简介：Weintroducemultilevelaugmentationmethodsforsolvingoperatorequationsbasedondirectsumdecompositionsoftherangespaceoftheoperatorandthesolutionspaceoftheoperatorequationandamatrixsplittingscheme.Weestablishageneralsettingfortheanalysisofthesemethods,showingthatthemethodsyieldapproximatesolutionsofthesameconvergenceorderasthebestapproximationfromthesubspace.Theseaugmentationmethodsallowustodevelopfast,accurateandstablenonconventionalnumericalalgorithmsforsolvingoperatorequations.Inparticular,forsecondkindequations,specialsplittingtechniquesareproposedtodevelopsuchalgorithms.Thesealgorithmsarethenappliedtosolvethelinearsystemsresultingfrommatrixcompressionschemesusingwavelet-likefunctionsforsolvingFredholmintegralequationsofthesecondkind.Forthisspecialcase,acompleteanalysisforcomputationalcomplexityandconvergenceorderispresented.Numericalexamplesareincludedtodemonstratetheefficiencyandaccuracyofthemethods.IntheseexamplesweusetheproposedaugmentationmethodtosolvelargescalelinearsystemsresultingfromtherecentlydevelopedwaveletGalerkinmethodsandfastcollocationmethodsappliedtointegralequationsofthesecondkind.Ournumericalresultsconfirmthatthisaugmentationmethodisparticularlyefficientforsolvinglargescalelinearsystemsinducedfromwaveletcompressionschemes.
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