23 research outputs found
(SI10-077) A Novel Collocation Method for Solving Second-order Volterra Integro-differential Equations
In this article, we present an efficient numerical methodology to solve second-order linear Volterra integro-differential equations. Further, the modified Chebyshev collocation method is used at the Gauss-Lobatto collocation points. In that context, some theoretical investigation related to error analysis is suggested through residual function. Numerical examples are also encountered to study the applicability of the present method. In order to get a vivid illustration of the efficiency, we present a comparative survey with three existing collocation methods
Some Aspects of Mathematical Programming in Statictics
The Almighty has created the Universe and things present in it with an
order and proper positions and the creation looks unique and perfect. No one
can even think much better or imagine to optimize these further. People
inspired by these optimum results started thinking about usage of
optimization techniques for solving their real life problems. The concept of
constraint optimization came into being after World War II and its use
spread vastly in all fields. However, in this process, still lots of efforts are
needed to uncover the mysteries and unanswered questions, one of the
questions always remains live that whether there can be a single method that
can solve all types of nonlinear programming problems like Simplex
Method solves linear programming problems. In the present thesis, we have
tried to proceed in this direction and provided some contributions towards
this area.
The present thesis has been divided into five chapters, chapter wise
summary is given below:
Chapter-1 is an introductory one and provides genesis of the
Mathematical Programming Problems and its use in Statistics.
Relationship of mathematical programming with other statistical
measures are also reviewed. Definitions and other pre-requisites are
also presented in this chapter. The relevant literature on the topic has
been surveyed.
Chapter-2 deals with the two dimensional non-linear programming
problems. We develop a method that can solve approximately all type
of two dimensional nonlinear programming problems of certain class.
The method has been illustrated with numerical examples.
Chapter-3 is devoted to the study of n-dimensional non-linear
programming problems of certain types. We provide a new method
based on regression analysis and statistical distributions. The method
can solve n-dimensional non-linear programming problems making
use of regression analysis/co-efficient of determination.
In chapter-4 we introduce a filtration method of mathematical
programming. This method divides the constraints into active and non
active and try to eliminate the less important constraints (non-active
constraints) and solve the problem with only active constraints. This
helps to find solution in less iterations and less in time while retaining
optimality of the solution.
The final chapter-5 deals with an interesting relationship between
linear and nonlinear programming problems. Using this relationship,
we can solve linear programming problems with the help of non-linear
programming problems. This relationship also helps to find a better
alternate solutions to the linear programming problems.
In the end, a complete bibliography is provided
Optimization design of mth-band FIR filters with application to image processing
Cone programming (CP) is a class of convex optimization technique, in which a linear objective function is minimized over the intersection of a set of affine constraints. Such constraints could be linear or convex, equalities or inequalities. Owing to its powerful optimization capability as well as flexibility in accommodating various constraints, the cone programming finds wide applications in digital filter design. In this thesis, fundamentals of linear-phase M th-band FIR filters are first introduced, which include the time-domain interpolation condition and the desired frequency specifications. The restriction of the interpolation matrix M for linear-phase two-dimensional (2-D) M th-band filters is also discussed by considering both the interpolation condition and the symmetry of the impulse response of the 2-D filter. Based on the analysis of the M th-band properties, a semidefinite programming (SOP) optimization approach is developed to design linear-phase 1-0 and 2-D M th-band filters. The 2-D SOP optimization design problem is modeled based on both the mini-max and the least-square error criteria. In contrast to the 1-D based design, the 2-D direct SDP design can offer an optimal equiripple result. A second-order cone programming (SOCP) optimization approach is then presented as an alternative for the design of M th-band filters. The performances as well as the design complexity of these two design approaches are justified through numerical design examples. Simulation results show that the performance of the SOCP approach is better than that of the SDP approach for 1-D M th-band filter design due to its reduced computational complexity for the worst-case, whereas the SDP approach is more appropriate for the 2-D M th-band filter design than the SOCP approach because of its efficient and simple optimization structure. Moreover, the designed M th-band filters are proved useful in image interpolation according to both the visual quality and the peak signal-to-noise ratio (PSNR) for the images with different levels of details
A product space reformulation with reduced dimension for splitting algorithms
In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra’s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments