CARNICER,J .M.,DAHMEN, W., AND PE NA, J. M. 1994. Local decompositions of refinable spaces. Tech. Rep. Institut fu r Geometrie und Praktische Mathematik, RWTH Aachen, Aachen, Germany.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....defines wavelets (especially Riesz bases in L 2(##8 via their two scale relations (1.16) But, for instance, this straightforward construction does not imply fixed and finite masks for the wavelets. Hence, in order to define suitable wavelet bases we utilize the concept of the stable completion [3]. This concept is universal and often employed in the sequel. Definition 1.1. Let # j = # j,k ] k## j V j 1 be a given collection of functions satisfying # j = # j 1 , such that [M j,0 , M j,1 ] is invertible. We define the matrix [G j,0 , G j,1 ] with G j,0 and G ....

J. Carnicer, W. Dahmen, and J. Pena. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

....orders d; e d, respectively. It is known that the respective regularity indices ; e (inside each patch) satisfy = d 1=2 while e 0 is known to increase proportionally to e d. Appropriate wavelet bases are constructed by projecting a stable completion into the correct complement spaces (see [3, 13, 28] for details) 4 The Wavelet Galerkin scheme This section presents a fully discrete wavelet Galerkin scheme for boundary integral equations. In the first subsection we discretize the given boundary integral equation. In Subsection 4.2 we introduce the a priori matrix compression which reduces ....

J. Carnicer, W. Dahmen, and J. Pena. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

....d, d, respectively. It is known that the respective regularity indices #, # (inside each patch) satisfy # = d 1 2 while # 0 is known to increase proportionally to d. Appropriate wavelet bases are constructed by projecting a stable completion into the correct complement spaces (see [3, 13, 28] for details) 4 The Wavelet Galerkin scheme This section presents a fully discrete wavelet Galerkin scheme for boundary integral equations. In the first subsection we discretize the given boundary integral equation. In Subsection 4.2 we introduce the a priori matrix compression which reduces ....

J. Carnicer, W. Dahmen, and J. Pe na. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

....upper diagonal and D= 1 2 I. The inverses of these expressions are easy to write down as well # P m Q m # = # I 0 AI # U T P m 1 and # P m Q m # = # I 0 AI ## I A 0 I ## D 1 0 0 I # U T P m 1 . This scheme of factoring subdivision [5] as well as wavelet transforms [1] is quite general and it can be shown that any finitely supported wavelet transform can be written as a sequence of lifting steps [7] Consequently writing a given subdivision scheme as a wiring diagram immediately gives us access to all associated bi orthogonal wavelet transforms [30] In ....

....wiring diagrams the wavelets live on the odd wires. There are many possible choices of wavelets for a given set of scaling functions. In the lifting framework all bi orthogonal choices can be described as long as one initial choice is given. This is often referred to as the initial completion [32, 1, 28]. In terms of linear algebra, finding an initial completion is equivalent to asking for a matrix Y such that the matrix (F Y) is invertible. Posed as a linear algebra question it is not immediately obvious how to find such a Y.Inthe lifting framework it is trivial: simply consider the odd wire ....

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CARNICER,J.M.,DAHMEN,W.,AND PE NA, J. M. Local decompositions of refinable spaces. Appl. Comput. Harmon. Anal. 3 (1996), 127--153.

....j 1 f M [0;1] j;1 : 6) But, for instance, this straightforward construction does not imply fixed and finite masks in the two scale relations of the collections Psi [0;1] j and e Psi [0;1] j . Hence, in order to define suitable wavelet bases we utilize the concept of the stable completion [3]. This concept is universal and often employed in the sequel but to avoid confusion we add the suffix [0; 1] Definition 1.1. Let Psi [0;1] j = Phi [0;1] j;k : k 2 r [0;1] j Psi ae V [0;1] j 1 be a given collection of functions satisfying Psi [0;1] j = Phi [0;1] ....

J. Carnicer, W. Dahmen, and J. Pe~na. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

.... orthogonal and biorthogonal multiwavelet bases characterize Sobolev norm k:k W s ( 0;1] where W s ( 0; 1] is the restriction of W s to [0; 1] For the multiwavelets on [0; 1] constructed in the above theorem based on OE; e OE; e constructed in the preceding section, we have (see [2], 5] 6] the following theorem. Theorem 5.3. For any f 2 W s ( 0; 1] k(f; e Phi j0 ) 0;1] k 2 2 (4 j 0 ) 1 X j=j0 k(f; e Psi j ) 0;1] k 2 2 (5 j ) 8 : kfk 2 W s ( 0;1] s 2 [0; 3:63298) kfk 2 (W Gammas ( 0;1] s 2 ( Gamma1:75833; 0) where 4 j = f0; 1; ....

J. M. Carnicer, W. Dahmen and J. M. Pena, Local decomposition of refinable spaces, Appl. Comput. Harm. Anal., 3 (1996), 127--153.

....in the first generation setting. It turns out that the same lemma was also used for the construction of filter banks in [49] and in [31] ffl Dahmen and collaborators, independently of lifting, worked on stable completions of multiscale transforms, a setting similar to second generation wavelets [9, 17]. Again independently, both of Dahmen and of lifting, Harten developed a general multiresolution approximation framework based on prediction [26] ffl In [16] Dahmen and Micchelli propose a construction of compactly supported wavelets that generate complementary spaces in a multiresolution ....

J. M. Carnicer, W. Dahmen, and J. M. Pe~na. Local decompositions of refinable spaces. Journal of Appl. and Comput. Harmonic Analysis, 3:127--153, 1996.

....of the refinement matrices which leads to refinement matrices characterizing complement spaces. With this method we achieve an initial stable completion. A desired stable completion can be gained by lifting the initial stable completion according to the method of Carnicer, Dahmen and Pena [4]. The result ist a biorthogonal wavelet basis leading to C 1 functions on the sphere. Key words: Multiresolution analysis, wavelets, exponential splines, tensor splines, surfaces on surfaces AMS subject classification: 42C15, 41A15, 41A63 1 Introduction During the last years there has been a ....

....generalization of a matrix factorization method from Dahmen and Micchelli [12] to non stationary splines. This method yields a so called initial stable completion. A suitable stable completion corresponding to the biorthogonal system is obtained with a general method from Carnicer, Dahmen and Pena [4]. We proceed as follows: In section 2 we introduce non stationary multiresolution analysis with E splines and the concept of biorthogonal interval wavelets from [10] In section 3 we present our approach for the sphere. We consider a parametrization of the sphere by polar coordinates. The claim ....

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J.M. Carnicer, W. Dahmen and J.M. Pe~na. Local Decomposition of refinable Spaces. Appl. Comput. Harmon. Anal., 3:127--153, 1996.

....filters can greatly magnify any numerical errors present in an initial set of coefficients. Typically, the initial filter banks are lifted such that applying the resulting sequence of analysis filters is numerically stable. For a detailed description of this problem, we refer the reader to [2]. 3 A construction for bi infinite subdivision matrices Let S j denote a bi infinite 2 slanted subdivision matrix with column height 2n 1. We describe a local construction that produces a 2 slanted Q j of column height 2n Gamma 1 whose inverse filters A j and B j are the transpose of ....

J. Carnicer, W. Dahmen, and J. Pena, 1994: Local decomposition of refinable space. Appl. Comp. Harm. Anal. 3 127--153.

....called wavelets. Moreover, for bivariate polygonal domains the restriction of a spline as generator to the boundary is a univariate spline, generally on a non uniform grid. Consequently, one can compute spline pre wavelets on each segment of the boundary explicitly by following the lines in [CDP] or [LyM] giving equations like (3.4.7) below. Thus, whenever stable bases of W and X are known as in this case, the construction of the preconditioner can be performed by a change of bases as follows. Let Phi m : f m i : i 2 I mg and Phi m Gamma : f m i; Gamma : i 2 I m; Gamma g, I ....

....the coefficients in the refinement equations j fi; Gamma (x) X ff2I j 1; Gamma a j 1 ff;fi j 1 ff; Gamma (x) fi 2 I j; Gamma ; j = 0; m Gamma 1 ; 3.4. 7) j fi; Gamma (x) X ff2I j 1; Gamma b j 1 ff;fi j 1 ff; Gamma (x) fi 2 J j; Gamma ; j = 1; m ; cf. [CDP]) are known. Writing any hm 2 Tm with respect to the basis Phi m Gamma1 Gamma [ Psi m Gamma , inserting (3.4.7) and comparing with the representation of hm relative to the basis Phi m Gamma , the change of basis from Tm Gamma1 Phi Xm to Tm can be expressed as y Phi; Gamma = y ....

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, manuscript, 1994.

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3, 1996, pp. 127-153. 43

....ffl Given a biorthogonal pair Phi j ; Phi j , e.g. the one given by (3.1.24) and (3.2.10) with refinement matrices M j;0 ; M j;0 , the stable completions required in (3.3. 1) are given by M j;1 = I Gamma M j;0 j;0 ) M j;1 K j ; j;1 K (3:3:8) where K j is any invertible matrix [7]. In the following we will mainly use K j = I (for corresponding concrete examples see [18] The nonzero pattern of a pair of stable completions satisfying (3.3.8) for j = j 0 = 6 and d = 4; d = 8 which corresponds to the pair in Figure 1 is illustrated in Figure 2. x4. Quantitative Stability ....

J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3, 1996, 127--153.

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J. Carnicer, W. Dahmen, and J. Pena. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

....on prolongation and restriction matrices as known from multigrid theory. Two different concepts are presented how to determine the transfer matrices. The first one is based on Harten s general framework [H96] applied to cell averages. The second strategy examined by Carnicer, Dahmen and Pe na [CDP ] is based on bi orthogonal wavelets. The relation between the two concepts is presented. Numerical calculations for the rotational symmetric Euler equations are presented. We consider the unsteady shock focussing problem. The results show a significant reduction in computational time. ....

....V k 1;l ; D k;j ae I k 1 ; j 2 I k ; 10) i.e. a coarse grid cell is decomposed by #D k;j fine grid cells. Herein, we have to impose D 1 #D k;j D 2 independently of the level k and the positions j in order to obtain a uniform decomposition and to satisfy certain stability requirements (cf. CDP ] Analogously to (10) we obtain Gamma m;k;j = r;l)2D m;k;j Gamma r;k 1;l ; D m;k;j ae f(r; l) r 2 f1; P k;j g; l 2 D k;j g ; i.e. a cell interface Gamma m;k;j ; 1 m P k;j of volume V k;j can be decomposed by #D m;k;j interfaces corresponding to volumes of the finer ....

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Carnicer, J. M., Dahmen, W., Pe~na, J. M.: "Local Decomposition of Refinable Spaces", Appl. Comp. Harm. Anal., 3, 1996, pp. 127--153.

....with a whole ascending sequence of trial spaces, often referred to as multiresolution analysis. This permits the interaction of different scales of discretizations. In basis or transform oriented methods this is effected with the aid of appropriate multiscale bases of hierarchical type. Following [37, 59, 60, 61] a general framework of multiresolution and multiscale decompositions of trial spaces is described next in a form which will later host all the required specializations. The examples in Sections 1.2 and 1.4 can be used as a conceptual as well as a notational orientation. 21 3.1 Multiresolution ....

....any two successive trial spaces. One may think of orthogonal complements as in Section 1.2 or of the hierarchical complements in Section 1.4 induced by Lagrange interpolation (1.4.10) Depending on the case at hand, different choices will be seen to be preferable. So at this point we follow [37] and keep the specific choices open. Thus one looks for collections Psi j = f j;k : k 2 r j g ae S ( Phi j 1 ) such that S( Phi j 1 ) S( Phi j ) Phi S( Psi j ) 3.2.1) and f Phi j [ Psi j g is still uniformly stable in the sense of (3.1.3) Like refinability such decompositions may be ....

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3 (1996), 127-153.

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3 (1996), 127--153.

....this was studied in [JL] where successive projections are used. In the following we will present an approach that avoids glueing the wavelets across segment boundaries and where corresponding filters are obtained by local operations. Remark 3.2. 5 To this end, we will employ concepts developed in [CDP]. The main idea is to determine first some initial complement spaces between two successive spaces S ( Theta j ) and S ( Theta j 1 ) from which the desired complements spanned by biorthogonal wavelets will be generated with the aid of certain projections. The point is to perform this latter ....

....to biorthogonal bases they still should exhibit certain stability properties which we briefly describe first. 3.3 Some Auxiliary Facts As indicated above, we shall have to manipulate and vary complements between two successive multiresolution spaces. The necessary tools have been developed in [CDP] (see also [Sw] Since we shall have to apply them several times for different settings, we find it worth briefly stating them here in sufficient generality. Thus we will consider for the time being some Hilbert space H with inner product h Delta; Deltai and norm k Delta k = h Delta; Deltai ....

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3, 1996, 127-153.

....j ; Phi j , e.g. the one given by (3.1.24) and (3.2.10) with refinement matrices M j;0 ; M j;0 , the stable completions required in (3.3. 1) are given by M j;1 = I Gamma M j;0 M T j;0 ) M j;1 K j ; M j;1 = G T j;1 K Gamma1 j ; 3:3:8) where K j is any invertible matrix [7]. In the following we will mainly use K j = I (for corresponding concrete examples see [18] The nonzero pattern of a pair of stable completions satisfying (3.3.8) for j = j 0 = 6 and d = 4; d = 8 which corresponds to the pair in Figure 1 is illustrated in Figure 2. x4. Quantitative Stability ....

J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3, 1996, 127--153.

....are known, some are implicit in various studies and some are simply folklore. Nevertheless, we hope that the reader will benefit from putting them briefly together since we feel that they help making several somewhat technical developments more transparent. The main tools are stable completions [CDP] and associated stability criteria as well as a mechanism for generating from some initial multiscale decomposition of a multiresolution space other complements which correspond to biorthogonal wavelets. Moreover, we recall a general criterion for establishing the Riesz basis property and Sobolev ....

....from Section 1.2 to the interval [0; 1] we have to give up on translation invariance. The arguments which we will employ actually hold in greater generality. Since we will use several variants, we will formulate the main facts in sufficiently general terms. For later use we record a few facts from [CDP]. Let H be some Hilbert space with inner product h Delta; Deltai and norm k Delta kH . We are interested in spaces of the form S( Phi j ) j j 0 , j 0 2 IN fixed, where Phi j = fOE j;k : k 2 Delta j g ae H are uniformly stable in the sense that kck 2 ( Delta j ) k Phi T j ckH ; c 2 2 ....

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3, 1996, 127--153.

....n ) rests to a great deal on the possibility of reducing the computation to manipulating Laurent polynomials (see (5.15) 5.16) As soon as one has to deal with function spaces defined on domains other than IR n or the torus, such techniques are usually not applicable. The common theme of [D1, D2, D3, CDP] is to develop an appropriate general framework for multiscale techniques which are applicable to more realistic problems. Two major issues arise in this context. On one hand, one has to bring out the relevant facts pertaining to stability properties and norm equivalences as indicated in Section 3 ....

....to stability properties and norm equivalences as indicated in Section 3 above. On the other hand, one has to develop new tools for actually realizing these properties for concrete constructions. We briefly review next one general concept that has proven to be quite useful in several applications [CDP]. To describe this for the general setting in Section 3, let us first note that the nestedness of the spaces S( Phi j ) and (uniform) stability of the bases Phi j imply the existence of refinement matrices M j;0 = m j ;k ) 2 Delta j 1 ;k2 Delta j representing (uniformly) bounded mappings ....

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J.M. Carnicer, W. Dahmen and J.M. Pe~na, Local decomposition of refinable spaces, IGPM Report #112, RWTH Aachen, October 1994, to appear in ACHA.

....q = 1, hv; wi : Z IR s v(x)w(x) dx; where we write for any N Theta n matrix C C : C T : Any function g 2 L N 2 which satisfies hg; h( Delta Gamma ff)i = ffi 0;ff I ; is called dual to h. The following observations extend corresponding known facts for the case N = 1 (see e.g. [CDP]) Proposition 2.4 Suppose g and h are dual. Then the following holds: i) If h has compact support then its integer translates are lineraly independent. ii) If both g and h have compact support and h is A refinable, then A has finite support. iii) If h 2 L N 2 has compact support, linearly ....

....in Proposition 2.6. Of course, it is not clear whether the completion A(z) of the first block row given by A 0 gives rise to a B 0 such that SB 0 ;M converges in 2 . A possible strategy for dealing with this difficulty can be sketched as follows. One can try to employ the concept from [CDP] to modify the masks A e ; e 2 E , so as to ensure convergence of SB 0 ;M . It remains then to show that if SB 0 ;M converges in 2 then the limits actually belong to some Sobolev space H t (IR s ) of positive index t 0. One can then resort to the general stability criterion from ....

J.M. Carnicer, W. Dahmen, J.M. Pe~na, Local decompositions of refinable spaces, IGPM Report 112, Oct. 1994.

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CARNICER,J .M.,DAHMEN, W., AND PE NA, J. M. 1994. Local decompositions of refinable spaces. Tech. Rep. Institut fu r Geometrie und Praktische Mathematik, RWTH Aachen, Aachen, Germany.

No context found.

J. Carnicer, W. Dahmen, and J. Pena. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

No context found.

J. Carnicer, W. Dahmen, and J. Pena. Local decompositions of refinable spaces. Appl. Comp. Harm. Anal., 3:127--153, 1996.

No context found.

CARNICER,J .M.,DAHMEN, W., AND PE NA, J. M. 1994. Local decompositions of refinable spaces. Tech. Rep. Institut fu r Geometrie und Praktische Mathematik, RWTH Aachen, Aachen, Germany.

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