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14
Conformal mapping in linear time
, 2006
"... Abstract. Given any ɛ> 0 and any planar region Ω bounded by a simple ngon P we construct a (1 + ɛ)quasiconformal map between Ω and the unit disk in time C(ɛ)n. One can take C(ɛ) = C + C log 1 ɛ log log ..."
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Abstract. Given any ɛ> 0 and any planar region Ω bounded by a simple ngon P we construct a (1 + ɛ)quasiconformal map between Ω and the unit disk in time C(ɛ)n. One can take C(ɛ) = C + C log 1 ɛ log log
ON MODULI OF RINGS AND QUADRILATERALS: ALGORITHMS AND EXPERIMENTS
, 2009
"... Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of ..."
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Cited by 7 (3 self)
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Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new hpFEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the SchwarzChristoffel Toolbox of Driscoll and Trefethen.
History and Recent Developments in Techniques for Numerical Conformal Mapping
 PROCEEDINGS OF THE INTERNATIONAL WORKSHOP ON QUASICONFORMAL MAPPINGS AND THEIR APPLICATIONS (IWQCMA05)
, 2005
"... A brief outline is given of some of the main historical developments in the theory and practice of conformal mappings. Originating with the science of cartography, conformal mappings has given rise to many highly sophisticated methods. We emphasize the principles of mathematical discovery involved i ..."
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Cited by 6 (0 self)
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A brief outline is given of some of the main historical developments in the theory and practice of conformal mappings. Originating with the science of cartography, conformal mappings has given rise to many highly sophisticated methods. We emphasize the principles of mathematical discovery involved in
OPTIMAL ANGLE BOUNDS FOR QUADRILATERAL MESHES
"... Abstract. We show that any simple planar ngon can be meshed in linear time by O(n) quadrilaterals with all new angles bounded between 60 and 120 degrees. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: Key words and phrases. Quadrilateral meshes, Riemann mapping, thick/thin decom ..."
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Cited by 5 (3 self)
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Abstract. We show that any simple planar ngon can be meshed in linear time by O(n) quadrilaterals with all new angles bounded between 60 and 120 degrees. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: Key words and phrases. Quadrilateral meshes, Riemann mapping, thick/thin decomposition, linear
A fast QCmapping theorem for polygons
, 2007
"... Abstract. Given a simple ngon P in the plane, normalized to contain the unit disk, we define a map from P to the unit circle so that (1) the map extends to be 8quasiconformal on the interior, (2) it contracts arclength on the boundary and (3) the images of all n vertices can be computed in time O( ..."
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Cited by 4 (4 self)
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Abstract. Given a simple ngon P in the plane, normalized to contain the unit disk, we define a map from P to the unit circle so that (1) the map extends to be 8quasiconformal on the interior, (2) it contracts arclength on the boundary and (3) the images of all n vertices can be computed in time O(n). Thus we obtain a fast KQCapproximation to the Riemann map with constant K independent of the domain.
A fast mapping theorem for polygons
, 2008
"... Given a simple ngon P in the plane, normalized to contain the unit disk, we define a map from P to the unit circle so that (1) the map extends to be 8quasiconformal on the interior, (2) it contracts arclength on the boundary and (3) the images of all n vertices can be computed in time O(n). Thus ..."
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Cited by 2 (2 self)
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Given a simple ngon P in the plane, normalized to contain the unit disk, we define a map from P to the unit circle so that (1) the map extends to be 8quasiconformal on the interior, (2) it contracts arclength on the boundary and (3) the images of all n vertices can be computed in time O(n). Thus we obtain a fast QCapproximation to the Riemann map with constant independent of the domain.
COMPUTATION OF EXTERIOR MODULI OF QUADRILATERALS
"... Abstract. We study the problem of computing the exterior modulus of a bounded quadrilateral. We reduce this problem to the numerical solution of the DirichlétNeumann problem for the Laplace equation. An algorithm and its implementation with hpFEM is documented. Several experimental results, with ..."
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Abstract. We study the problem of computing the exterior modulus of a bounded quadrilateral. We reduce this problem to the numerical solution of the DirichlétNeumann problem for the Laplace equation. An algorithm and its implementation with hpFEM is documented. Several experimental results, with error estimates, are reported. 1.
ON THE USE OF CONFORMAL MAPS FOR THE ACCELERATION OF CONVERGENCE OF THE TRAPEZOIDAL RULE AND SINC NUMERICAL METHODS
"... Abstract. We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the sinhmap, and allows the treatment of a finite number of singularities in the complex plane. In the case where ..."
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Abstract. We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the sinhmap, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the socalled SincPadé approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals. Key words. Trapezoidal rule. Sinc numerical methods. Conformal maps. AMS subject classifications. 30C30, 41A30, 65D30, 65L10. 1. Introduction. The
NEW TECHNIQUES IN NUMERICAL INTEGRATION: THE COMPUTATION OF MOLECULAR INTEGRALS OVER EXPONENTIALTYPE FUNCTIONS
"... The numerical evaluation of challenging integrals is a topic of interest in applied mathematics. We investigate molecular integrals in the B function basis, an exponentially decaying basis with a compact analytical Fourier transform. The Fourier property allows analytical expressions for molecular ..."
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The numerical evaluation of challenging integrals is a topic of interest in applied mathematics. We investigate molecular integrals in the B function basis, an exponentially decaying basis with a compact analytical Fourier transform. The Fourier property allows analytical expressions for molecular integrals to be formulated in terms of semiinfinite highly oscillatory integrals with limited exponential decay. The semiinfinite integral representations in terms of nonphysical variables stand as the bottleneck in the calculation. To begin our numerical experiments, we conduct a comparative study of the most popular numerical steepest descent methods, extrapolation methods and sequence transformations for computing semiinfinite integrals. It concludes that having asymptotic series representations for integrals and applying sequence transformations leads to the most efficient algorithms. For threecenter nuclear attraction integrals, we find an analytical expression for the semiinfinite integrals. Numerical experiments show the resulting algorithm is approximately 102.5 times more efficient than the stateoftheart. For the fourcenter twoelectron Coulomb integrals, we take a different approach. The integrand has singularities in the complex plane that can be near the path of integration, making standard quadrature routines unreliable. The trapezoidal rule with double exponential variable transformations has been shown to have very promising properties as a generalpurpose integrator. We investigate the use of conformal maps to maximize the convergence rate, resulting in a nonlinear program for the optimized variable transformation. ii Preface