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چكيده انگليسي :

Solving Monge Ampere equation using a B spline collocation method Application to optics Masoomeh Ahmadi masoomeh ahmadi@math iut ac ir 07 05 2017 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir Advisor Dr Asghar Gholami gholami@cc iut ac ir 2010 MSC 35J66 35J96 35Q60 65N21 65N35 Keywords inverse reflector problem Monge Ampere equation B spline collocation method AbstractIn this thesis we consider both the inverse refractor and the inverse reflector problem These problems goal is to design a free form lens or a free form mirror those when illuminated by a point light source produces a given illumination pattern on a target Both problems are modeled by a strongly nonlinearsecond order partial differential equation PDE of Monge Ampere type We explore a different simplebut very flexible approach for solving such Monge Ampere type equations i e a collocation methodcan directly be applied to the strong formulation corresponding to the Monge Ampere equation Thesolution is approximated in a finite dimensional trial space which we consider the space of splinefunctions because of its good approximation properties The spline space is spanned by B splinefunctions which form an advantageous basis due to its flexible manageability Moreover this basis iswell known to be numerical stable and these basis functions are of minimal support which lead to asparse collocation matrix First of all we formulate the collocation method for a general second order PDE Next the trial spaceand a modified B spline basis that is suited for our particular choice of the collocation points areconstructed in one spatial dimension Finally the trial space is extended to the two dimensional viathe tensor product We aim to investigate some recent works in the literature related to solving theMonge Ampere equation for the inverse reflector problem For this purpose at first some preliminarieswhich will be used in the sequel are prepared Some important preliminaries are some useful materialssuch as optical topics some concepts of functional analysis and topics in PDEs After applying themethod of B spline collocation we have a system of nonlinear equations which we use the iterativemethod of Newton to solve it Evidently we obtain a free surface that has a good reflection Thenumber of equations and unknowns are not equal in this system One way of solving this problem isto reduce the B spline space dimension So we modify the B spline basis functions and solve a systemassociated to the modified basis functions For solving the Monge Ampere equation using the Newton iterative method we consider five examplesof the standard type with Dirichlet boundary conditions We report some errors with norm infinityas well as overall computing time For the first four test cases as suggested we find an initial guessby solving a Poisson boundary value problem with the same Dirichlet boundary conditions and forthe same right hand side function as for the corresponding Monge Ampere equation Solution of theinverse reflection problem is carried out using a B spline collocation method which is performed usingthe Newton type method and Picard iteration and also the Newton method needs an initial guessthat we obtain from the supporting ellipsoid method We are now ready to calculate the reflectorsurfaces for three gray scale test images from the USC SIPI image database

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