Local Improvements to Reduced-Order Approximations of Optimal Control Problems Governed by Diffusion-Convection-Reaction Equation

We consider the optimal control problem governed by diffusion convection reaction equation without control constraints. The proper orthogonal decomposition(POD) method is used to reduce the dimension of the problem. The POD method may be lack of accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. We are interested in the perturbation of diffusion term. To increase the accuracy and robustness of the basis, we compute three bases additional to the baseline POD. The first two of them use the sensitivity information to extrapolate and expand the POD basis. The other one is based on the subspace angle interpolation method. We compare these different bases in terms of accuracy and complexity and investigate the advantages and main drawbacks of them.Comment: 19 pages, 5 figures, 2 table

Structure preserving reduced order modeling for gradient systems

Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg-Landau equation and Swift-Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state

Streamline upwind/Petrov Galerkin solution of optimal control problems governed by time dependent diffusion-convection-reaction equations

The streamline upwind/Petrov Galerkin (SUPG) finite element method is studied for distributed optimal control problems governed by unsteady diffusion-convectionreaction equations with control constraints. We derive stability and convergence estimates for fully-discrete state, adjoint and control and discuss the choice of the stabilization parameter by applying backward Euler method in time. We show that by balancing the error terms in the convection dominated regime, optimal convergence rates can be obtained. The numerical results confirm the theoretically observed convergence rates.Publisher's Versio

Okul Öncesi Dönem Çocuklarının Duygusal Yüz İfadelerini Yansıtan İnsan Figürü Çizimlerinde Kullandıkları Renklerin İncelenmesi

The aim of this study is to examine the facial expressions and color preferences used by orphaned children, children who have experienced an earthquake and children living with their parents.The sample of the study included 85 children, 27 of whom had experienced the great earthquakes in Bolu and Gölcük in 1999, 21 orphans living in Keçiören Atatürk Child Care Center and 37 children living with their families. Regular A4 white paper and coloring pens were used as data collecting tools, and children were asked to draw human figures with happy, scared, surprised, sad and aggravated facial expressions, respectively. Children were told that they were allowed to use whichever colors they wanted.. The children’s drawings, each reflecting different facial expressions, were examined to ascertain which, and how many, colors had been used. The results of the study revealed that the majority of the children who had experienced an earthquake and the orphaned children used only one or two colors, while the majority of the children living with their families used at least two or more different colors. It was found that the children who had experienced an earthquake particularly chose black as the color to reflect all of the facial expressions. It was seen that the colors used by orphans and by the children living with their families showed differences according to the facial expressions illustrated.Bu araştırmanın amacı, korunmaya muhtaç çocukların, depremi yaşayan çocukların ve ailesi ile yaşayan çocukların duygularını dışa vurmalarında, yüz ifadeleri ve renk tercihlerini incelemektir. Çalışmanın örneklemini, Bolu ve Gölcük depremini yaşamış 27 çocuk ile Ankara ilinde Keçiören Atatürk Çocuk Yuvası’nda kalan 21 korunmaya muhtaç çocuk ve ailesi yanında yaşayan 37 çocuk olmak üzere toplam 85 çocuk oluşturmuştur. Veri toplama aracı olarak A4 boyutunda beyaz kağıtlar ve renkli boya kalemleri çocuklara dağıtılmış ve çocuklardan sırasıyla mutlu, korkmuş, şaşırmış, üzgün ve öfkeli bir insan figürü çizmeleri istenmiştir. Çocukların renk tercihlerinde serbest oldukları belirtilmiştir. Çocukların yaptığı her biri farklı bir yüz ifadesini yansıtan resimler, kullanılan renk sayısına ve kullanılan renklere göre incelenmiştir. Araştırma sonuçları, korunmaya muhtaç ve depremi yaşayan çocukların çoğunluğunun tek veya iki renk kullandığını, aileleri ile yaşayan çocukların çoğunluğunun ise en az iki ya da daha fazla renk kullandığını ortaya koymaktadır. Özellikle depremi yaşamış çocukların tüm yüz ifadelerini yansıtan insan figürü çizimlerinde siyah rengi tercih ettikleri bulunmuştur. Korunmaya muhtaç çocukların ve ailesi ile yaşayan çocukların en yüksek oranda kullandıkları renklerin yüz ifadelerine göre değişiklik gösterdiği belirlenmiştir. Her iki grup incelendiğinde; ailesi ile yaşayan çocukların kullandıkları renklerin kullanım yüzdelerinin, korunmaya muhtaç çocukların kullanım yüzdelerine göre daha yüksek olduğu da ulaşılan önemli bulgulardandır

Modeling of mouse experiments suggests that optimal anti-hormonal treatment for breast cancer is diet-dependent

Estrogen receptor positive breast cancer is frequently treated with anti-hormonal treatment such as aromatase inhibitors (AI). Interestingly, a high body mass index has been shown to have a negative impact on AI efficacy, most likely due to disturbances in steroid metabolism and adipokine production. Here, we propose a mathematical model based on a system of ordinary differential equations to investigate the effect of high-fat diet on tumor growth. We inform the model with data from mouse experiments, where the animals are fed with high-fat or control (normal) diet. By incorporating AI treatment with drug resistance into the model and by solving optimal control problems we found differential responses for control and high-fat diet. To the best of our knowledge, this is the first attempt to model optimal anti-hormonal treatment for breast cancer in the presence of drug resistance. Our results underline the importance of considering high-fat diet and obesity as factors influencing clinical outcomes during anti-hormonal therapies in breast cancer patients.Comment: 44 pages, 21 figure

A new mathematical model for tuberculosis epidemic under the consciousness effect

Compared to many infectious diseases, tuberculosis has a high mortality rate. Because of this, a great deal of illustrative research has been done on the modeling and study of tuberculosis using mathematics. In this work, a mathematical model is created by taking into account the underlying presumptions of this disease. One of the main novelties of the paper is to consider two different treatment strategies namely protective treatment for the latent populations from the disease and the main treatment applied to the infected populations. This situation can be regarded as the other novelty of the paper. The susceptible, latent, infected, and recovered populations, as well as the two mentioned treatment classes, are all included in the proposed six-dimensional model's compartmental framework. Additionally, a region that is biologically possible is presented, as well as the solution's positivity, existence, and uniqueness. The suggested model's solutions are carried out as numerical simulations using assumed and literature-based parameter values and analyzing its graphics. To get the results, a fourth-order Runge-Kutta numerical approach is used

The Gamma-Butyrolactone Model of Absence Epilepsy: Acute and Chronic Effects in Wistar Rats

Objectives: We studied the electroencephalographic (EEG) and behavioral changes of the chemical model of generalized absence epilepsy induced by acute and chronic administration of gamma-butrolactone (GBL), a prodrug of gamma-hydroxybutyric acid.Methods: Adult male Wistar rats under anesthesia were implanted with bilateral cortical recording electrodes. The rats were administered 30 intraperitoneal injections of GBL twice daily from Monday to Friday and EEG was recorded 20 min before and 40 min after GBL injections. In order to monitor spontaneous spike-and-wave discharges (SWDs), the baseline EEGs on the subsequent Monday mornings after the first, second and third weekends were recorded for 90 min.Results: The intraperitoneal administration of GBL caused a rapid onset of bilaterally synchronous SWDs in the cortical EEG accompanied by behavioral immobility, vacant-staring and vibrissal twitching. By repeated GBL injections, animals displayed spontaneous bilateral synchronous SWDs in the baseline EEG on the Monday morning session after the GBL-free weekend period (60 h after the Friday afternoon injection).Conclusion: The present study reports the acute and chronic effects of the systemic administration of GBL. The chronic systemic application of GBL may represent a model of epileptogenesis for absence epilepsy