DISCRETE DIFFERENTIAL OPERATORS IN MULTIDIMENSIONAL HAAR WAVELET SPACES

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1 IJMMS 2004: PII. S Hindawi Publishing Corp. DISCRETE DIFFERENTIAL OPERATORS IN MULTIDIMENSIONAL HAAR WAVELET SPACES CARLO CATTANI and LUIS M. SÁNCHEZ RUIZ Received 30 July 2003 We consider a class of discrete differential operators acting on multidimensional Haar wavelet basis with the aim of finding wavelet approximate solutions of partial differential problems. Although these operators depend on the interpolating method used for the Haar wavelets regularization and the scale dimension space they can be easily used to define the space of approximate wavelet solutions Mathematics Subject Classification: 35A35 42C40 41A Introduction. Advantage has extensively taken upon wavelets in order to analyze a number of different applications such as image processing signal detection geophysics medicine or turbulent flows [ ]. More mathematically focussed differential equations and even nonlinear problems have also been studied with this cargo of possibilities that wavelets have brought out into scene [ ]. In this paper we study a Cauchy problem with the PDE u t = Lu where the unknown function utx L 2 [01] for each t [0T] x [01] d d N and L is a nonnecessarily linear partial differential operator whose derivatives act on the space variable. Situations like this happen quite commonly in evolution processes such as fluid dynamics heat transfer elasticity traffic flow shock propagation and so on [5 12]. According to the nature and properties of L different approaches have been developed and even simple difference approximations may provide good results [3 11]. However in the presence of discontinuities some difficulties arise. The development of Haar wavelets with its extreme suitability for dealing with experimental problems having piecewise constant functions as initial conditions or functions with sharp discontinuities such as those that appear in the Riemann problem or within smooth environments seems to provide a natural scheme to study these problems and wavelet approximate solutions have been obtained by using some regular mostly Daubechies bases of wavelets [11]. In the following we propose an algorithm based upon Haar wavelet basis in order to define a discrete operator that maps piecewise constant functions into piecewise constant functions which projects the continuous operator L therein and enables to get wavelet approximate solutions in L 2 [01] just working with Haar wavelets. 2. Haar wavelet basis. Haar wavelets are piecewise constant functions with compact support and finite jumps at their extremal points. The drawback of not being smooth enough has been overcome by a smoothing process based on spline interpolation [3].

2 2348 C. CATTANI AND L. M. SÁNCHEZ RUIZ Haar functions ϕ n k x := 2n/2 ϕ2 n x k are generated from the scaling function ϕx which is given by the characteristic function χ [01]. Then the Haar family of wavelets ψ n k x :={2n/2 ψ2 n x k} kn Z fulfills [ k 2 n/2 x 2 k+1/2 n 2 n ψ n k x = [ k+1/2 2 n/2 x k+1 2 n 2 n 0 elsewhere ψ n k x 2 = 1 and the multiresolution axioms [7] as well as the recursive conditions n+1/2 ϕ n k x = ϕn+1 k x+ϕ n+1 k+1 x 2n+1/2 ψ n k x = ϕn+1 k x ϕ n+1 k+1 x kn Z. 2.2 Haar wavelets form a complete orthonormal system for the finite-energy L 2 R- functions [7] so that L 2 R = W n = V q W j q Z n Z j q V n+1 = V n W n. 2.3 Here for each fixed n Z V n is the subspace { V n := f L 2 R : f = nk Z c n k χ D n k cn k = constant } 2.4 where D n k = [k/2n k+1/2 n and W n are orthogonal subspaces in L 2 R the so-called wavelet spaces. As usual we consider the scalar product + fg := fxgxdx fg L 2 R. 2.5 According to the first equation of 2.3 we have that fx= nk Z β n k ψn k x. Ifwe fix a resolution value N = 2 M M Nin2.3 then an approximation of the L 2 R-space is obtained for example by L 2 N R V 0 n=0 W n thatis M 1 fx π N fx:= α0 0 + n=0 2 n 1 k=0 β n k ψn k x 2.6 π N being a projection operator into V N so that π N : L 2 R V N. In general the coefficients αn k β n k are given by α n k = fϕ n k β n k = fψ n k. 2.7 The dyadic discretization is the operator N : L 2 R K N l 2 where K stands for the complex or real field and N fx = f N = f 0 f 1...f N 1 T K N with f k := fk/n 1 0 k N 1.

3 DIFFERENTIAL OPERATORS IN MULTIDIMENSIONAL HAAR WAVELET The above notation may be simplified if we denote β N := α0 0 β0 0 β1 0...βM and Ψ N x := ϕxψ0 0 xψ1 0 x...ψm 1 2 x so that 1 π N fx= β N ΨN T x fn = β N N ΨN T x. 2.8 If P 2 i denotes the permutation or shuffle matrix cf. [3 4 7] and I is the identity matrix then the discrete Haar wavelet transform is the one-to-one linear operator of K N onto K N such that [ M f N = β T ] N = P2 k I 2 M 2 k H2 k I 2 M 2 k f N. 2.9 k=1 The matrices H 2 i have size 2 i 2 i and are the direct sum of the lattice coefficients of the system /2 2 1/2 2 1/2 2 1/2 H 2 i := 2 1/2 2 1/2 2 1/2 2 1/2 where for 1 j 2 i 1 and O pq standing for the p q null matrix 1 2 i O 2j 22j 2 O 2j 22 O 2j /2 2 1/2 i 2j 2 1/2 2 1/2 = 2 1/2 2 1/2 O 22j 2 j 2 1/2 2 1/2 O 22 i 2j O 2 i 2j2j 2 O 2 i 2j2 O 2 i 2j2 i 2j Thus having in mind that V N+1 K N by identifying α0 0 + M 1 2 n 1 n=0 k=0 β n k ψn k x with {α0 0 βn k }n=0...m 1 k= the projection operator π N : L 2 R V N+1 may be factorized by means of the following diagram: L 2 R N K N π N 2.12 V N+1 K N The operator π N maps each f L 2 R into a piecewise constant function π N f V N+1 being easy to check lim N π N f = f pointwise cf. [7]. Note that the inverse discrete Haar transform 1 of 2.9 is univocally defined [7] but the inverse N := N 1 of the operator N is not because there are infinitely many functions interpolating a finite set of values. 3. Haar series regularization. In what follows we will consider the interpolating operator p : K N C p 1 Ω which maps the vector f N = f i N 1 i=0 based on the dyadic nodes of Ω R into the p-differentiable function sx := p f N where C 1 Ω stands for the piecewise continuous functions defined on Ω. Hencesx is an interpolating function of order p through i/n 1f i i = 0...N 1.

4 2350 C. CATTANI AND L. M. SÁNCHEZ RUIZ Definition 3.1 interpolating p-space. For a given resolution N the interpolating p-space σ p is the space σ p L 2 Ω of interpolating functions with nonredundant fixed conditions at the boundary { σ p := sx = p f N with p := p 1 = N f N K N}. 3.1 There follows that fixing the resolution N and the interpolating method each function in σ p corresponds one-to-one to a discrete set f N making the following diagrams commutative: L 2 Ω σ p N K N L 2 Ω σ p p K N π N π N V N+1 V N+1 Theorem 3.2. For a fixed resolution N and a given interpolating method the inverse π N of the projection operator π N is univocally defined when acting on the σ p space functions by means of π N = p 1 π N = N. 3.3 Proof. Note that N p = p N = I and 1 is univocally defined [7]. Definition 3.3. The q-derivative of a p-interpolating function 0 q p atagiven dimension N is the linear operator δ pq : K N V N+1 such that dq δ pq := π N dx q p. 3.4 This definition provides an algorithm that maps piecewise constant functions into piecewise constant functions. If we look at it we may note that firstly it smoothens piecewise constant functions with a suitable interpolation; secondly it finds out the derivative of the interpolating function and then this derivative is transformed into a piecewise constant function by the discrete Haar wavelet transform. From [7 page 13] and since d q /dx q p N f belongs to L 2 Ω for each f L 2 Ω it follows that d q /dx q p N f δ pq N f 2 L 2 2 2M 1 2 µω K + ε where µ stands for the Lebesgue measure K = M 1 k=0 m 0kN 2 + m 1kN 2 1/2 withm 0kN and m 1kN being the average value of d q /dx q p N f over the first and the second half of D N k respectively and ε may be taken as small as desired since any function in L 2 Ω may be approximated by functions with compact support which are piecewise constant. It is easy to see that if f = M 1 k=0 f k χ Dk g = M 1 k=0 g k χ Dk V N where f k g k are constants then there are unique c k such that f k = g k +c k. Hence there exists a matrix A pq depending on N that makes δ pq N f = A pq N f δ pq f N = A pq β N N Ψ T N x. 3.5

5 DIFFERENTIAL OPERATORS IN MULTIDIMENSIONAL HAAR WAVELET At the end of this paper we provide an appendix with some of these matrices A for different spline interpolations. 4. Multidimensional wavelet solution. In a d-dimensional variable space x = x 1...x d we restrict ourselves to considering the same dyadic discretization on each coordinate of x so that the natural generalization of the dyadic discretization operator N : L 2 R d d k=1 K N N = 2 M is given by N i1 fx := f N 1 i 2 N 1... i d N 1 0 i j N 1 1 j d. 4.1 By using tensor products we define V N x := d i=1 V N x i where the orthonormal wavelet basis is {ψ n k x 1 ψ n k x d} 0 k N 1cf.[17]. Thus V N x is the space of the piecewise constant functions on the d-dimensional product D N k DN h 0 h k 2 N 1 which can be represented as π N fx := + 1 i d α i ϕ x i + 0 k i 2 n i 1 0 n i M 1 i d 1 ij d i j 0 k i 2 n i 1 0 n i M 1 i d 1 i d β n i k i ψ n i k i xi ω n i k i j ϕ x j ψ n i k i xi 4.2 in short π N fx := 0 i j M 1 1 j d β i 1i 2 i d N N ψ N x i1 ψ N x id. Then the discrete d-dimensional Haar wavelet transform is the one-to-one linear operator d = d of K N d onto K N d and the discrete derivative of N fx is the tensor product of the corresponding 1-dimensional discrete derivative. Hence in the 2-dimensional case this leads to a matrix A pq which in fact is the same as the one of the 1-dimensional case such that δ pq i1 x f δ pq y f N 1 i 2 N 1 i1 N 1 i 2 N 1 = A pq f = A pq i1 N 1 i 2 0 i 2 N 1 N 1 0 i 1 N proving that if x 1 = i 1 /N 1 x 2 = i 2 /N 1 δ pq x N f x 1 x 2 = A pq β N N Ψ T N x1 N Ψ T N x2 4.4 and analogously for δ pq y. Taking this into account we obtain the following result.

6 2352 C. CATTANI AND L. M. SÁNCHEZ RUIZ Proposition 4.1. problem An approximate wavelet solution ũtx=π N utx of the Cauchy u t = Lu u0x = u 0 x x [01] d t [0T] 4.5 is obtained by taking ũtx := 0 i j M 1 1 j d β i 1i 2 i d N t N ψ N xi1 ψn xid 4.6 where the wavelet coefficients are explicit functions of t. Proof. From 4.5 it follows that π N u t = π N Lu. HenceifL δ denotes the operator acting on piecewise functions obtained by replacing the partial derivatives that appear on L with the corresponding δ pq i j N for example if L = L / x 2 / x y then L δ = Lδ p1 x N δ p1 y δ p1 x N it happens that the projection in V N x must verify ũ t tx = L δ ũtx and 0 i j M 1 1 j d β i 1i 2 i d N 0 N ψ N xi1 ψn xid = N fx. 4.7 Orthonormality of the scaling functions ϕ and ψ n k gives an easily solved first-order ordinary system in the unknown wavelet coefficients. Note that due to the definition of Haar wavelets the boundary conditions are automatically fulfilled. 5. Spline Haar solution of a parabolic equation. In this section we will compare the Haar regularized solution of the following 1-dimensional parabolic problem with the exact solution and the one obtained with Crank-Nicholson s method. Example 5.1. Compareattime0.04 and 0.08 the above-mentioned methods in the heat equation without exchanges at the boundary and a periodic function as initial function given by u t = u xx 0 x 1 t 0 ux0 = sinπx u0t= u1t= In Figure 5.1 the exact solution uxt = sinπxe π2t at t = 0.01 and the one obtained by the Haar regularized method based on 64 nodes are shown as functions of the variable distance x. Table 5.1 compares the set of values that we obtain HR at time t = 0.04 t = 0.08 with the exact ones E and those obtained by Crank-Nicholson s method CN.

7 DIFFERENTIAL OPERATORS IN MULTIDIMENSIONAL HAAR WAVELET u heat x distance 1 Figure 5.1. Exact and Haar regularized solution with 64 nodes. Table 5.1 x t = 0.04 t = 0.08 E HR CN E HR CN Annex. Finally we enclose a list of matrices A pq announced at the end of Section 3 which satisfy δ pq N f = A pq N f. We fix our attention on the case p = 3 q = 1. For this and different resolutions N = 2 M M N the matrices A pq are computed according to the following steps: 1 discretize the interval [01] at the dyadic points k/n 1 0 k N 1 2 consider the points whose first coordinates are the dyadic points and whose second coordinates are arbitrary values a 0 a 1...a N 1 that is 0a 0 1/N 1a 1...1a N 1 3 build up the cubic spline that goes through the aforementioned set of points 4 compute the first derivative of this spline and evaluate it at the dyadic points 5 assuming the above evaluations generate the points 0a 0 1/N 1a a N 1 thena31 is the N N matrix that satisfies a 0 a 1.. a N 1 = A 31 a 0 a 1. a N

8 2354 C. CATTANI AND L. M. SÁNCHEZ RUIZ With a bound error of 10 5 routine calculations make A 31 become N = 2 1 = N = 2 2 = N = 2 3 = Acknowledgment. The second author is supported by Generalitat Valenciana. References [1] E. Bacry S. Mallat and G. Papanicolaou A wavelet based space-time adaptive numerical method for partial differential equations RAIRO Modél. Math. Anal. Numér no [2] W. Cai and J. Wang Adaptive multiresolution collocation methods for initial-boundary value problems of nonlinear PDEs SIAM J. Numer. Anal no [3] C. Cattani Haar wavelet splines J. Interdiscip. Math no [4] Haar wavelet-based technique for sharp jumps classification Math. Comput. Modelling no [5] R. Courant and K. O. Friedrichs Supersonic Flow and Shock Waves Interscience Publishers New York [6] W. Dahmen Wavelet methods for PDEs some recent developments J.Comput.Appl.Math no [7] I. Daubechies Ten Lectures on Wavelets CBMS-NSF Regional Conference Series in Applied Mathematics vol. 61 Society for Industrial and Applied Mathematics Pennsylvania [8] L. Debnath Wavelet transforms and their applications Proc. Indian Nat. Sci. Acad. Part A no [9] R. A. DeVore B. Jawerth and B. J. Lucier Image compression through wavelet transform coding IEEE Trans. Inform. Theory no [10] D. L. Donoho Nonlinear wavelet methods for recovery of signals densities and spectra from indirect and noisy data Different Perspectives on Wavelets San Antonio Tex

9 DIFFERENTIAL OPERATORS IN MULTIDIMENSIONAL HAAR WAVELET Proc. Sympos. Appl. Math. vol. 47 American Mathematical Society Rhode Island 1993 pp [11] S. Lazaar P. Ponenti J. Liandrat and P. Tchamitchian Wavelet algorithms for numerical resolution of partial differential equations Comput. Methods Appl. Mech. Engrg no [12] R. J. LeVeque Numerical Methods for Conservation Laws 2nd ed. Lectures in Mathematics ETH Zürich Birkhäuser Basel [13] P. J. Oonincx A wavelet method for detecting S-waves in seismic data Comput. Geosci no [14] W. Schempp Wavelet modelling of high resolution radar imaging and clinical magnetic resonance tomography Results Math no [15] J. Segman and Y. Y. Zeevi Image analysis by wavelet-type transforms: group-theoretic approach J. Math. Imaging Vision no [16] O. V. Vasilyev and S. Paolucci A fast adaptive wavelet collocation algorithm for multidimensional PDEs J. Comput. Phys no [17] P. Wojtaszczyk A Mathematical Introduction to Wavelets London Mathematical Society Student Texts vol. 37 Cambridge University Press Cambridge [18] V. Zimin and F. Hussain Wavelet based model for small-scale turbulence Phys. Fluids no Carlo Cattani: Dipartimento di Scienze Farmaceutiche DiFARMA Università degli Studi di Salerno Via Ponte Don Melillo Invariante 11/C Fisciano Salerno Italy address: ccattani@unisa.it Luis M. Sánchez Ruiz: Departamento de Matemática Aplicada Escuela Técnica Superior de Ingeniería de Diseño ETSID Universidad Politécnica de Valencia Valencia Spain address: lmsr@mat.upv.es