Spelling suggestions: "subject:"approximation theory"" "subject:"eapproximation theory""
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Hierarchical aggregation of linear systems with multiple time scalesJanuary 1979 (has links)
M. Coderch ... [et al.]. / Bibliography: leaf 6. / "September, 1981." / Supported in part by the DOE under Grant ET-76-C-01-2295
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Piecewise polynomial functions on a planar region: boundary constraints and polyhedral subdivisionsMcDonald, Terry Lynn 16 August 2006 (has links)
Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines
of degree at most k forms a vector space Crk() Moreover, a nice way to study
Cr
k()is to embed n Rd+1, and form the cone b of with the origin. It turns
out that the set of splines on b is a graded module Cr b() over the polynomial ring
R[x1; : : : ; xd+1], and the dimension of Cr
k() is the dimension o
This dissertation follows the works of Billera and Rose, as well as Schenck and
Stillman, who each approached the study of splines from the viewpoint of homological
and commutative algebra. They both defined chain complexes of modules such that
Cr(b) appeared as the top homology module.
First, we analyze the effects of gluing planar simplicial complexes. Suppose
1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate
pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the
Mayer-Vietoris sequence to obtain a natural relationship between the spline modules
Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2).
Next, given a simplicial complex , we study splines which also vanish on the
boundary of. The set of all such splines is denoted by Cr(b). In this case, we will
discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b).
Finally, we consider splines which are defined on a polygonally subdivided region
of the plane. By adding only edges to to form a simplicial subdivision , we will
be able to find bounds for the dimensions of the vector spaces Cr
k() for k 0. In
particular, these bounds will be given in terms of the dimensions of the vector spaces
Cr
k() and geometrical data of both and .
This dissertation concludes with some thoughts on future research questions and
an appendix describing the Macaulay2 package SplineCode, which allows the study
of the Hilbert polynomials of the spline modules.
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Rational fraction approximations for passive network functionsJohnson, William Joel Dietmar 01 June 2005 (has links)
In electrical engineering, the designer is often presented with the problem of synthesizing a circuit for which the mathematical specifications are unsuitable for physical realization. Hence, the engineer must approximate as well as possible the prescribed network function by another function which is realizable. This paper describes a new approximation method for solving the problem of realizing passive network transfer functions, where the realization is carried out through the use of passive, reciprocal,lumped, linear, and time-invariant elements.
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Approximation and interpolation employing divergence-free radial basis functions with applicationsLowitzsch, Svenja 30 September 2004 (has links)
Approximation and interpolation employing radial basis functions has
found important applications since the early 1980's in areas such
as signal processing, medical imaging, as well as neural networks.
Several applications demand that certain physical properties be
fulfilled, such as a function being divergence free. No such class
of radial basis functions that reflects these physical properties
was known until 1994, when Narcowich and Ward introduced a family of
matrix-valued radial basis functions that are divergence free. They
also obtained error bounds and stability estimates for interpolation
by means of these functions. These divergence-free functions are
very smooth, and have unbounded support. In this thesis we
introduce a new class of matrix-valued radial basis functions that are
divergence free as well as compactly supported. This leads to the
possibility of applying fast solvers for inverting interpolation
matrices, as these matrices are not only symmetric and positive
definite, but also sparse because of this compact support. We develop
error bounds and stability estimates which hold for a broad class of
functions. We conclude with applications to the numerical solution of
the Navier-Stokes equation for certain incompressible fluid flows.
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Polynomial approximations to functions of operators.Singh, Pravin. January 1994 (has links)
To solve the linear equation Ax = f, where f is an element of Hilbert space H and A
is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate
-1
the inverse operator A by an operator V which is a polynomial in A. Using the
spectral theory of bounded normal operators the problem is reduced to that of
approximating a function of the real variable by polynomials of best uniform
approximation. We apply two different techniques of evaluating
A-1 so that the
operator V is chosen either as a polynomial P (A) when P (A) approximates the
n n
function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn
(A)
approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A)
satisfy three point recurrence relations, thus the approximate solution vectors P (A)f
n
and Q (A)f can be evaluated iteratively. We compare the procedures involving
n
Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite.
We also show that the technique can be applied to an operator which is not selfadjoint,
but close, in the sense of operator norm, to a selfadjoint operator. The iterative
techniques we develop are used to solve linear systems arising from the discretization of
Freedholm integral equations of the second kind. Both smooth and weakly singular
kernels are considered. We show that earlier work done on the approximation of linear
functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse
operator and are thus special cases of a general result involving an approximation of
arbitrary degree to A -1 . / Thesis (Ph.D.)-University of Natal, 1994.
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Efficient algorithms for geometric pattern matchingCardoze, David Enrique Fabrega January 1999 (has links)
No description available.
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Algorithmic aspects of connectivity, allocation and design problemsChakrabarty, Deeparnab 23 May 2008 (has links)
Most combinatorial optimization problems are
NP -hard, which imply that under well- believed complexity assumptions, there exist no polynomial time
algorithms to solve them. To cope with the NP-hardness, approximation algorithms which return solutions close to
the optimal, have become a rich field of study. One successful method for designing approx-
imation algorithms has been to model the optimization problem as an integer program and
then using its polynomial time solvable linear programming relaxation for the design and
analysis of such algorithms. Such a technique is called the LP-based technique.
In this thesis, we study the algorithmic aspects of three classes of combinatorial optimization problems
using LP-based techniques as our main tool.
Connectivity Problems:
We study the Steiner tree problem and devise new linear pro-
gramming relaxations for the problem. We show an equivalence of our relaxation with the
well studied bidirected cut relaxation for the Steiner tree problem. Furthermore, for a class
of graphs called quasi-bipartite graphs, we improve the best known upper bound on the
integrality gap from 3/2 to 4/3. Algorithmically, we obtain fast and simple approximation
algorithms for the Steiner tree problem on quasi-bipartite graphs.
Allocation Problems:
We study the budgeted al location problem of allocating a set of
indivisible items to a set of agents who bid on it but possess a hard budget constraint more
than which they are unwilling to pay. This problem is a special case of submodular welfare
maximization. We use a natural LP relaxation for the problem and improve the best known
approximation factor for the problem from ~ 0.632 to 3/4. We also improve the inapprox-
imability factor of the problem to 15/16 and use our techniques to show inapproximability
results for many other allocation problems.
We also study online allocation problems where the set of items are unknown and appear one at a time.
Under some necessary assumptions we provide online algorithms for
many problems which attain the (almost) optimal competitive ratio. Both these works have
applications in the area of budgeted auctions, the most famous of which are the sponsored
search auctions hosted by search engines on the Internet.
Design Problems:
We formally define and study design problems which asks how the
weights of an input instance can be designed, so that the minimum (or maximum) of
a certain function of the input can be maximized (respectively, minimized). We show
if the function can be approximated to any factor $alpha$, then the optimum design can be
approximated to the same factor.
We also show that (max-min) design problems are dual to packing problems. We use
the framework developed by our study of design problems to obtain results about fraction-
ally packing Steiner trees in a "black-box" fashion. Finally, we study integral packing of
spanning trees and provide an alternate proof of a theorem of Nash-Williams and Tutte
about packing spanning trees.
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A methodology for ballistic missile defense systems analysis using nested neural networksWeaver, Brian Lee January 2008 (has links)
Thesis (M. S.)--Aerospace Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Mavris, Dimitri; Committee Member: Biltgen, Patrick; Committee Member: Ender, Tommer.
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Methods for increased computational efficiency of multibody simulationsEpple, Alexander. January 2008 (has links)
Thesis (Ph. D.)--Aerospace Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Olivier A. Bauchau; Committee Member: Andrew Makeev; Committee Member: Carlo L. Bottasso; Committee Member: Dewey H. Hodges; Committee Member: Massimo Ruzzene. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Computer experiments [electronic resource] : design, modeling and integration /Qian, Zhiguang. January 2006 (has links)
Thesis (Ph. D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2007. / Lu, Jye-Chyi, Committee Member ; Shaprio, Alexander, Committee Member ; Amemiya, Yasuo, Committee Co-Chair ; Wu, C. F. Jeff, Committee Chair ; Vengazhiyil, Roshan Joseph, Committee Member.
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