%ADykes, Laura [University School Hunting Valley OH 44022 USA]%ADykes, Laura [University School; Hunting Valley OH 44022 USA]%AHuang, Guangxin [Geomathematics Key Laboratory of Sichuan, Department of Information and Computing Science; Chengdu University of Technology; Chengdu 610059 China]%AHuang, Guangxin [Geomathematics Key Laboratory of Sichuan, Department of Information and Computing Science Chengdu University of Technology Chengdu 610059 China]%ANoschese, Silvia [Department of Mathematics SAPIENZA Università di Roma P.le A. Moro, 2 Roma I‐00185 Italy]%ANoschese, Silvia [Department of Mathematics; SAPIENZA Università di Roma; P.le A. Moro, 2 Roma I-00185 Italy]%AReichel, Lothar [Department of Mathematical Sciences Kent State University Kent OH 44242 USA]%AReichel, Lothar [Department of Mathematical Sciences; Kent State University; Kent OH 44242 USA]%BJournal Name: Numerical Linear Algebra with Applications; Journal Volume: 25; Journal Issue: 4; Related Information: CHORUS Timestamp: 2023-09-13 19:29:30 %D2018%IWiley Blackwell (John Wiley & Sons) %JJournal Name: Numerical Linear Algebra with Applications; Journal Volume: 25; Journal Issue: 4; Related Information: CHORUS Timestamp: 2023-09-13 19:29:30 %K %MOSTI ID: 10055011 %PMedium: X %TRegularization matrices for discrete ill‐posed problems in several space dimensions %XSummary

Many applications in science and engineering require the solution of large linear discrete ill‐posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right‐hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill‐posed problem by a penalized least‐squares problem, whose solution is less sensitive to the error in the right‐hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions.

%0Journal Article