Numerical Linear Algebra for Signal Processing and Integral Equations
{'Code': '03040109', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Alvin I. Thaler', 'PO_EMAI': None, 'PO_PHON': None}
Many questions in signal processing can be formulated as                problems for positive definite Toeplitz and unitary (or                         orthogonal) Hessenberg matrices.  For example, systems of linear                filtering, recursive estimation, time series analysis, as well as               in image processing.  Furthmore, Pisarenko frequency estimates,                 also of interest in signal processing, can be computed using                    Toeplitz and unitary Hessenberg matrices as follows:  a Toeplitz                eigenvalue problem is solved; the Schur parameters associated                   with the orthogonal polynomials defined implicitly by the                       Toeplitz matrix are then used to construct a certain unitary                    Hessenberg matrix; the eigenvalues of this matrix are the                       frequency estimates sought.                                                             Using the close connection between the theory of polynomials            orthogonal on the unit circle and these problems of linear                      algebra, the investigators have already developed competitive                   numerical methods for computing eignevalues and eigenvectors of                 unitary Hessenberg matrices for solving the inverse eigenvalue                  problem for unitary matrices, and for solving systems of linear                 equations with positive definite symmetric Toeplitz matrices.                           The purpose of this work is to further speed up the                     algorithms for the above problems and to investigate how they can               be implemented efficiently on computers with vector and parallel                architectures. The investigators also wish to gain a better                     understanding about the numerical properties, such as stability,                of some of the algorithms mentioned so that fast reliable                       computer programs can be developed.  The work should result in                  published fast stable algorithms for the above problems.                                Another application of fast Toeplitz solvers is the solution            of integral equations.  Iterative schemes with Toeplitz matrices                as preconditioners will also be investigated.  Encouraging                      preliminary results for an integral equation of plane potential                 theory on a piecewise smooth curve currently exist.                             This is a model problem for integral equations on a piecewise                   smooth curve or surface with a displacement kernel.  Such                       integral equations are common in applications of the boundary                   element method.  The project will investigate when Toeplitz                     matrices make good preconditioners.  More generally, it will                    continue the investigation in and develop fast solution methods                 for integral equations by exploiting the structure of the system                of equations obtained after discretization.  Both direct and                    iterative schemes will be considered.
[{'FirstName': 'William', 'LastName': 'Gragg', 'PI_MID_INIT': 'B', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'William B Gragg', 'EmailAddress': None, 'NSF_ID': '000439137', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}, {'FirstName': 'Lothar', 'LastName': 'Reichel', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Lothar Reichel', 'EmailAddress': 'reichel@math.kent.edu', 'NSF_ID': '000341820', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Co-Principal Investigator'}]
{'Name': 'University of Kentucky Research Foundation', 'CityName': 'LEXINGTON', 'ZipCode': '405260001', 'PhoneNumber': '8592579420', 'StreetAddress': '500 S LIMESTONE', 'StreetAddress2': '109 KINKEAD HALL', 'CountryName': 'United States', 'StateName': 'Kentucky', 'StateCode': 'KY', 'CONGRESSDISTRICT': '06', 'CONGRESS_DISTRICT_ORG': 'KY06', 'ORG_UEI_NUM': 'H1HYA8Z1NTM5', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF KENTUCKY RESEARCH FOUNDATION, THE', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1271', 'Text': 'COMPUTATIONAL MATHEMATICS'}
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