Semester Final Report

Transcription

1 CSUMS SemesterFinalReport InLaTex AnnKimball 5/20/2009 ThisreportisageneralsummaryoftheaccumulationofknowledgethatIhavegatheredthroughoutthis semester. I was able to get a birds eye view of many different topics to help prepare me for my future research topic. Since the beginning of the semester, I have learned about Wavelets, Fourier Transforms, Finite Difference Decomposition Method, Stencils, Interpolation, Grids or Meshes, Euler's method for forward/backward/centered difference equations, Taylor expansions to determine order and accuracy, Partial Differential Equations, Governing Equations, Poisson s equation, Laplace s equation, Boundary Conditions, Convergence, Explicit vs. Implicit schemes, algorithms like Crank Nicholson, Gauss Seidel iteration, Fast Fourier transforms, SOR, Fast Sine Transform, and Schwarz Saturation Method, and also more Linear Algebra Banded matrices, Tri diagonal matrices, Dense LU, BandLU,Latticesandmostrecently serialcomputationvs.parallelcomputation theefforttofind better approximations,faster and more efficient coding techniques tolower costs and minimize time. TheHolyGrail thequestforexactsolutions.

2 WAVELETS Awaveletisbasicallyamathematicaltransformation likeafouriertransform.thereisthediscrete wavelet transform which is used for data compression and there is the continuous wavelet transform whichisusedinscientificresearchforsignalanalysis.awaveletisbasicallyafamilyoffunctionsandthe goalistochoosea motherwavelet orananalyzingwaveletfromthisfamily.westartbywritingthe signal out as a sum of weighted sine and cosine waves, we then use a Fourier transform to alter the domain. This basically gives us a way to look at the information. But Fourier transforms are not necessarilythebestrepresentationfor asignal.thedrawbackisthatwecangraphamplitudeversus time or frequency versus time with them but, what would really be nice isto have both pieces of informationinsteadofoneoftheother. TheHeisenberguncertaintyprincipledealswiththeconceptoftimefrequencyresolution.Itbasically states that we could have good frequency resolution or good time resolution but not both. Wavelet transformsareabletosatisfytheheisenberguncertaintyprincipleandprovideuswithtimefrequency multi resolutionanalysis.thefirststepinthisanalysisistocreatea motherwavelet whichisthen translatedthroughthesignalusingconvolution thescalecanbecompressedtorepresentahighrateof change or it can be dilated to represent a low rate of change. A wavelet then quantifies how well it matches the signal. If there is a high correlation the transform returns a high index value at that particular scale and position in time if there is a low correlation the transform returns a low index value. Once all the data from the index values are collected a 3 D image can then be constructed. Frequency and time will be seen in 2 D but the 3 D picture adds amplitude. This representation is shownbycontoursincolor. ItwasataboutthistimeinmyresearchthatIrealizedthattocontinueoninwavelets,Ineededto learnalotmoremathandphysicsanddevelopbetterprogrammingskillstonotonlypracticewavelet analysisbuttobeabletoexplainthingsgraphicallytobeabetterpresenter.atthispointistartedto studyfinitedifferencemethod,thepoissonequationandprogramming. FINITEDIFFERENCEDECOMPOSITIONMETHOD Thefinitedifferencemethodisbasically replacingthederivativefromadifferentialequationwith analgebraicequationinordertocreateadiscretesectionofacontinuousfunction.wecanthenuse iterative or direct methods to approximate a solution. If possible we can compare the approximate solutionfromthealgebraicequationstotheanalyticalsolutionobtainedbycalculustechniques.ifirst

3 beganthismethodbystudyinglagrangepolynomialsandcreatingacoefficientmatrixtosolvealinear system of equations. Then I learned about deriving forward/backward/centered difference equations fromeuler'smethodandthenperformingataylorexpansiononthecentereddifference.afivepoint stencilwascreatedandauniformgridormeshwasusedtodiscretizethearea. Figure1.2Examplesofastencil Acentereddifferenceequation h (, ) h =12h h 1 +h +1 h

4 WhenaPartialDifferentialEquationoranOrdinaryDifferentialEquationcanbesolveddirectly(by integration) weareleftwithaninfinitenumberofsolutionstotheequationbecausewhenintegrating thereisalwaysa+c(aconstant)involved.wethereforegetafamilyofcurvesasasolutioninsteadof just one curve for a solution. In order to solve for c we must evaluate a value (h) at a known position(x). Plot a parametric region: In[1]:= Out[1]= Figure2 eachline(solution)dependsonxvalue It is also very helpful to have Boundary Conditions. There are three types of boundary conditions Dirichlet,NeumannandMixed(acombinationofboth). A partial differential equation can contain a large number of continuous variables each of which depends on a number of continuous parameters. The equation that describes how each variable changesasafunctionofalltheothersiscalledagoverningequation.basically asetofrulesthatthe systemobeys. TherearethreekindsofPartialDifferentialEquations elliptical,parabolicandhyperbolic.poisson's equation is an elliptical PDE. It has many applications such as heat flow, electro magnetic fields and fluiddynamicstonamejustafew. Myprojectfocusedonthe2 DPoissonequation.Wecreatedameshoveraregularsquareregion first.thenwesettheboundaryconditionequaltozeroandapproximatedthefunctiononthisgridby

5 computing at the grid points. Poisson's equation is = (, ) f is considered a forcing function.whenaforcingfunctionissetequaltozero,wethenhavelaplaceisequation = Whenwespecifyboundaryconditionsandwecanguessattheinitialvaluesof(i,j),wethenbegin our iterations. We update the values at each (i, j) with the calculation from the previous equation. Whenconvergenceismettheiterationsarecompleted. I had difficulty understanding the problem initially without an application problem. Once I understoodaphysicalproblemlikeheatdistributiononametalplate.iwasabletohaveabirdproblem andafrogproblem thebestofbothworlds. OncewecalculatedthePoissonequationoveraregularsquareregionwethentookthenextstepand created a square hole in the middle of the original square region this led to some computational difficulties.ithenbegantostudytheschwarzsaturationmethod whichcombinessorandafastsine Transform.Thismethodbasicallycalculatesinblocksandupdatesthevaluesasitrotates.Thismethod convergesquitequickly. Since my poster presentation in Amherst, I have looked at how to parallelize these calculations to improve speed and accuracy. Because of the efficiency, we can calculate a larger number of mesh points thedrawbackofeuler'smethodisthatiftimestepsaretakenthataretoolarge thesolution becomesunstable.wethenseeoscillatorybehaviorinsteadofconvergence.thebestalternativeisto use implicit methods. This requires solving a linear system at each step. One such method is Crank Nicholson. Going back to our Poisson equation we originally used Gaussian elimination to solve the linear systemonaserialmachine thismethoddoesnotstoreordoarithmeticwithzeroentriesoutsidethe nonzerobead.onaparallelmachineweareabletoseparatethegridpointsintosubsetsandsendeach sub set to different processors. Since the boundary points are a set value and do not have to be computed onlytheinteriorpointsneedtobecalculated thereforethetwooutsideprocessorshave onefewergridpointtoupdate Iamstillpuzzledathowglitchesdonotoccurwithinthecomputeror thecalculationsbecauseeachprocessorisnotdoingequalnumber ofcalculations Iunderstandthat the processors all wait to return to serial programming together but I am unsure as to how timing issuesmayormaynotberesolved.hopefullythesequestionswillbeansweredasilearnmore. HereisthematrixthatwascreatedforthediscretePoissonproblem

6 MoretocomewiththeLinearAlgebraaspect BIBLIOGRAPHY (Iserles,2009) (Faires,1993) Wavelets,AmericanScientist82(1994) orange.fr/polyvalens/clemens/wavelets/wavelets.html

7 TysonStrand,Introductiontogroundwaterflowmodeling:finitedifferentialmethod(can tfindoriginal article)