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<p class="MsoNormal"><span style="background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;"><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Sayin
Liste Uyeleri,</span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"></span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Gebze Teknik Üniversitesi (GTU</span></span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">)
</span><span style="background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;"><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Matematik
Bölümü Genel Seminerleri kapsamında,</span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"></span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">23</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">
</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Aralık</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"> Cuma günü saat 14:00'da
</span></span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Emre Mengi</span><span style="background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;"><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">
(</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Koç </span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Üniversitesi) bir seminer verecektir. Seminerin detayları
</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">a</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">şağıda olup tüm ilgilenenler davetlidir.</span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"></span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Saygılarımızla.</span><o:p></o:p></span></p>
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<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"></span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Dear all,</span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"></span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">There will be a seminar in Gebze Technical University (GTU) on
</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">23</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">th of</span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;"></span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Dec</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">ember
by Emre Mengi </span></span><span style="font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
mso-ascii-theme-font:minor-latin;mso-fareast-font-family:Calibri;mso-fareast-theme-font:
minor-latin;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:minor-bidi;mso-ansi-language:TR;mso-fareast-language:EN-US;
mso-bidi-language:AR-SA"><span style="background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;"><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">(</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Koç
University</span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">)</span><br>
<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">Time and place: At 14:00 in Department of Mathematics, Building I, Seminar room.</span><br>
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<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif; color: rgb(33, 33, 33);">TITLE: Large-Scale Optimization of Eigenvalues<o:p></o:p></span></p>
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<span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">ABSTRACT: This talk introduces a subspace framework for the optimization of the jth largest eigenvalue of a large, Hermitian and analytic matrix-valued function depending on several
parameters for a prescribed j. The range of the large matrix-valued function </span><span style="font-size: 12pt; font-family: Calibri, Arial, Helvetica, sans-serif;">is projected onto a small subspace orthogonally, and its domain is restricted to </span><span style="font-size: 11pt; font-family: "Times New Roman", Times, serif;"><span style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;">the
same subspace. This leads to reduced eigenvalue optimization problems involving small matrix-valued functions. The subspace is expanded with the addition of the eigenvectors for the optimal parameter values of the small problem. In the infinite dimensional
setting and for the minimization of the jth largest eigenvalue, we prove that the optimal solution of the reduced problem converges to the optimal solution of the infinite dimensional problem as the subspace dimension grows to infinity. Furthermore, we establish
that the rate of convergence is at least superlinear with respect to the subspace dimension for both the minimization and the maximization problems. These theoretical results are in harmony with what we observe in practice, that the problems involving matrices
of size on the scale of thousands are approximated very accurately with the reduced problems involving matrices of size on the scale of tens.</span></span><o:p></o:p></span></p>
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