wavelets and two-point boundary value problems - Semantic Scholar

WAVELETS AND TWO-POINT BOUNDARY VALUE PROBLEMS* I-LIANG CHERN1 AND WEI-CHANG SHANN2

Abstract. We develop a stable direct method with linear complexity for solving large banded sparse linear systems which are typical for the numerical solution of one-dimensional dierential equations. This method was derived on the basic idea of multiresolutional decompositions that are carried out by the theory of wavelets.

1. Introduction. Because of their similarities with Fourier transforms and nite elements, wavelets are studied for the applicability of being a tool in scienti c computation and numerical analysis, such as the numerical solution of dierential equations. Among the recent articles on the later subject, we refer to the works by Glowinski, Lawton, Ravachol and Tenenbaum [1]. However, some authors found that the \usual" wavelets, which are orthogonal among the original functions instead of the rst derivatives, do not eciently localize the dierential operator. Recently there is a new set of biorthogonal wavelets that diagonalize the dierential operator. The L2 -orthonormality is helpful to represent \nice" integral operators like the Calderon-Zygmund operator, as demonstrated by Beylkin, Coifman and Rokhlin [2]. But the band-width will be wider when applied to dierential operators, compared with the matrices induced by an approximation of same order in nite dierence or nite element. Besides the locality problem mentioned above, wavelets on the real line cause tedious problems for those \crossing" the boundary of a domain. Wavelets that \live" in intervals can be derived from those \live" on real lines by, basically, applying the Gram-Schmidt process on those crossing the boundary. They are rst constructed by Meyer, then improved by Andersson, Hall, Jawerth and Peters [3]. The numerical experiences of these new wavelets are yet to be explored. There are other works looking for alternatives that are orthogonal among the derivatives and are better adapted for bounded domains. Among them we refer to Xu and Shann [4]. In this article, we also do not directly use wavelets. We found that the fundamental ideas of locality, orthogonal among the derivatives and multiresolution decomposition are the essential parts. We explored the connections of these ideas and derived this class of direct methods for solving large banded and sparse linear systems, which are typical when a dierential equation is discretized by nite element or nite dierence methods. 2. State of problem. The second order nite element or nite dierence approximation of the dierential equation ?u00 = f (1) is basically of the form

?uj?1 + 2uj ? uj+1 = fj :

* Submitted to the Conference on High-Performance Computings, April 1994. This work was supported by the National Science Council in Taiwan; grant #83-0208-M-008-052. 1 Department of Mathematics, National Taiwan University, Taipei, Taiwan, R.O.C. E-mail address: [email protected]. 2 Department of Mathematics, National Central University, Chung-Li, Taiwan, R.O.C. E-mail address: [email protected] 1

However, the bandedness may be aected by R the associated boundary conditions. For instance the periodic condition with the normalization u dx = 0. Then the matrix induced by (1) is 0 2 ?1 ?1 1 BB ?1 2 ?1 C C ... BB C (2) C : @ ?1 2 ?1 A 1 1


1 1 N N If the system with matrix (2) is to be solved by Gaussian elimination, the last two equations will be 8 N N u =f < u ? N ? 1 N ? 1 N N ?1 (3) :N?1 NuN = fN : We shall solve uN by (3), then solve backward for uN ?1 ; : : :; u1 . In our case juk j is independent of N and hence (3) is a typical sign of instability. For instance, if N = 106 , then we will loss 6 signi cant digits in this calculation. This situation is generally resolved by the Gaussian elimination with pivoting. But clearly this is not a desired approach because we will loss the sparsity and hence increase the computational complexity up to O(N 2 ). The goal in our article is to solve this kind of problem by a direct method with O(N ) computational complexity, and yet covers a wide range of applications like the foregoing examples. 3. Basic idea. The basic idea is the multiresolutional decomposition. Suppose the numerical solution of a dierential equation is done is a nite dimensional space VL , and the solution is uL 2 VL. Then we decompose uL = uL?1 + (uL ? uL?1 ) = uL?2 + (uL?1 ? uL?2) + (uL ? uL?1 ) =


:= u0 + w0 +


+ wL?1 : Here u` is considered the resolution of u at level `; while w` = u`+1 ? u` is the dierence, or \detail," between the two resolutions. We want to design a sequence of spaces

V0  V 1 


 V L

such that

V` = V0 + W0 +


+ W`?1 : Suppose u` is solved by the linear system A` u` = f` ; here AL = A. After one step of

decomposition, we shall have the linear system

A

`?1 C`?1

B`?1 D`?1

 u   f  `?1 w`?1

= g`?1 : (4) `?1 For simplicity we assume that A is symmetric, so B`?1 = C`T?1 . For our purpose, we would like the decomposition to be done in spaces V`?1 and W`?1 such that, ideally, we hope (i) B`?1 = 0, (ii) D`?1 = I , and (iii) A`?1 has the same banded diagonal terms as A` has. Let `k (x) and k` (x) be the basis functions of V ` and W ` respectively. For the foregoing purpose we conclude that the following characteristics are important for `k and k` : (10 ) supp k` intersects with only a few of supp`k , no more than the band-width of A; and we like to have (`k )0 orthogonal to ( k` )0 . (20 ) supp k` are mutually disjoint; or ( k` )0 are orthonormal. (30 ) `k are compactly supported to produce a sparse matrix. 2

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Figure 1. Piecewise linear hierarchical basis at levels ` = 0, 1, 2. 4. Derivation from hierarchical bases. Now we suppose the dierential equation is on

the spatial domain (0; 1), on which we de ne a sequence of partitions. For simplicity of notations (which is not theoretically essential), we let h` = 2?` , for ` = 1; : : :; L; here L > 0 is the prescribed highest resolution. The degree of freedom for each level ` is N` = 2` ? 1. De ne the partitions T` = fx`k g to be x`k = k h` , where k = 1; : : :; N`. Let `k (x) be the piecewise linear nodal basis function associated with x`k in T`. Suppose we want to obtain the Galerkin solution uh in the highest resolution. That is, let Th = TL and Vh = spanfLk (x) : k = 1; : : :; NLg. We denote the nodal basis for Vh by N . The hierarchical basis associated with this set of nodal basis is de ned by ` k (x) =

r

h` `+1 (x); for ` = 0; : : :; ` ? 1 and k = 1; : : :; 2`: 2 2k?1

We denote the hierarchical basis by H. One can verify that jNj = jHj and Vh = spanH. That is, H consists of another basis for Vh. The rst levels of basis functions in H are plotted in Figure 1. Observe that the rst generalized derivatives of k` (x) are exactly the Haar functions on [0; 1]. They form an orthonormal set of functions. Therefore, in the special case of (1) with Dirichlet boundary condition, the matrix A is the identity. Here we do not view the hierarchical basis as another set of basis for Vh , neither change the basis from N to H all at once. Let V` be the subspace of Vh spanned by the nodal basis functions on T`, W` be that spanned by the hierarchical basis k` at the level `. We focus on the relations between V` and V`?1 and W`?1 . First of all, it is clear that dim V` = dim V`?1 + dim W`?1 . We drop the scaling coecient for k` at this moment. The correct scale can be restored later. Now we have `?1 ` `k?1 = 21 `2k?1 + `2k + 21 `2k+1; k = 2k?1 : Also, notice that < ( k` )0 ; ( m` )0 >= km ; < ( k` )0 ; (`m?1)0 >= 0: Here <
;
> is the L2 inner product. 3

Since V` = V`?1 + W`?1 , there is a change-of-basis matrix between the two sets of basis functions. That is, for any u` 2 V` , we have

u` =

X

u`k `k (x) =

X

u`k?1 `k?1 (x) +

X

wk`?1 k`?1(x):

There is a matrix to convert between fu`k g and fu`k?1 ; wk`?1 g. We illustrate the pattern of the matrix by the case of ` = 3:

0 1=2 BB 1 BB 1=2 BB B@

1 1

1=2

1

1=2 1=2

1

1 1

1=2

1 0 u21 1 0 u31 1 u22 C u32 C C B B C B C B 2 3C u u C B C B 3 3C C B B 2C 3 C: w = u C B C B 1C B 4C C B 2 3C C AB @ ww232 C A B @ uu536 C A w42

u37

Observe that u`k?1 = u`2k . That is, when the solution uh is obtained in V`?1 , the dierences between that of V` and V`?1 are totally presented in W`?1 . This feature makes the reconstruction easier. With this change of basis, we can solve the lienar system Au = f where A is given as in (2) by the decomposition as given in (4). All three wishes under (4) can be made true. And thus we have a direct method of computational complexity O(N ), which is also stable. 5. Conclusion. The \usual" wavelets do not show much advantage for solving dierential equations. What essential are the concepts of orthogonal on the derivative level, the multiresolutional decomposition and the compact supportness (locality) which can also t the boundary conditions. Here we demonstrate a result in this way of thinking. The idea of multiresolution is used to reduce the linear system to half size. If the band-width of the matrix is independent of N , usually this means the problem is one-dimensional, this method gives a direct method of complexity O(N ) and it is stable for many applicable physical problems. REFERENCES

[1] R. Glowinski, W. M. Lawton, M. Ravachol and E. Tenenbaum, Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension, in \Computing methods in applied sciences and engineering," SIAM, Philadelphia, 1990, pp. 55{120. [2] G. Beylkin, R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Comm. Pure Appl. Math., 44 (1991), pp. 141{183. [3] L. Andersson, N. Hall, B. Jawerth and G. Peters, Wavelets on closed subsets of the real line, in \Recent advances in wavelet analysis," Academic Press, Boston, 1994, pp. 1{61. [4] J. Xu and W. Shann, Galerkin-wavelets methods for two-point boundary value problems, Numer. Math., 63 (1992), pp. 123{144.

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