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Wavelets

Fukui Naoto

Scho ol of Information Science,

Japan Advanced Institute of Science and Technology

February 13, 1998

Keywords: wavelet, preconditioning,conjugate gradient method,,incomplete wavelet

transform, matrixsolver, dierential equation.

The numerical simulationin CFD(computationaluid dynamics)is a large-scale cal-

culation which needs degree of freedom that most is large. In such a simulation, the

partial dierential equation which rules the phenomenon of the object is calculated by

discreting by the methodlikethe niteelementmethodand the nitedierence metho d,

etc. Itisaparttohavethemajorityofcomputingtimeofthecalculationwherethe linear

system is solved.

Ingeneral,iftheproblem becomeslarge-scale,thedistributionoftheeigenvalueofthe

coecientmatrix ismadeill-conditioned, and whenthe scaleof theequationis large,the

increaseofcomputingtimewillbeinvitedthoughiterativemetho disused. Thereasonfor

this isthatthe distributionof theeigenvalue ofcoecientmatrix ismadeill-conditioned,

and the iterative numb er increases. The increase of such a computing time is one of

factors to makea detailed numerical analysis dicult.

On the otherhand,inrecentyears, thewaveletanalysiscame tobeusedbythesignal

analysis and image processing, etc. as a technique which took the place of the Fourier

analysis. In addition, the range extended by wavelet with a compact supp ort having

been proposed by I.Daubechies, to oand the application became possible also inthe eld

of the numerical analysis. The metho ds using wavelet for the numerical solution of the

partial dierential equation are methods by which the grid at each levelin the adaptive

grid scheme, as the basis function in the Galerkin method, using the scaling function of

wavelet,the matrix metho dsby the wavelet transform,etc.

WhenTanakatakeadvantageofthefactthatthepartialdierentialequationdiscreted

andusedwaveletforcoecientmatrixoftheobtainedlinearsystem,theconditionnumb er

donot depend on the numb ergrid p oits and calculated the Poisson's equationas elliptic

problem, using wavelet forpreconditioning the conjugategradient method.

Copyright c

1998byFukuiNaoto

using the iterative methods, as the problem becomes large-scale, the distribution of the

eigenvalue of the coecient matrix ismade ill-conditioned, the numb erof iteration is in-

creased,and,inaword,theincreaseofcomputingtimeisinvited. However,wetransforms

the coecient matrix by discrete wavelet transform, and the coecientmatrix with dis-

cretewavelettransformisrescaledsuitably,sothatthe conditonnumb erofthecoecient

matrix do not depned on the numb er of grid points. So if the numb er of grid points is

increased, thenumb erofiteration donot depend onthe numb erof gridpoints. However,

as usual,the matrix solver taking advantage of this merit has the followingproblems.

(a) It is necessaryto convertioninto the one that the problem wasperiodic.

(b) DWTshould be made p eriodic.

(c) The eigenvalue whichbecomes 0 appears.

(d) The metho d becomes verycomplex.

Then, Tanaka proposed an easily executable, imcomplete wavelet transform to solve

these problems takeing advantage of the merit of wavelet, and applied tonot the matrix

solver but the preconditioning. The reason for this is that, the matrix solver requires

strictness, but as preconditioning ofthe matrix solver,approximating techniquewhich is

ecient, and which procedure is simple, is more suitable than the solver which is strict

andcomplex. Moreover,becausethemeritof waveletisthelocalityofdata, weonlytreat

local data for discretewavelet transform. and this methods is suitableto parallelize and

to vectorize.

Navier-Stokes equation represent a phenomenon of ow. It is combined the elliptic

problem and the parabolic problem. The elliptic problem is Poisson's equation, and the

parabolic problem is nonlinear.

Inthis study,wecalculate basicpartialdierentialequationsbythe metho dusingim-

completewavelettransformandverifytheeectofthismethod,andexaminetheeciency

of this method. The method used in this studyis, for solving the linear system obtained

thatthe partialdierentialequationisdiscreted bythe nitedierencemethods,thepre-

conditioningconjugategradientmetho dusingtheimcompletediscretewavelettransform.

Asa result,forthe Poisson'sequation,the diusionequation,and the Burger's equation,

we obtained the numb er of the iteration and accuracy, and veried the performance of

this metho ds.