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Publication Metadata only A dynamical formulation of one-dimensional scattering theory and its applications in optics(Academic Press Inc Elsevier Science, 2014) NA; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.Publication Open Access A note on a strongly damped wave equation with fast growing nonlinearities(American Institute of Physics (AIP) Publishing, 2015) Zelik, Sergey; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved, the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function. (C) 2015 AIP Publishing LLCPublication Open Access A statistical subgrid scale model for large eddy simulations(American Institute of Physics (AIP) Publishing, 2013) Kara, Rukiye; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131Çinlar velocity is a promising subgrid velocity model for large eddy simulation. The energy spectrum plays a central role for modeling the subgrid stress term in filtered Navier-Stokes equations. Considering a truncated Gamma distribution for radius of eddies, the subgrid scale energy spectrum has been computed analytically. In this study, we develop a new subgrid stress model for representing the small scale effects in LES by defining the parameters of the energy spectrum.Publication Open Access A subspace method for large-scale eigenvalue optimization(Society for Industrial and Applied Mathematics (SIAM), 2018) Meerbergen, Karl; Michiels, Wim; Department of Mathematics; Kangal, Fatih; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; N/A; 113760We consider the minimization or maximization of the Jth largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM T. Matrix Anal. Appl., 35, pp. 699-724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale of thousands.Publication Metadata only A support function based algorithm for optimization with eigenvalue constraints(Siam Publications, 2017) N/A; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. the quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. the convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the practical applicability of the algorithm on the pseudospectral functions.Publication Open Access Active invisibility cloaks in one dimension(American Physical Society (APS), 2015) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We outline a general method of constructing finite-range cloaking potentials which render a given finite-range real or complex potential, v(x), unidirectionally reflectionless or invisible at a wave number, k(0), of our choice. We give explicit analytic expressions for three classes of cloaking potentials which achieve this goal while preserving some or all of the other scattering properties of v(x). The cloaking potentials we construct are the sum of up to three constituent unidirectionally invisible potentials. We discuss their utility in making v(x) bidirectionally invisible at k(0) and demonstrate the application of our method to obtain antireflection and invisibility cloaks for a Bragg reflector.Publication Open Access Addendum to 'Unidirectionally invisible potentials as local building blocks of all scattering potentials'(American Physical Society (APS), 2014) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231In [Phys. Rev. A 90, 023833 (2014)], we offer a solution to the problem of constructing a scattering potential v(x) which possesses scattering properties of one's choice at an arbitrarily prescribed wave number. This solution involves expressing v(x) as the sum of n <= 6 finite-range unidirectionally invisible potentials. We improve this result by reducing the upper bound on n from 6 to 4. In particular, we show that we can construct v(x) as the sum of up to n = 3 finite-range unidirectionally invisible potentials, unless if it is required to be bidirectionally reflectionless.Publication Open Access Adiabatic approximation, semiclassical scattering, and unidirectional invisibility(Institute of Physics (IOP) Publishing, 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(tau). We show that the application of the adiabatic approximation to H(tau) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(tau) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.Publication Open Access Adiabatic series expansion and higher-order semiclassical approximations in scattering theory(Institute of Physics (IOP) Publishing, 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231The scattering properties of any complex scattering potential, nu: R -> C, can be obtained from the dynamics of a particular non-unitary two-level quantum system. S-nu. The application of the adiabatic approximation to S-nu yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of S-v and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of S-nu. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.Publication Metadata only Alternative polynomial and holomorphic Dunford-Pettis properties(Scientific and Technical research Council of Turkey - TUBITAK/Türkiye Bilimsel ve Teknik Araştırma Kurumu, 1999) Department of Mathematics; Freedman, Walden; Faculty Member; Department of Mathematics; College of Sciences; N/AAlternatives to the Polynomial Dunford-Pettis property and the Holomorphic Dunford-Pettis property, called the PDP1 and HDP1 properties, respectively, are introduced. These are shown to be equivalent to the DP1 property, an alternative Dunford-Pettis property previously introduced by the author, thus mirroring the equivalence of the three original properties. © TÜBİTAK.