School of Mathematical Sciences
College of Science
School of Mathematical Sciences
College of Science
MS, Technical University Kaiserslautern (Germany); Ph.D., Michigan Technological University
Most models in applied and social sciences are formulated using the broad spectrum of linear and nonlinear partial differential equations involving parameters characterizing specific physical characteristics of the underlying model. Inverse problems seek to determine such parameters from the measured data and have many applications in medicine, economics, and engineering. This course will provide a thorough introduction to inverse problems and will equip students with skills for solving them. The topics of the course include existence results, discretization, optimization formulation, and computational methods.
This course covers numerical techniques for the solution of systems of linear equations, eigenvalue problems, singular values and other decompositions, applications to least squares, boundary value problems, and additional topics at the discretion of the instructor.
1 - 3 Credits
This course is a faculty-directed project that could be considered original in nature. The level of work is appropriate for students in their final two years of undergraduate study.
0 - 9 Credits
Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.
Khan, Akhtar A. "An Optimization Framework for the Elasticity." Variational Analysis And Applications. International Centre for Scientific Culture "E. Majorana" School of Mathematics "G. Stampacchia". Erice, Italy. 28 Aug. 2015. Conference Presentation.
Khan, Akhtar A. "Stability of the Elasticity Imaging Inverse Problem." Applied Inverse Problems. Inverse Problems Society. Helsinki, Finland. 25 May 2015. Conference Presentation.
Khan, Akhtar A. "Computational Methods for Elastography Inverse Problem,." High Performance Computing in Science and Engineering. University of Ostrava. Ostrava, zech Republic. 25 May 2015. Conference Presentation.
Khan, Akhtar A. "On Evolutionary and Elliptic Quasi Variational Inequalities." The 22nd International Symposium on Mathematical Programming,. Carnegie Mellon University and University of Pittsburgh. Pittsburgh,, USA. 12 Jul. 2015. Conference Presentation.
Khan, Akhtar A. "Some Aspects of Elastography Inverse Problem." The Modeling and Optimization: Theory and Applications (MOPTA). Lehigh University. Bethlehem, PA. 20 Jul. 2015. Conference Presentation.
Khan, Akhtar A. "Parameter Identification in Variational Problems,." Recent Developments in Applied Mathematics. Palacky University Olomouc. Palacky, Czech Republic. 2 Feb. 2015. Conference Presentation.
Khan, Akhtar A. "Heavy Ball with Friction Methods for Inverse Problems." Joint Mathematics Meetings. SIAM/MAA. San Antonio, USA. 10 Jan. 2015. Conference Presentation.
Khan, Akhtar A. "A Convex Optimization Problem in Identifying Tumor Location." The 21st International Symposium on Mathematical Programming. N/A. Berlin, Germany. 19 Aug. 2012. Lecture.
Khan, Akhtar A. "Ill Posed Quasi-Variational Inequalities with Multi-valued Maps." The 6th International Conference on Inverse Problems: Modeling and Simulation. IP. Antalya, Turkey. 21 May 2012. Lecture.
Khan, Akhtar A. "Inverse Problems in Linear Elasticity: Compressible and Incompressible Cases." The fourth International Workshop on Variational Analysis and Applications. Stampacchia School of Mathematics, Erice. Sicily, Italy. 14 May 2012. Lecture.
Khan, Akhtar A. "Conical Regularization for Optimal Control Problems." Invited Talk. University of Catania. Catania, Italy. 3 Mar. 2012. Lecture.
Khan, Akhtar A. "Conical Regularization for Abstract Constrained Optimization Problems in Hilbert Spaces." Joint Mathematics Meetings. AMS/SIAM. Boston, MA. 4 Jan. 2012. Lecture.
Khan, Akhtar A. "Conical Derivatives for Optimization Problems." Applied Inverse Problems. University of Texas at A&M. University of Texas at A&M, College Station, TX. 24 May 2011. Conference Presentation.
Khan, Akhtar A. "Toward Second-Order Sensitivity Analysis in Set-Valued Optimization." Joint Mathematics Meetings. AMS/MAA. Sheraton, New Orleans, LA. 8 Jan. 2011. Conference Presentation.
Khan, Akhtar A. "Inverse Problem for Quasi Variational Inequalities." 7th International Congress on Industrial and Applied Mathematics. ICIAM. Vancouver Convention Centre, Vancouver, British Columbia, Canada. 18 Jun. 2011. Conference Presentation.
Khan, Akhtar A. and Dumitru Motreanu. "Existence Theorems for Elliptic and Evolutionary." J Optim Theory Appl 167. (2015): 1136—1161. Print.
Jadamba, B., et al. "Identification of Flexural Rigidity in a Kirchhoff Plates Model." Mathematical Problems in Engineering ID 290301. (2015): 1--11. Print.
Khan, Akhtar A., Christiane Tammer, and Constantin Zalinescu. "Regularization of quasi-variational inequalities." Optimization 64. (2015): 1703--1724. Print.
Bush, Nathan, et al. "Identification Of A Parameter in Fourth-Order Partial Differential Equations By An Equation Error Approach." Mathematica Slovaca 65. (2015): 1--13. Print.
Khan, Akhtar A. and Doug Ward. "Toward Second-Order Sensitivity Analysis in Set-Valued Optimization." Journal of Nonlinear and Convex Analysis 13. (2012): 65-83. Print.
Khan, Akhtar A. and M. Sama. "A Multiplier Rule for Stable Problems in Vector Optimization." Journal of Convex Analysis 19. (2012): 525-539. Print.
Khan, Akhtar A. and M. Sama. "Optimal Control of Multivalued Quasi Variational Inequalities." Nonlinear Analysis 75. (2012): 1419-1428. Print.
Khan, Akhtar A., B. Jadamba, and M. Sama. "Regularization for State Constrained Optimal Control Problems by Half Spaces Based Decoupling." Systems Control Letters 61. (2012): 707-713. Print.
Khan, Akhtar A. and D. Motreanu. "Local Minimizers Versus X-Local Minimizers." Optimization Letters 10.1007/s11590-012-0474-8. (2012): 1-7. Web.
Khan, Akhtar A. and C. Tammer. "Second-Order Optimality Conditions in Set-valued Optimization via Asymptotic Derivatives." Optimization 10.1080/02331934.2012.674948. (2012): 1-16. Web.
Khan, Akhtar A., et al. "Proximal Point Methods for the Inverse Problem of Identifying Parameters in Beam Models." Emerging Applications of Wavelet Methods 1463. (2012): 16-38. Print.
Khan, Akhtar A. and C. Tammer. "Generalized Dubovitskii-Milyutin Approach." Vietnam Journal of Mathematics 40. (2012): 285-304. Print.
Khan, Akhtar A., Baasansuren Jadamba, and Miguel Sama. "Generalized Solutions of Quasi Variational Inequalities." Optimization Letters doi:10.1007/s11590-011-0363-6. (2011): 1-11. Print.
Khan, Akhtar A., Elisabeth Koebis, and Christiane Tammer. "Scalarization Methods in Multiobjective Optimization, Robustness, Risk Theory and Finance." Multiple Criteria Decision Making in Finance. Ed. M. Al-Shammari. Berlin, Germany: Springer, 2015. 135-157. Print.
Gockenbach, M., et al. "Proximal Methods for the Elastography Inverse Problem of Tumor Identification Using an Equation Error Approach." Advances in Variational and Hemivariational Inequalities. Berlin, Germany: Springer, 2015. 169--192. Print.
Khan, Akhtar A., Baasansuren Jadamba, and Miguel Sama. "Inverse Problems on Parameter Identification in Partial Differential Equations." Mathematical Methods, Models and Algorithms in Science and Technology. Singapore: World Scientific, 2011. 228- 258. Print.
Grecksch, Wilfried, et al, ed. JOTA. New York: Springer, 2015. Print.
Khan, Akhtar, B. Jadamba, M.S. Gockenbach. “A Comparative Numerical Study ofOptimization Approaches for ellipticInverse Problems.” JMI International Journal of Mathematical Sciences,1 (2010): 1-20. Print. *
Khan, Akhtar, B.Jadamba, B.D. Rouhani, F. Raciti.“Generalized solutions of multi-valuedmonotone quasi variational inequalities.” Optimization and Optimal Control: Theoryand Applications, 2010. 227-240. Print. *
Khan, Akhtar. “Inverse Problems for Variational Equationsand Quasi-variational Inequalities.” International Congress of Mathematicians2010. Hyderabad, India. August 2010.Presentation.
Khan, Akhtar.”Ill-posed Quasi-Variational Inequalities.” International Conference on Mathematicsand Applications. New Delhi, India. 15-17Aug. 2010. Presentation.
Khan, Akhtar. “Stability Analysis of the Modified Output Least Squares for the Elliptic Inverse Problems.” Satellite Conference onInverse Problems. New Delhi, India.14 Aug. 2010. Presentation.
Khan, Akhtar. “Numerical Methods for Elliptic Inverse Problems.” International Conference on Optimization, Simulation and Control. Ulannbaatar, Mongolia. 25-28 July 2010.Presentation.