A comparison of regularizations for an ill-posed problem

• Published in: Mathematics of computation Vol. 67; no. 224; pp. 1451 - 1471
• Main Authors: Ames, Karen A., Clark, Gordon W., Epperson, James F., Oppenheimer, Seth F.
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-10-1998
• Subjects: Approximation; Calculus of variations and optimal control; Differential equations; Eigenfunctions; Eigenvalues; Error rates; Exact sciences and technology; Heat equation; Ill posed problems; Information retrieval noise; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; ill-posed problems; Freholm equations; numerical methods; final value problems; Quasi-reversibility; Regularization method; Parabolic equation; Boundary value problems; Partial differential equations; Error estimation; Numerical methods; Initial value problem; Ill posed problem; Fredholm integral equations
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-98-01014-X

We consider numerical methods for a ``quasi-boundary value'' regularization of the backward parabolic problem given by \[ \left{ \begin{array}{ll} u_t+Au=0, & 0<t<T u(T)=f, & \end{array}\right. ] where A is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value u(T) by adding \alpha u(0), where \alpha is a small parameter. We show how this leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson. Error estimates and examples are provided. We also compare the regularization used here with that from Ames and Epperson.