Invited Speakers
Invited Speakers
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Title : Subspace Iteration Techniques for Approximating Eigenpairs of Large Matrices
Abstract : Many application problems demand computing the eigenvalues and eigenvectors of large matrices. Since this problem cannot be solved exactly, many heuristics have been developed to approximate the eigenvalues and eigenvectors. One of the important heuristics is to select a suitable small subspace and the projection of the given large matrix on this subspace. Then eigenvalues and eigenvectors of such small projected matrices are taken as approximations to those of the large matrix. In this talk we will review some of these approximation methods which are currently in use. Also, we will highlight the difficulties encountered and their solutions giving rise to problems that need attention of researchers.
Title : Spectral Methods for Computing Discontinuous Solutions
Abstract : Spectral and pseudo-spectral methods are well known global methods providing approximation that converges exponentially to the exact function, if it is infinitely smooth. In practical applications, the benefit of the spectral method is that extraordinary accuracy can be achieved with moderate number of data/nodal points. These methods have been useful in successfully simulating incompressible turbulent flows and also applied in applications in geophysics and time domain electromagnetics. However, if piecewise smooth or discontinuous solutions are approximated by spectral methods, they exhibit only linear or sub-linear convergence rate. This is because the approximate solution is contaminated by spurious oscillations, known as Gibbs phenomenon, near the discontinuities. In this talk, an overview on spectral methods and some of the techniques to resolve Gibbs phenomenon (thus improving the order of convergence) will be discussed.
Title : A H1-Galerkin Mixed finite element method for fourth order differential equations
Abstract : In this talk, a H1-Galerkin mixed finite element method is introduced for fourth order differential equations. Using suitable splitting technique, the fourth order problem is split into a system of coupled problems and then, this method is applied to the resultant couple system. The method described in this talk may also be considered as a Petrov-Galerkin method with cubic spline space as trial space and piecewise linear space as test space. Optimal order error estimates are obtained without any restriction on the mesh for both semi-discrete and fully discrete schemes. Linear/ non-linear, time independent / time dependent differential equations are considered. This method is compared with orthogonal spline collocation method.
Title : Numerical Methods and its Applications in Real life Problems
Abstract : Mathematics plays a major role in the development of modern sciences, engineering, management and many other important areas of scientific activities. On the other hand, real life situations give rise to topics in mathematics such as linear equations, systems of linear equations, matrix theory, differential equations, interpolation and optimization. This lecture focusses on few examples that exhibit the interaction between the abstract mathematics and real life applications, and then we discuss about how Numerical Methods plays a key role in treating the problems arising in the real life situations. In addition, we shall discuss about a specially designed numerical technique called Single-Term Walsh Series method with an application.
Title : Fractional Filters for Retinal Blood Vessel Segmentation
Abstract : In this talk, we will discuss a new fractional filter and an algorithm for retinal blood vessel segmentation. The proposed fractional filter is designed with the help of a weighted fractional derivative and an exponential weight factor. We have utilized the fractional filter and the eigenvalue maps of a local covariance matrix to develop the algorithm for retinal vessel segmentation. The local covariance matrix is formed by a second-order image moment. Experiments are performed on two well-studied evaluation databases named STARE and DRIVE and the simulation results are discussed.
Title : Spectral Collocation Method and its Applications to Differential Equations
Abstract : Spectral Methods are a class of mathematical techniques to numerically solve differential equations. It is a collective name for spatial discritization methods. In this lecture, the essence of spectral method will be explained. Basic idea is to write the solution of a problem as a truncated series of global basis functions. It rely on the expansion of solution as coefficients for basis functions. The coefficients are considered as SPECTRUM of the solutions, hence the name. The method will be illustrated with an example.
Title : On designing Scalable Numerical Algorithms
Abstract : In this talk, we will see a few discoveries around addressing the scalability issues of numerical computations abstractly and numerically. We also present a few key open problems around scalability and mathematics of computation. For demonstration, we present the design and analysis of a scalable numerical algorithm for a problem which is a special case of a generalised viscous Burger equation.
Title : Numerical Solution of Functional Differential Equations having Boundary Layer
Abstract : Functional Differential Equations have a richer mathematical framework when compared with Ordinary Differential Equations for the analysis of bio-system dynamics and display better consistency with the nature of the underlying processes and predictive results. In general, an ordinary differential equation in which the highest order derivative is multiplied by a small positive parameter and containing at least one delay/advance is popularly known as Singularly Perturbed Functional Differential Equations. Such types of differential equations arise in the modelling of various practical phenomena in bioscience, engineering, control theory, such as in variational problems in control theory, in describing the human pupil-light reflex, in a variety of models for physiological processes or diseases and first exit time problems in the modelling of the determination of expected time for the generation of action potential in nerve cells by random synaptic inputs in dendrites. In ODE’s, the future behaviour of many phenomena are assumed to be determined by the present and is independent of the past. In Functional Differential Equations, the past exerts its influence in a significant manner upon the future.
To solve Functional Differential Equations, perturbation methods such as WKB method together with matched asymptotic expansions are used extensively. These asymptotic expansions of solutions require skill, insight and experimentation. Further, matching of the coefficients of the inner and outer regions solution expansions is also a demanding process. Hence, researchers started developing numerical methods. If we use the existing numerical methods with the step size more than the perturbation parameters, for solving these problems we get oscillatory solutions due to the presence of the boundary layer. Existing numerical methods will produce good results only when we take step size less than perturbation parameters. This is very costly and time-consuming process. Hence, the researchers are concentrating on developing the methods, which can work with a reasonable step size. In fact, the efficiency of a numerical method is determined by its accuracy, simplicity in computing the numerical solution and its sensitivity to the parameters of the given problem.
With this motivation, we, present here some simple, easy and efficient numerical methods which are readily adaptable for computer implementation with a modest amount of problem preparation.
To solve Functional Differential Equations, perturbation methods such as WKB method together with matched asymptotic expansions are used extensively. These asymptotic expansions of solutions require skill, insight and experimentation. Further, matching of the coefficients of the inner and outer regions solution expansions is also a demanding process. Hence, researchers started developing numerical methods. If we use the existing numerical methods with the step size more than the perturbation parameters, for solving these problems we get oscillatory solutions due to the presence of the boundary layer. Existing numerical methods will produce good results only when we take step size less than perturbation parameters. This is very costly and time-consuming process. Hence, the researchers are concentrating on developing the methods, which can work with a reasonable step size. In fact, the efficiency of a numerical method is determined by its accuracy, simplicity in computing the numerical solution and its sensitivity to the parameters of the given problem.
With this motivation, we, present here some simple, easy and efficient numerical methods which are readily adaptable for computer implementation with a modest amount of problem preparation.
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