Finansal krizlerin dinamiklerinin elipsosoidal analiz ve kümeleme metodları aracılığıyla modellenmesi ve anlaşılması

TÜBİTAK TBAG15.02.201

Ters özdeğer problemleri

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.İnvers özdeğer problemleri kendi başlarına önemli oldukları gibi pratik başka önemliuygulamalara da sahiptir. Parabolik ve hiperbolik diferansiyel denklemlerinparametre belirlemeleri ve grating teori de bunların kullanım alanlarındanbazılarıdır.Biz bu çalışmamızda kanonik sturm-liouville özdeğer problemini çeşitli sınırkoşulları için analiz ettik. Sayılabilir sayıda λn özdeğerinin bulunduğunu, bunlaraait asimptotik formülleri teoremleriyle beraber ispatladık. Bundan sonra da tersprobleme yani özdeğerlerin bilinmesi halinde potansiyel fonksiyonun belirlenmesiişini yaptık.Son kısımda da nümerik işlemlerle q potansiyelinin nasıl elde edileceğini örneklerlegösterdik.viiiInverse eigenvalue problems are not only interesting in their own right but also haveimportant practical applications. Other applications appear in paramateridentification problems for parabolic or hyperbolic differantial equations and ingrating theory.We will study the canonical Sturm-Liouville eigenvalue problem some differentboundary conditions. We will prove that there exists a countable number ofeigenvalues λn of this problem and also obtain some asymptotic formula foreigenvalues.We also study inverse problem , i.e, we are given the eigenvalues λn and thendetermine the potential function q (x) .Finally in last section we use some numerical procedure and obtain some resultswith examples.i

Signal representation and recovery under measurement constraints

Ankara : The Department of Electrical and Electronics Engineering and the Graduate School of Engineering and Science of Bilkent University, 2012.Thesis (Ph. D.) -- Bilkent University, 2012.Includes bibliographical references.We are concerned with a family of signal representation and recovery problems under various measurement restrictions. We focus on finding performance bounds for these problems where the aim is to reconstruct a signal from its direct or indirect measurements. One of our main goals is to understand the effect of different forms of finiteness in the sampling process, such as finite number of samples or finite amplitude accuracy, on the recovery performance. In the first part of the thesis, we use a measurement device model in which each device has a cost that depends on the amplitude accuracy of the device: the cost of a measurement device is primarily determined by the number of amplitude levels that the device can reliably distinguish; devices with higher numbers of distinguishable levels have higher costs. We also assume that there is a limited cost budget so that it is not possible to make a high amplitude resolution measurement at every point. We investigate the optimal allocation of cost budget to the measurement devices so as to minimize estimation error. In contrast to common practice which often treats sampling and quantization separately, we have explicitly focused on the interplay between limited spatial resolution and limited amplitude accuracy. We show that in certain cases, sampling at rates different than the Nyquist rate is more efficient. We find the optimal sampling rates, and the resulting optimal error-cost trade-off curves. In the second part of the thesis, we formulate a set of measurement problems with the aim of reaching a better understanding of the relationship between geometry of statistical dependence in measurement space and total uncertainty of the signal. These problems are investigated in a mean-square error setting under the assumption of Gaussian signals. An important aspect of our formulation is our focus on the linear unitary transformation that relates the canonical signal domain and the measurement domain. We consider measurement set-ups in which a random or a fixed subset of the signal components in the measurement space are erased. We investigate the error performance, both We are concerned with a family of signal representation and recovery problems under various measurement restrictions. We focus on finding performance bounds for these problems where the aim is to reconstruct a signal from its direct or indirect measurements. One of our main goals is to understand the effect of different forms of finiteness in the sampling process, such as finite number of samples or finite amplitude accuracy, on the recovery performance. In the first part of the thesis, we use a measurement device model in which each device has a cost that depends on the amplitude accuracy of the device: the cost of a measurement device is primarily determined by the number of amplitude levels that the device can reliably distinguish; devices with higher numbers of distinguishable levels have higher costs. We also assume that there is a limited cost budget so that it is not possible to make a high amplitude resolution measurement at every point. We investigate the optimal allocation of cost budget to the measurement devices so as to minimize estimation error. In contrast to common practice which often treats sampling and quantization separately, we have explicitly focused on the interplay between limited spatial resolution and limited amplitude accuracy. We show that in certain cases, sampling at rates different than the Nyquist rate is more efficient. We find the optimal sampling rates, and the resulting optimal error-cost trade-off curves. In the second part of the thesis, we formulate a set of measurement problems with the aim of reaching a better understanding of the relationship between geometry of statistical dependence in measurement space and total uncertainty of the signal. These problems are investigated in a mean-square error setting under the assumption of Gaussian signals. An important aspect of our formulation is our focus on the linear unitary transformation that relates the canonical signal domain and the measurement domain. We consider measurement set-ups in which a random or a fixed subset of the signal components in the measurement space are erased. We investigate the error performance, both We are concerned with a family of signal representation and recovery problems under various measurement restrictions. We focus on finding performance bounds for these problems where the aim is to reconstruct a signal from its direct or indirect measurements. One of our main goals is to understand the effect of different forms of finiteness in the sampling process, such as finite number of samples or finite amplitude accuracy, on the recovery performance. In the first part of the thesis, we use a measurement device model in which each device has a cost that depends on the amplitude accuracy of the device: the cost of a measurement device is primarily determined by the number of amplitude levels that the device can reliably distinguish; devices with higher numbers of distinguishable levels have higher costs. We also assume that there is a limited cost budget so that it is not possible to make a high amplitude resolution measurement at every point. We investigate the optimal allocation of cost budget to the measurement devices so as to minimize estimation error. In contrast to common practice which often treats sampling and quantization separately, we have explicitly focused on the interplay between limited spatial resolution and limited amplitude accuracy. We show that in certain cases, sampling at rates different than the Nyquist rate is more efficient. We find the optimal sampling rates, and the resulting optimal error-cost trade-off curves. In the second part of the thesis, we formulate a set of measurement problems with the aim of reaching a better understanding of the relationship between geometry of statistical dependence in measurement space and total uncertainty of the signal. These problems are investigated in a mean-square error setting under the assumption of Gaussian signals. An important aspect of our formulation is our focus on the linear unitary transformation that relates the canonical signal domain and the measurement domain. We consider measurement set-ups in which a random or a fixed subset of the signal components in the measurement space are erased. We investigate the error performance, both in the average, and also in terms of guarantees that hold with high probability, as a function of system parameters. Our investigation also reveals a possible relationship between the concept of coherence of random fields as defined in optics, and the concept of coherence of bases as defined in compressive sensing, through the fractional Fourier transform. We also consider an extension of our discussions to stationary Gaussian sources. We find explicit expressions for the mean-square error for equidistant sampling, and comment on the decay of error introduced by using finite-length representations instead of infinite-length representations.Özçelikkale Hünerli, AyçaPh.D

Lineer olmayan ve ek koşulları integral operatörler ile verilmiş parabolik problemler

Fizik ve mühendislik gibi birçok alanda karşılaşılan lineer olmayan problemler son yıllarda matematikçilerin en yaygın çalışma alanlarını oluşturmaktır. Bu problemler genelde düz problemler olsalar da bazen ters problemler olarak da ele alınabilir. Bu tezde, ters problemler ve ters problemlerin özel bir hali olan ek koşulları integral operatörler ile verilmiş lineer olmayan parabolik problem tanıtıldı. Bu tip problemlerin indirgeme metodu kullanılarak bilinmeyen katsayının ek koşul yardımı ile ortadan kaldırılabileceği ve problemin başlangıç ve sınır değer problemi olarak ifade edilebileceği gösterilmiştir. Ayrıca problemin nümerik çözümleri için sonlu fark denklemleri elde edilmiştir. Ele alınan örnek problem çeşitli sonlu fark şemaları ile ele alınan örnek problem çözülmüş ve elde nümerik sonuçlar değerlendirilmiştir

Hafızalı yarı doğrusal ısı denklemi için ters problemler

In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.Ph.D. - Doctoral Progra