Mathematical Sciences: Preconditioning by Fast Transforms
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{'SignBlockName': 'Michael Steuerwalt', 'PO_EMAI': None, 'PO_PHON': None}
The analytical problem in this proposal is the relation between a           given operator and its preconditioner.  Given is a differential or              integral or convolution operator with general boundary conditions.  The         preconditioner has particularly convenient domain and boundary                  conditions. After discretization it can be inverted by the Fast Fourier         Transform, or in the case of the Dirichlet problem by a Fast Sine               Transform.                                                                          In the matrix case one applies the preconditioned conjugate gradient        method and the distribution of the eigenvalues is crucial.  A small             number of extreme eigenvalues is acceptable if all other eigenvalues are        clustered - that is what appeared in numerical experiments.  The                conjecture on clustering depends on establishing that the eigenvectors          decay exponentially, away from the boundary.  The analysis should               connect problems in operator theory to problems in numerical analysis,          and may give a new direction in the class of equations for which                preconditioning is essential.                                                       This research falls into the general area of numerical analysis,            numerical computing, computational mathematics and software design. The         problem of Toeplitz operators is important in signal processing.                The problem of numerical computations associated with Toeplitz matrices         arises in many problems of communications engineering and its solution          is of considerable importance.
{'FirstName': 'Gilbert', 'LastName': 'Strang', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Gilbert Strang', 'EmailAddress': 'gs@math.mit.edu', 'NSF_ID': '000476330', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Massachusetts Institute of Technology', 'CityName': 'CAMBRIDGE', 'ZipCode': '021394301', 'PhoneNumber': '6172531000', 'StreetAddress': '77 MASSACHUSETTS AVE', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Massachusetts', 'StateCode': 'MA', 'CONGRESSDISTRICT': '07', 'CONGRESS_DISTRICT_ORG': 'MA07', 'ORG_UEI_NUM': 'E2NYLCDML6V1', 'ORG_LGL_BUS_NAME': 'MASSACHUSETTS INSTITUTE OF TECHNOLOGY', 'ORG_PRNT_UEI_NUM': 'E2NYLCDML6V1'}
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[{'Code': '1266', 'Text': 'APPLIED MATHEMATICS'}, {'Code': '1271', 'Text': 'COMPUTATIONAL MATHEMATICS'}, {'Code': '2865', 'Text': 'NUMERIC, SYMBOLIC & GEO COMPUT'}]
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