linear approximation. It is well known that the order of approximation for linear methods to recover the solution u of (1. 2) is determined by the regularity of u in the usual Sobolev scale H s( Omega Gamma ; s 1; see, e.g. [7, 20] for a further discussion. If the domain Omega Gamma the right hand side f and the coefficients of the operator A are smooth, then this Sobolev regularity is sufficiently high so that linear methods are appropriate, see [14, 17] The situation changes completely in the nonsmooth case, for then ....

S. Dahlke, W. Dahmen, and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: "Multiscale Wavelet Methods for Partial Differential Equations" (W. Dahmen, A. Kurdila, and P. Oswald, Eds.), Academic Press, San Diego, 1997, 237--283.

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