講義案例講稿syllabus on applied pu math probability and statistics

ComputationalMathematics,AppliedMathematics,ProbabilityandStatistics(theseconddraft)ComputationalMathematicsInterpolationandapproximationPolynomialinterpolationandleastsquareapproximation;interpolationandapproximation,fastFouriertransform;approximationsbyrationalfunctions;splines.NonlinearequationConvergenceofi tivemethods(bisection,secantmethod,Newtonmethod,otheritivemethods)forbothscalarequationsandsystems;findingrootsofpolynomials.LinearsystemsandeigenvalueDirectsolvers(Gausselimination,LU position,pivoting,operationcount,bandedmatrices,round-offerroraccumulation);itivesolvers(Jacobi,Gauss-Seidel,successiveover-relaxation,conjugategradientmethod,multi-gridmethod,Krylovmethods);numericalsolutionsforeigenvaluesandNumericalsolutionsofordinarydifferentialOnestepmethods(TaylorseriesmethodandRunge-Kuttamethod);stability,accuracyandconvergence;absolutestability,longtimebehavior;multi-stepNumericalsolutionsofpartialdifferentialFinitedifferencemethod;stability,accuracyandconvergence,Laxequivalencetheorem;finiteelementmethod,boundaryvalueproblemsC.deBoorandS.D.Conte,ElementaryNumerical ysis,analgorithmicapproach,McGraw-Hill,2000.G.H.GolubandC.F.vanLoan,MatrixComputations,thirdedition,JohnsHopkinsUniversityPress,1996.E.Hairer,P.SyvertandG.Wanner,SolvingOrdinaryDifferentialEquations,Springer,1993.B.Gustafsson,H.-O.KreissandJ.Oliger,TimeDependentProblemsandDifferenceMethods,JohnWileySons,1995.G.StrangandG.Fix,AnysisoftheFiniteElementMethod,edition,Wellesley-CambridgePress,AppliedODEwithconstantcoefficients;NonlinearODE:criticalpoints,phasespace&stabilityysis;Hamiltonian,gradient,conservativeODE’s.CalculusofVariations:Euler-LagrangeEquations;BoundaryConditions,parametricformulation;optimalcontrolandHamiltonian,Pontryagin umprinciple.Firstorderpartialdifferentialequations(PDE)andmethodofcharacteristics;Heat,wave,andLace’sequation;Separationofvariablesandeigen-functionexpansions;Stationaryphasemethod;HomogenizationmethodforellipticandlinearhyperbolicPDEs;HomogenizationandfrontpropagationofHamilton-Jacobiequations;GeometricopticsfordispersivewaveW.D.BoyceandR.C.DiPrima,ElementaryDifferentialEquations,Wiley,F.Y.M.Wan,IntroductiontoCalculusofVariationsandItsApplications,Chapman&Hall,G.Whitham,“LinearandNonlinearWaves”,John-WileyandSons,J.Keener,“PrinciplesofAppliedMathematics”,Addison-Wesley,A.Benssousan,P-LLions,G.Papanicolaou, ysisforPeriodicNorth-HollandPublishingCo,V.Jikov,S.Kozlov,O.Oleinik,“HomogenizationofdifferentialoperatorsandintegralSpringer,J.Xin,“AnIntroductiontoFrontsinRandomMedia”,SurveysandTutorialsinAppliedMathSciences,No.5,Springer,2009.RandomVariables;ConditionalProbabilityandConditionalExpectation;MarkovChains;TheExponentialDistributionandthePoissonProcess;Continuous-TimeMarkovChains;RenewalTheoryandItsApplications;QueueingTheory;ReliabilityTheory;BrownianMotionandStationaryProcesses;Simulation.Reference:SheldonM.Ross,IntroductiontoProbabilityDistributionTheoryandBasicFamiliesofcontinuousdistributions:Chi-sq,t,F,g ,beta;Familiesofdiscretedistributions:Multinomial,Poisson,negativebinomial;Basicstatistics:Mean,median,les,orderLikelihoodLikelihoodfunction,parametricinference,sufficiency,factorizationtheorem,ancillarystatistic,conditionallikelihood,marginallikelihood.Neyman-Pearsonparadigm,nullandalternativehypotheses,simpleandcompositehypotheses,typeIandtypeIIerrors,power,mostpowerfultest,likelihoodratiotest,Neyman-PearsonTheorem,monotonelikelihoodratio,uniformlymostpowerfultest,generalizedlikelihoodratioParameterestimation,methodofmoments,umlikelihoodestiation,unbiasedness,quadraticandothercriterionfunctions,Rao-BlackwellTheorem,Fisherinformation,Cramer-Raobound,confidenceinterval,dualitybetweenconfidenceintervalandhypothesisBayesianPrior,posterior,conjugatepriors,BayesianNonparametricPermutationtest,permutationdistribution,rankstatistics,Wilcoxon-Mann-Whitneytest,log-ranktest,Kolmogorov-Smirnov-typetests.Linearregression,leastsquaresmethod,Gauss-MarkovTheorem,logisticregression, LargesampleConsistency,asymptoticnormality,chi-sqapproximationtolikelihoodratiostatistic,large-samplebasedconfidenceinterval,as