Researcher:
Mengi, Emre

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Emre

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Mengi

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2011 - 2019152020 - 20227

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    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; 113760
    Optimization 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.
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    Quantitative analysis of structural alterations in the choroid of patients with active Behçet uveitis
    (Lippincott Williams and Wilkins (LWW), 2018) Oray, Merih; Herbort, Carl P.; Akman, Mehmet; Tugal-Tutkun, Ilknur; Department of Mathematics; N/A; N/A; N/A; N/A; Mengi, Emre; Önal, Sumru; Uludağ, Günay; Metin, Mustafa Mert; Akbay, Aylin Koç; Faculty Member; Other; Doctor; Undergraduate Student; Doctor; Department of Mathematics; College of Sciences; School of Medicine; N/A; School of Medicine; N/A; N/A; N/A; Koç University Hospital; N/A; Koç University Hospital; 113760; 52359; N/A; N/A; N/A
    Purpose: To quantitatively analyze in vivo morphology of subfoveal choroid during an acute attack of Behcet uveitis. Methods: In this prospective study, 28 patients with Behcet uveitis of <= 4-year duration, and 28 control subjects underwent enhanced depth imaging optical coherence tomography. A novel custom software was used to calculate choroidal stroma-to-choroidal vessel lumen ratio. Subfoveal choroidal thickness was measured at fovea and 750 mu m nasal, temporal, superior, and inferior to fovea. Patients underwent fluorescein angiography and indocyanine green angiography. Receiver operating characteristic curve and area under the curve were computed for central foveal thickness. The eye with a higher Behcet disease ocular attack score 24 was studied. The main outcome measures were choroidal stromato-choroidal vessel lumen ratio and choroidal thickness. Results: The mean total Behcet disease ocular attack score 24, fluorescein angiography, and indocyanine green angiography scores were 7.42 +/- 4.10, 17.42 +/- 6.03, and 0.66 +/- 0.73, respectively. Choroidal stroma-to-choroidal vessel lumen ratio was significantly higher in patients (0.413 +/- 0.056 vs. 0.351 +/- 0.063, P = 0.003). There were no significant differences in subfoveal choroidal thickness between patients and control subjects. Choroidal stroma-tochoroidal vessel lumen ratio correlated with retinal vascular staining and leakage score of fluorescein angiography (r = 0.300, P = 0.036). Central foveal thickness was significantly increased in patients (352.750 +/- 107.134 mu m vs. 263.500 +/- 20.819 p.m, P < 0.001). Central foveal thickness showed significant correlations with logarithm of minimum angle of resolution vision, Behcet disease ocular attack score 24, total fluorescein angiography score, retinal vascular staining and/or leakage and capillary leakage scores of fluorescein angiography, and total indocyanine green angiography score. At 275 mu m cutoff, diagnostic sensitivity and specificity of central foveal thickness for acute Behcet uveitis were 89% and 72%, respectively (area under the curve = 0.902; 95% CI = 0.826-0.978, P < 0.001). Conclusion: There was choroidal stromal expansion which was not associated with thickening of the choroid. Central foveal thickness may be used as a noninvasive measure to assess inflammatory activity in early Behcet uveitis.
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    Nonsmooth algorithms for minimizing the largest eigenvalue with applications to inner numerical radius (vol 40, pg 2342, 2020)
    (Oxford Univ Press, 2020) N/A; N/A; Department of Mathematics; Kangal, Fatih; Mengi, Emre; PhD Student; Faculty Member; Department of Mathematics; Graduate School of Sciences and Engineering; College of Sciences; N/A; 113760
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    Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity
    (Springer, 2011) N/A; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev's formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.
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    Generalized eigenvalue problems with specified eigenvalues
    (Oxford University Press (OUP), 2014) Kressner, Daniel; Nakic, Ivica; Truhar, Ninoslav; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms.
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    Matrix polynomials with specified eigenvalues
    (Elsevier Science Inc, 2015) Karow, Michael; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    This work concerns the distance in the 2-norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Initially, we consider perturbations of the constant coefficient matrix only. A singular value optimization characterization is derived for the associated distance. We also consider the distance in the general setting, when all of the coefficient matrices are perturbed. In this general setting, we obtain a lower bound in terms of another singular value optimization problem. The singular value optimization problems derived facilitate the numerical computation of the distances.
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    PublicationOpen Access
    Certifying global optimality for the L-infinity-norm computation of large-scale descriptor systems
    (Elsevier, 2020) Schwerdtner, P.; Voigt, M.; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    We present a method for the certification of algorithms that approximate the L-infinity or H-infinity-norm of transfer functions of large-scale (descriptor) systems. This certification is needed because such algorithms depend heavily on user input, and may converge to a local maximizer of the related singular value function leading to an incorrect value, much lower than the actual norm. Hence, we design an algorithm that determines whether a given value is less than the L-infinity-norm of the transfer function under consideration, and that does not require user input other than the system matrices. In the algorithm, we check whether a certain structured matrix pencil has any purely imaginary eigenvalues by repeatedly applying a structure-preserving shift-and-invert Arnoldi iteration combined with an appropriate shifting strategy. Our algorithm consists of two stages. First, an interval on the imaginary axis which may contain imaginary eigenvalues is determined. Then, in the second stage, a shift is chosen on this interval and the eigenvalues closest to this shift are computed. If none of these eigenvalues is purely imaginary, then an imaginary interval around the shift of appropriate length is removed such that two subintervals remain. This second stage is then repeated on the remaining two subintervals until either a purely imaginary eigenvalue is found or no critical subintervals are left. We show the effectiveness of our method by testing it without any parameter adaptation on a benchmark collection of large-scale systems.
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    PublicationOpen Access
    Approximation of stability radii for large-scale dissipative Hamiltonian systems
    (Springer, 2020) Aliyev, Nicat; Mehrmann, Volker; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    A linear time-invariant dissipative Hamiltonian (DH) system (x) over dot = (J-R)Qx, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625-1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + B Delta C-H for given matrices B, C, and another with respect to Hermitian perturbations in the form R + B Delta B-H, Delta = Delta(H). We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.
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    PublicationOpen Access
    Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems
    (Oxford University Press (OUP), 2017) Meerbergen, Karl; Michiels, Wim; Van Beeumen, Roel; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.
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    PublicationOpen Access
    Subspace methods for three-parameter eigenvalue problems
    (Wiley, 2019) Hochstenbach, Michiel E.; Meerbergen, Karl; Plestenjak, Bor; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    We propose subspace methods for three-parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for two-parameter eigenvalue problems exist, their extensions to a three-parameter setting seem challenging. An inherent difficulty is that, while for two-parameter eigenvalue problems, we can exploit a relation to Sylvester equations to obtain a fast Arnoldi-type method, such a relation does not seem to exist when there are three or more parameters. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a Jacobi-Davidson-type method for three or more parameters, which we generalize from its two-parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi-Davidson approach is devised to locate eigenvalues close to a prescribed target; yet, it often also performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. The proposed approaches are suitable especially for problems where the computation of several eigenvalues is required with high accuracy. MATLAB implementations of both methods have been made available in the package MultiParEig (see https://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig).