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Numerical conformal mapping of exterior domains
The work of the present paper is closely related to the two numerical procedures described in [11], for determining approximations to the function which maps conformally a bounded simply-connected domain Ω1 , with boundary âΩ, onto the unit disc. Here, we consider the use of these procedures for the solution of the corresponding exterior problem, i.e. the problem of determining approximations to the mapping function which maps conformally the exterior domain Ω = compl(ΩIââΩ) onto the unit disc
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Numerical solution of steady state diffusion problems containing singularities
Efficient Methods for Multidimensional Global Polynomial Approximation with Applications to Random PDEs
In this work, we consider several ways to overcome the challenges associated with polynomial approximation and integration of smooth functions depending on a large number of inputs. We are motivated by the problem of forward uncertainty quantification (UQ), whereby inputs to mathematical models are considered as random variables. With limited resources, finding more efficient and accurate ways to approximate the multidimensional solution to the UQ problem is of crucial importance, due to the âcurse of dimensionalityâ and the cost of solving the underlying deterministic problem.
The first way we overcome the complexity issue is by exploiting the structure of the approximation schemes used to solve the random partial differential equations (PDE), thereby significantly reducing the overall cost of the approximation. We do this first using multilevel approximations in the physical variables, and second by exploiting the hierarchy of nested sparse grids in the random parameter space. With these algorithmic advances, we provably decrease the complexity of collocation methods for solving random PDE problems.
The second major theme in this work is the choice of efficient points for multidimensional interpolation and interpolatory quadrature. A major consideration in interpolation in multiple dimensions is the balance between stability, i.e., the Lebesgue constant of the interpolant, and the granularity of the approximation, e.g., the ability to choose an arbitrary number of interpolation points or to adaptively refine the grid. For these reasons, the Leja points are a popular choice for approximation on both bounded and unbounded domains. Mirroring the best-known results for interpolation on compact domains, we show that Leja points, defined for weighted interpolation on R, have a Lebesgue constant which grows subexponentially in the number of interpolation nodes. Regarding multidimensional quadratures, we show how certain new rules, generated from conformal mappings of classical interpolatory rules, can be used to increase the efficiency in approximating multidimensional integrals. Specifically, we show that the convergence rate for the novel mapped sparse grid interpolatory quadratures is improved by a factor that is exponential in the dimension of the underlying integral
Boundary algebras and Kac modules for logarithmic minimal models
Virasoro Kac modules were initially introduced indirectly as representations
whose characters arise in the continuum scaling limits of certain transfer
matrices in logarithmic minimal models, described using Temperley-Lieb
algebras. The lattice transfer operators include seams on the boundary that use
Wenzl-Jones projectors. If the projectors are singular, the original
prescription is to select a subspace of the Temperley-Lieb modules on which the
action of the transfer operators is non-singular. However, this prescription
does not, in general, yield representations of the Temperley-Lieb algebras and
the Virasoro Kac modules have remained largely unidentified. Here, we introduce
the appropriate algebraic framework for the lattice analysis as a quotient of
the one-boundary Temperley-Lieb algebra. The corresponding standard modules are
introduced and examined using invariant bilinear forms and their Gram
determinants. The structures of the Virasoro Kac modules are inferred from
these results and are found to be given by finitely generated submodules of
Feigin-Fuchs modules. Additional evidence for this identification is obtained
by comparing the formalism of lattice fusion with the fusion rules of the
Virasoro Kac modules. These are obtained, at the character level, in complete
generality by applying a Verlinde-like formula and, at the module level, in
many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 71 pages. v3: version published in Nucl. Phys.
The string of variable density: perturbative and non-perturbative results
We obtain systematic approximations for the modes of vibration of a string of
variable density, which is held fixed at its ends. These approximations are
obtained iteratively applying three theorems which are proved in the paper and
which hold regardless of the inhomogeneity of the string. Working on specific
examples we obtain very accurate approximations which are compared both with
the results of WKB method and with the numerical results obtained with a
collocation approach. Finally, we show that the asymptotic behaviour of the
energies of the string obtained with perturbation theory, worked to second
order in the inhomogeinities, agrees with that obtained with the WKB method and
implies a different functional dependence on the density that in two and higher
dimensions.Comment: 28 pages, 3 tables, 6 figure
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