Research Areas

Major research areas: 

• Spectral Approximation 

One would like to obtain approximations for an  unknown isolated part of the spectrum of an operator and the corresponding spectral subspace. The operator may be a bounded operator on a Banach space or a densely defined closed operator with domain and range in a Banach space. The spectral set and the corresponding spectral subspace are approximated with the help of a nearby operator or by a sequence of operators.  

Some of the noteworthy contributions:

• A new procedure is developed for establishing an approximated invariant subspace to be a spectral subspace, which as a particular case, results in showing simplicity of an approximated eigenvalue. 

• A paper on approximately solving a Riccatti operator equation (appeared in Studia Mathematica) concerning a newton-type method, which is of independent interest as well, has been studied by others and has been named as "M.T. Nair method". 

• A procedure adopted  for strongly stable approximations has influenced the research of many in the area of spectral approximations.

• Operator Equations 

One would like to obtain approximations for the solution of an operator equation Tx- k x = y, where T is a bounded operator on a Banach space X and k  is a scalar not in the spectrum of T. This is done usually  with the help of a  sequence of operators which is an approximation of  T  in some sense. Such procedures included the well-known projection methods. In case of  T  is a Fredholm integral operator, then the quadrature based approximations  such as Nystrom method and Fredholm methods are examples. 

Some of the noteworthy contributions:

• Introduced and analyzed  a new method of projection, named as "modified projection method", for  second  kind operator equations, which is as simple as Galerkin method, but yields better order of convergence in certain favourable conditions.  

• A feature called  "arbitrary slow convergence" introduced by  Eberhard Schock has been further studied and showed how an arbitrary slow convergent method can be suitably modified so as to obtain a uniformly convergent method. 

• Inverse and Ill-Posed Problems

Many of the inverse problems that appear in applications in science and engineering are ill-posed, in the sense that a unique solution which depends continuously on the data does not exist. So, special procedures, the so called regularization methods, are to be employed. To obtain stable approximate solutions using such procedures suitable parameter choice strategies to be adopted to choose the parameters occurring in the methods. One is also interested in knowing whether such procedures yield error  estimates which are order optimal with respect to certain source conditions. 

Some of the noteworthy contributions:

• Proved the order optimality result for the "Arcangeli's discrepancy principle (1966)"  for Tikhonov regularization, which settled an open problem.  

• The paper appeared in ZAA and the papers in collaboration with  Pereverziev (Linz), Schock (Kaiserslautern), and Tautenhahn (Zittau) have unified many of the results known in the literature for mildly and severely ill-posed problems. 

• Introduced a unified treatment for Tikhonov regularization in a paper appeared in "Analysis and its Applications" which covers stability results for  ordinary Tikhonov regularization and the results in Hilbert scale settings. 

• A new procedure based on linear regularization theory  has been introduced and studied  for  (nonlinear)  parameter identification problems in elliptic and parabolic PDE in collaboration with  Ph.D students Samprita Das Roy and Subhankar Mondal. 

Some of the journals where papers appeared:

(The number in bracket indicates the number of papers appeared in that journal)

• Advances in Computational Mathematics (1)

• Analysis and Applications (1)

• Annales Polonici Mathmatici (1)

• ANZIAM (1)

• Applicable Analysis (2)

• Bulletin of Australian Mathematical Society (3)

• Computational Methods in Applied Mathematics (3)

• Houston Journal Mathematics (1)

• Indian Journal Pure and Applied Mathematics (2)

• Integral Equations and Operator Theory (6)

• Inverse Problems (2)

• Inverse Problems in Science and Engineering (1)

• Journal of Australian Mathematical Society (Series A & B) (2 & 1)

• Journal of Complexity (2)

• Journal of Indian Mathematical Society (2)

• Journal of Integral Equations and Applications (1)

• Journal of Inverse and Ill-Posed Problems (9)

• Journal of Mathematical Analysis and Applications (2)

• Journal of Nonlinear Analysis and Optimization: Theory & Applications (1)

• Journal of Optimization Theory and Application (1)

• Journal of Partial Differential Equations (1)

• Mathematics of Computation (3)

• Numerical Functional Analysis and Optimization (9)

• Proceedings of American Mathematical Society (2)

• Proceedings of Indian Academy of Sciences: Mathematical Sciences (3)

• Studia Mathematica (1)

• Zeitschrift f ̈ur Analysis und ihre Anwendungen (3)