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Regularization Of Inverse Problems

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

Diabetic Retinopathy (DR) is a complication of diabetes that affects the eyes. It is caused by blood vessel damage of the light-sensitive tissue at the back of the retina. Neovascularization are emerged and the small blood vessels are blocked. The prevention or delaying vision loss can be obtained by DR early detection. The retinal microvascular network as a biological system has its own multifractal features as generalized dimensions, lacunarity and singularity spectrum. In this study, a novel approach for DR early detection based on the multifractal geometry has been proposed in some details. Analyzing the macular optical coherence tomography angiography (OCTA) images for diagnosing early non-proliferative diabetic retinopathy (NPDR). Using a supervised machine learning method as a Support Vector Machine (SVM) algorithm to automate the diagnosis process and improving the resultant accuracy. The classification technique had achieved 98.5 % accuracy. This approach also can classify easily other diabetic retinopathy stages or other retinal diseases, which affect the vessels or neovascularization distribution.

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A Novel Approach of Diabetic Retinopathy Early Detection Based on Multifractal Geometry Analysis for OCTA Macular Images Using Support Vector Machine

In this paper, we detail a method for domain specific, multi-category emotion recognition, based on human computation. We create an Amazon Mechanical Turk 1 task that elicits emotion labels and phrase-emotion associations from the participants. Using the proposed method, we create an emotion lexicon, compatible with the 20 emotion categories of the Geneva Emotion Wheel. GEW is the first computational resource that can be used to assign emotion labels with such a high level of granularity. Our emotion annotation method also produced a corpus of emotion labeled sports tweets. We compared the crossvalidated version of the lexicon with existing resources for both the positive/negative and multi-emotion classification problems. We show that the presented domain-targeted lexicon outperforms the existing general purpose ones in both settings. The performance gains are most pronounced for the fine-grained emotion classification, where we achieve an accuracy twice higher than the benchmark. 2

Fine-Grained Emotion Recognition in Olympic Tweets Based on Human Computation

We present in this paper a system of Amazigh handwriting recognition based on horizontal and vertical centerline of the character. After the image preprocessing, the text is segmented into lines and then into characters. The positions of the vertical and horizontal centerline of the character are used to obtain a set of independent and dependent features to those lines. These features are related to the densities of pixels and are extracted on binary images of characters using the sliding window technique. Finally, a multilayer perceptron is used for character classification. The system was tested on two bases of the Amazigh characters: on a printed database of Amazigh characters and on another one for handwritten characters created locally. The correct average recognition rate obtained using 10-crossvalidation was 99.28 % for the 19437 Amazigh printed characters and 96.32% for the 20150 Amazigh handwritten characters.

Amazigh Handwritten Character Recognition based on Horizontal and Vertical Centerline of Character

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

Loaded partial differential equations are solved numerically. For illustrative purposes, a boundary value problem for a parabolic equation with various point loads is considered. By applying difference approximations, the problems are reduced to systems of algebraic equations of special structure, which are solved using a parametric representation involving solutions of auxiliary linear systems with tridiagonal matrices. Numerical results are presented and analyzed.

Finite-difference methods for solving loaded parabolic equations

In this article we applied Support Vector Machines to acoustic model of Speech Recognition System based on MFCC and LPC features for Azerbaijani DataSet. This DataSet has been used for speech recognition by Multilayer Artificial Neural Network and achieved some results. The main goal of this work is applying SVM techniques to the Azerbaijan Speech Recognition System. The variety of results of SVM with different Kernel functions is analyzed in the training process. It is shown that SVM with radial basis and polynomial kernels give better recognition results that Multilayer Artificial Neural Network.

Speech recognition using Support Vector Machines

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.

On the solution of boundary value problems with nonseparated multipoint and integral conditions

Consideration was given to a class of the problems of optimal control of rod (plate) heating by controlling the furnace temperature. Control relies on the process information feedbacked continuously or discretely only from the individual points on the rod where the temperature sensors are installed. For the continuous and discrete observation, the mathematical model of the controlled process was reduced to the pointwise loaded parabolic equation. The paper established formulas for the functional gradient, proposed schemes of their numerical solution, and presented the results of the numerical experiments.

Kamil Aida-zade

Papers

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

Finite-difference methods for solving loaded parabolic equations

Speech recognition using Support Vector Machines

On the solution of boundary value problems with nonseparated multipoint and integral conditions

On an approach to designing control of the distributed-parameter processes