Multiwavelets based on hermite cubic splines

MULTIWAVELETS BASED ON HERMITE CUBIC SPLINES Dana ˇ Ce n´ a *V´ acla Finˇ ek **Ge a Plaˇ cko ´ a Technical Uni e si y o Libe ec Facul y o Science, Humani ies and Educa ion Depa men o Ma hema ics and Didac ics o Ma hema ics S uden sk´ a 2, 461 17, Libe ec, Czech Republic [email p o ec ed] *Technical Uni e si y o Libe ec Facul y o Science, Humani ies and Educa ion Depa men o Ma hema ics and Didac ics o Ma hema ics S uden sk´ a 2, 461 17, Libe ec, Czech Republic acla [email p o ec ed] **Technical Uni e si y o Libe ec Facul y o Science, Humani ies and Educa ion Depa men o Ma hema ics and Didac ics o Ma hema ics S uden sk´ a 2, 461 17, Libe ec, Czech Republic ge a.placko [email p o ec ed] Abs ac Fi s mul iwa ele s ha e appea ed a ound he ea ly 1990s. The basic idea behind mul iwa ele s is simple: o eplace he single scaling unc ion φby he mul iscaling unc ion Φ Φ Φ o ha e some addi ional desi ed p ope ies. I seems o be an in e es ing ade o because mul iwa ele s p o ide highe o de app oxima ion wi h sho e suppo han single scaling unc ion. Mo eo e , i is possible o ha e bo h symme ic and o hogonal mul iwa ele s while his is no possible o single wa ele s. In ecen yea s, se e al simple cons uc ions o wa ele bases based on He mi e cubic splines we e p oposed. In his con ibu ion, we sho ly e iew hese cons uc ions, use hese wa ele s o sol e nume ically di e en ial equa ions, and compa e hei pe o mance. Keywo ds: Wa ele ; He mi e cubic splines; ellip ic di e en ial equa ions. In oduc ion A ec o - alued unc ion Φ Φ Φ(x)=(φ1(x),φ2(x),...,φ (x))T,φ1(x),φ2(x),...,φ (x)∈L2(R) will be called a mul iscaling unc ion i i sa is ies he ollowing e inemen equa ion Φ Φ Φ(x) = ∑ k=s HkΦ Φ Φ(ax −k), whe e s, ∈Z,s< ,Hka e some × ma ices, a∈Nand a≥2. The mul iscaling unc ion gene a es a mul i esolu ion analysis o L2(R)in a simila way as o scala wa ele s. Subspaces Vja e de ined Vj=closL2(R){φl,j,k=aj/2φl(ajx−k): 1 ≤l≤ ,k∈Z},j∈Z. 46 Then Wj,j∈Zdeno es he o hogonal complemen a y subspaces o Vjin Vj+1and he ec o - alued unc ion Ψ Ψ Ψ(x)=(ψ1(x),ψ2(x),...,ψ(a−1) (x))T,ψ1(x),ψ2(x),...,ψ(a−1) (x)∈L2(R). The e o e he e exis ma ices Qksuch ha Ψ Ψ Ψ(x) = ∑ k=s QkΦ Φ Φ(ax −k), and as decomposi ion and econs uc ion algo i hms can be cons uc ed like o scala wa ele s. Mo eo e , mul iwa ele s usually ha e sho e suppo han co esponding single wa ele s, i is possible o cons uc bo h symme ic and o hogonal mul iwa ele s while his is no possible o single wa ele s, and inally, i is possible o ha e a basis o med by unc ions wi h di e en o de s (see Example 1). Howe e , mul iwa ele s ha e also some disad an ages: he disc e e mul iwa ele ans o m usually equi es p ep ocessing and pos p ocessing, and hei cons uc- ion is also mo e complica ed. The eason, why p ep ocessing is necessa y, is in he ac ha o scala wa ele s, we ha e 2−n/2 2−nk≈Z (x)φn,k(x)dx and his does no hold in gene al o mul iwa ele s. Mul iwa ele s which do no equi e p ep o- cessing a e o ins ance he so called balanced mul iwa ele s. They we e cons uc ed in [7, 8]. Fo mo e de ails on mul iwa ele s, we e e o [6]. Example 1 Simple example o piecewise linea mul iwa ele s is aken om [1]: φ1(x) = 1 0 ≤x≤1 0o he wise ,φ2(x) = 2√3x−1 20≤x≤1 0o he wise , ψ1(x) =    6x−1 0 ≤x<1 2 6x−51 2≤x≤1 0o he wise ,ψ2(x) =    2√32x−1 20≤x<1 2 −2√32x−3 21 2≤x≤1 0o he wise . The emainde o his pape is o ganized as ollows. In he nex sec ion, we e iew se - e al simple cons uc ions o wa ele s based on He mi e cubic splines. Speci ically, wa ele s wi h wo anishing wa ele momen s p oposed in [5], a hie a chical basis based on He mi e cu- bic splines, wa ele s wi h ou anishing wa ele momen s p oposed in [2], and wa ele s wi h espec o which bo h he mass and s i ness ma ices co esponding o he one-dimensional Laplacian a e spa se [3]. In he las sec ion, we use hese wa ele s o sol e nume ically di e - en ial equa ions and compa e hei pe o mance. 1 He mi e Cubic Spline Wa ele s In he yea 2000, W. Dahmen e al. [4] p oposed a cons uc ion o bio hogonal mul iwa ele s adap ed o he in e al [0,1]on he basis o He mi e cubic splines. They s a ed wi h He mi e cubic splines as he p imal scaling bases on R. Then, hey cons uc ed dual scaling bases on Rconsis ing o con inuous unc ions wi h small suppo s and wi h polynomial exac ness o o de 2. Consequen ly, hey de i ed p imal and dual bounda y scaling unc ions e aining he polynomial exac ness. This ensu es anishing momen s o he co esponding wa ele s. Finally, 47 Sou ce: Own Fig. 1.Piecewise linea mul iwa ele s hey applied he me hod o s able comple ions o cons uc he co esponding p imal and dual mul iwa ele s on he in e al. These He mi e cubic splines a e de ined by φ1(x) =    (x+1)2(1−2x)−1≤x≤0 (1−x)2(2x+1)0≤x≤1 0 o he wise ,φ2(x) =    (x+1)2x−1≤x≤0 (1−x)2x0≤x≤1 0 o he wise . They possess he ollowing in e pola ion p ope y: φ1(0) = 1,φ0 1(0) = 0,φ2(0) = 0,φ0 2(0) = 1 and hen o any unc ion ∈C1(R) u:=∑ j∈Z (j)φ1(x−j)+ ∑ j∈Z 0(j)φ2(x−j) is a He mi e in e polan o on Z. 1.1 Wa ele s p oposed by R. Q. Jia and S. T. Liu An in e es ing cons uc ion was p oposed in [5]. The au ho s cons uc ed mul iwa ele s based on He mi e cubic splines wi h con inuous de i a i es and wi h a suppo on he in e al [−1,1]. One o he cons uc ed wa ele s is symme ic, and he second one is an isymme ic. They also adap ed hem o he in e al [0,1]and hei cons uc ion o bounda y wa ele s is e y simple unlike he cons uc ion om [4]. In compa ison wi h he semi-o hogonal wa ele s, he wa ele s a di e en le els a e o hogonal wi h espec o hu0, 0iins ead o hu, i.They a e gi en by ψ1(x) = −2φ1(2x+1)+4φ1(2x)−2φ1(2x−1)−21φ2(2x+1)+21φ2(2x−1), 48 φ1φ2 Sou ce: Own Fig. 2.The He mi e cubic spline ψ1ψ2 Sou ce: Own Fig. 3.Wa ele s p oposed by R. Q. Jia and S. T. Liu ψ2(x) = φ1(2x+1)−φ1(2x−1)+9φ2(2x+1)+12φ2(2x)+9φ2(2x−1). Bo h wa ele s a e suppo ed on [−1,1]and i s adap a ion o he in e al [0,1]is pe o med by he es ic ion o he an isymme ic wa ele o he in e al [0,1]. Fo n≥1,le Vnbe he space o piecewise cubic splines ∈C1(0,1)∩C[0,1] o which (0) = (1) = 0.The dimension o Vnis 2n+1and he se Φn:={φ1(2nx−j):j=1,...,2n−1}∪φ2(2nx−j)|[0,1]:j=0,...,2n is he basis o Vn.Le Wnbe he complemen o Vnin Vn+1wi h he basis de ined by Ψn:={ψ1(2nx−j):j=1,...,2n−1}∪ψ2(2nx−j)|[0,1]:j=0,...,2n. Then, we ha e he ollowing decomposi ion o H1 0(0,1) H1 0(0,1) = V1+W1+W2+W3.... 1.2 Hie a chical Basis The second possibili y o de ine complemen wa ele spaces, we use he ollowing hie a chical app oach: ψ1(x) = φ1(2x+1)and ψ2(x) = φ2(2x+1). 49 Bo h unc ions a e suppo ed on [−1,0]and he e o e no bounda y unc ions a e necessa y. The complemen space Wnis hen gi en by Ψn:={ψ1(2nx−j):j=1,...,2n}∪{ψ2(2nx−j):j=1,...,2n}. ψ1ψ2 Sou ce: Own Fig. 4.Hie a chical wa ele s 1.3 Wa ele s wi h Momen s The hi d possibili y is o de ine wa ele s o ha e maximal numbe o anishing momen s o he gi en suppo [−1,1]. Wa ele s a e gi en hen by ψ1(x) = φ1(2x+1)−φ1(2x−1)+ 39 7φ2(2x+1)+ 132 7φ2(2x)+ 39 7φ2(2x−1), and ψ2(x) = −1 2φ1(2x+1)+φ1(2x)−1 2φ1(2x−1)−15 4φ2(2x+1)+ 15 4φ2(2x−1). ψ1ψ2 Sou ce: Own Fig. 5.Wa ele s wi h he maximal numbe o anishing momen s 50 Bounda y wa ele s a e cons uc ed o ha e he same numbe o anishing momen s as inne wa ele s. Fo mo e de ails, see [2]. Again le Wnbe he complemen o Vnin Vn+1wi h he basis de ined by Ψn:={ψ1(2nx−j),ψ2(2nx−j):j=1,...,2n−1} ∪ψ3(2nx−1)|[0,1]∪ψ4(2n(x−1)+1)|[0,1]. 1.4 Wa ele s P oposed by T. J. Dijkema and R. S e enson Fu he cons uc ion was p oposed in [3]. They i s p o ed ha i is no possible o cons uc con inuous piecewise smoo h wa ele s which ha e a compac suppo , o m Riesz basis o L2(I), p ope ly scaled wa ele s o m he Riesz basis o H1(I),and wi h Dψ0 λ,ψ0 µE=0,and ψλ,ψµ=0∀λ,µ. Consequen ly, hey cons uc ed cubic He mi e wa ele s wi h he con inuous i s de i a i e such ha among he p imal mul i esolu ion spaces Vjand he dual mul i esolu ion spaces ˜ Vj he ollowing ela ion holds Vj+V00 j⊂˜ Vj+1. As a consequence Dψ0 λ,ψ0 µE=0,ψλ,ψµ=0,∀λ,µ:|λ|>|µ|+1. Thei wa ele s a e hen gi en by ψ1(x) = φ1(2x+1)−φ1(2x−1)+ 39 7φ2(2x+1)+ 132 7φ2(2x)+ 39 7φ2(2x−1), ψ2(x) = −1 2φ1(2x+1)+φ1(2x)−1 2φ1(2x−1)−15 4φ2(2x+1)+ 15 4φ2(2x−1), ψ3(x) = 3 ∑ i=−3 ciφ1(2x−i)+diφ2(2x−i),ψ4(x) = 3 ∑ i=−3 eiφ1(2x−i)+ iφ2(2x−i) wi h c=−4595 13728,7 65,−18737 68640,1,−18737 68640,7 65,−4595 13728, d=−68741 22880,−69 40,−204701 22880 ,0,204701 22880 ,69 40,68741 22880, e=417 22880,−7 2340,5443 205920,0,−5443 205920,7 2340,−417 22880, =723 4576,1 8,8153 13728,1 2,8153 13728,1 8,723 4576. Again Wnis he complemen o Vnin Vn+1wi h he basis de ined by Ψn:=ψ1(2nx−2j−1),ψ2(2nx−2j−1):j=0,...,2n−1−1 ∪ψ3(2nx−2j):j=1,...,2n−1−1∪ψ4(2nx−2j)|[0,1]:j=0,...,2n−1. 51 ψ3ψ4 Sou ce: Own Fig. 6.Wa ele s p oposed by T. J. Dijkema and R. S e enson 2 Nume ical Expe imen s In his sec ion, he wa ele s in oduced in he p e ious sec ion a e used o sol e nume i-cally di - e en ial equa ions. We will es ic ou sel es o he equa ion −pu00+qu = wi h he Di ichle bounda y condi ions u(0) = u(1) = 0 and wi h posi i e cons an coe icien s. The co esponding Gale kin app oxima ion p oblem is he ollowing: Find un= 2n+2 ∑ i=1 ci isuch ha Z1 0 pu0 n 0+qun dx =Z1 0 dx ∀ ∈Vn. By he Lax-Milg am lemma, his app oxima ion p oblem has he unique solu ion. We also use he s anda d wa ele p econdi ioning consis ing in no malizing each basis unc ion wi h espec o he abo e bilinea o m. To sol e he a ising sys em o linea equa ions, we use he conjuga e g adien me hod. The i e a ions a e e mina ed i he di e ence o wo consecu i e i e a ions is less han 10−n−2/cond(An),whe e cond(An)deno es he condi ion numbe o he co esponding s i ness ma ix. Tab. 1.Ob ained esul s o he p oblem wi h p =q=1. JL H M DS n||un−u||L2NZ IT NZ IT NZ IT NZ IT 1 6.0e-02 48 4 40 7 54 7 54 5 2 1.7e-02 172 5 128 13 162 11 154 7 3 2.6e-03 552 6 368 20 418 16 410 9 4 2.7e-04 1580 7 976 31 1010 21 986 10 5 2.3e-05 4168 8 2448 41 2306 24 2202 13 6 1.8e-06 10396 10 5904 52 5042 25 4698 14 7 1.2e-07 24936 10 13840 74 10690 28 9754 16 8 8.3e-09 58283 11 31760 84 22194 31 19967 17 9 5.4e-10 135302 12 71696 110 46963 33 41627 19 Sou ce: Own 52 Tab. 2.Ob ained esul s o he p oblem wi h p =1and q =0. JL H M DS n||un−u||L2NZ IT NZ IT NZ IT NZ IT 1 6.0e-02 24 3 36 7 50 7 54 5 2 1.7e-02 56 5 116 12 154 11 154 7 3 2.6e-03 128 6 340 20 406 16 410 8 4 2.7e-04 280 6 916 31 994 20 986 10 5 2.3e-05 592 8 2324 41 2286 24 2202 12 6 1.8e-06 1224 10 5652 52 5018 25 4698 14 7 1.2e-07 2847 10 13332 69 10677 28 9762 15 8 8.3e-09 8818 10 30740 84 22347 30 20019 16 9 5.4e-10 35863 12 69652 110 46613 33 41421 19 Sou ce: Own Fi s we sol e he abo e p oblem wi h p=q=1 and he exac solu ion u=x(1−e50x−50) which exhibi s a s eep g adien nea he poin 1.Resul s a e summa ized in Table 1. Then we sol e he abo e p oblem wi h p=1,q=0 and wi h he exac solu ion u=x(1−e50x−50)which exhibi s a s eep g adien nea he poin 1.Resul s a e summa ized in Table 2. In bo h ables, NZ is he numbe o nonze o elemen s in s i ness ma ices, IT ep esen s he numbe o i e a ions, JL deno es wa ele s p oposed in [5], H deno es hie a chical basis, M deno es wa ele s p oposed in [2], and inally DS deno es wa ele s p oposed in [3]. Achie ed app oxima ion e o was he same o all bases. Conclusion P esen ed esul s a i m ha wa ele s p oposed in [5] ha e excellen condi ion numbe , and especially o he Poisson equa ion, he a ising s i ness ma ices a e e y spa se. Wa ele s p oposed in [3] a e bes sui ed o gene al di e en ial equa ions wi h cons an coe icien s. Resul s o es ed hie a chical basis con i m he well known ac ha hese basis do no o m he Riesz basis and he e o e hey need some addi ional p econdi ioning. Conce ning he basis p oposed in [2], we suppose ha i will be be e ( han es ed bases) sui ed o non-cons an di e en ial equa ions sol ed adap i ely. P esen ed esul s also sugges ha he e is p obably some space o imp o e he condi ion numbe o he basis p oposed in [2]. Acknowledgmen s This wo k has been suppo ed by he p ojec ESF No. CZ.1.07/2.3.00/09.0155 “Cons i u ion and imp o emen o a eam o demanding echnical compu a ions on pa allel compu e s a TU Libe ec”. Li e a u e [1] ALPER, B. 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