an interpolating polynomial method for numerical conformal mapping a simple algorithm is presented for numerical approximation of conformal mappings for simply connected planar regions the desired mapping is represented by a normalized polynomial of degree n determined by the boundary values which it is presumed to take at the nth roots of unity then its values at the n intermediate 2nth roots of unity are used to correct the initial guess each iteration which consists essentially of matrix multiplication and boundary projection costs o n log n arithmetic operations numerical results are provided to confirm linear convergence of the algorithm

maximal lattice free polyhedra finiteness and an explicit description in dimension three a convex set with nonempty interior is maximal lattice free if it is inclusion maximal with respect to the property of not containing integer points in its interior maximal lattice free convex sets are known to be polyhedra the precision of a rational polyhedron p in rd is the smallest natural number s such that sp is an integral polyhedron in this paper we show that up to affine mappings preserving zd the number of maximal lattice free rational polyhedra of a given precision s is finite furthermore we present the complete list of all maximal lattice free integral polyhedra in dimension three our results are motivated by recent research on cutting plane theory in mixed integer linear optimization

a fast conformal mapping algorithm with no fft abstract an algorithm is presented for the computation of a conformal mapping discretized on a non uniformly spaced point set useful for the numerical solution of many problems of fluid dynamics most existing iterative techniques both those having a linear and those having a quadratic type of convergence rely on the fast fourier transform fft algorithm for calculating a convolution integral which represents the most time consuming phase of the computation the fft however definitely cannot be applied to a non uniform spacing the algorithm presented in this paper has been made possible by the construction of a calculation method for convolution integrals which despite not using an fft maintains a computation time of the same order as that of the fft the new technique is successfully applied to the problem of conformally mapping a closely spaced cascade of airfoils onto a circle which requires an exceedingly large number of points if it is solved with uniform spacing

the riemann hilbert problem and the generalized neumann kernel on multiply connected regions this paper presents and studies fredholm integral equations associated with the linear riemann hilbert problems on multiply connected regions with smooth boundary curves the kernel of these integral equations is the generalized neumann kernel the approach is similar to that for simply connected regions see r wegmann a h m murid m m s nasser the riemann hilbert problem and the generalized neumann kernel j comput appl math 182 2005 388 415 there are however several characteristic differences which are mainly due to the fact that the complement of a multiply connected region has a quite different topological structure this implies that there is no longer perfect duality between the interior and exterior problems we investigate the existence and uniqueness of solutions of the integral equations in particular we determine the exact number of linearly independent solutions of the integral equations and their adjoints the latter determine the conditions for solvability an analytic example on a circular annulus and several numerically calculated examples illustrate the results

numerical conformal mapping based on the generalised conjugation operator an iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity the formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function the conventional approach is to use only one and ignore the other equation however the discrete version of the operator using the boundary element method bem leads to two non equivalent sets of linear equations forming an over determined system the generalised conjugation operator is introduced so that both sets of equations can be utilised and their least square solution determined without any additional computational cost a strategy largely responsible for the stability and efficiency of the proposed method numerical tests on various samples including problems with cracked domains suggest global convergence although this cannot be proved theoretically the computational efficiency appears significantly higher than that reported earlier by other investigators

a stable richardson iteration method for complex linear systems the richardson iteration method is conceptually simple as well as easy to program and parallelize this makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues we change the ordering of the relaxation parameters of a richardson iteration method proposed by eiermann niethammer and varga for the solution of such problems the new method obtained is shown to be stable and to have better convergence properties

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

numerical conformal mapping methods based on faber series methods are presented for approximating the conformal map from the interior of various regions to the interior of simply connected target regions with a smooth boundary the methods for the disk due to fornberg 1980 and the ellipse due to delillo and elcrat 1993 are reformulated so that they may be extended to other new computational regions the case of a cross shaped region is introduced and developed these methods are used to circumvent the severe ill conditioning due to the crowding phenomenon suffered by conformal maps from the unit disk to target regions with elongated sections while preserving the fast fourier methods available on the disk the methods are based on expanding the mapping function in the faber series for the regions all of these methods proceed by approximating the boundary correspondence of the map with a newton like iteration at each newton step a system of linear equations is solved using the conjugate gradient method the matrix vector multiplication in this inner iteration can be implemented with fast fourier transforms at a cost of o n log n it is shown that the linear systems are discretizations of the identity plus a compact operator and so the conjugate gradient method converges superlinearly several computational examples are given along with a discussion of the accuracy of the methods

the numerical solution of the biharmonic equation by conformal mapping the solution to the biharmonic equation in a simply connected region omega in the plane is computed in terms of the goursat functions the boundary conditions are conformally transplanted to the disk with a numerical conformal map a linear system is obtained for the taylor coefficients of the goursat functions the coefficient matrix of the linear system can be put in the form i k where k is the discretization of a compact operator k can be thought of as the composition of a block hankel matrix with a diagonal matrix the compactness leads to clustering of eigenvalues and the hankel structure yields a matrix vector multiplication cost of o n log n thus if the conjugate gradient method is applied to the system then superlinear convergence will be obtained numerical results are given to illustrate the spectrum clustering and superlinear convergence

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

necessary and sufficient conditions for bayes risk consistency of a recursive kernel classification rule corresp it is shown that for a nonparametric recursive kernel classification rule sum n i 1 h d i i h i epsilon sum n j 1 h d j rightarrow 0 rm as n rightarrow infty all epsilon 0 and sum infty i 1 h d i infty constitute a set of conditions which are not only sufficient but also necessary for weak and strong bayes risk consistency of the rule in this way weak and strong consistencies are shown to be equivalent

the accuracy of numerical conformal mapping methods a survey of examples and results this paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions especially fourier series methods both explicit examples of popular test cases and more general estimates are discussed the severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated remarks on various numerical methods are included

the conjugate function method and conformal mappings in multiply connected domains the conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains in this paper the conjugate function method is generalized for

conformal grid generation for high aspect ratio simply and doubly connected regions conformal co ordinate transformations are used to map rectangular computational domains onto arbitrary simply and doubly connected regions with smooth boundaries the efficient numerical schemes of wegmann involving the solution of the inverse boundary correspondence function problems associated with the mapping of the unit disc or circular annulus onto simply or doubly connected domains respectively are employed the numerical implementation of these schemes is emphasized examples are generated for regions with elliptic inner and outer boundaries additional examples are used to demonstrate the accuracy and convergence of the schemes and their practical limitations the techniques are found to converge well if holomorphic functions are used to describe the boundaries the use of preconditioning maps is also discussed

pattern recognition with ordered labels abstract we consider a multi class pattern recognition problem with linearly ordered labels and a loss function which is a sum of squared deviations of decisions from true classes it is proved that the optimal in the bayesian sense decision rule is the a posteriori mean which is then rounded to the nearest integer then the plug in type estimator of the optimal decision rule is proposed we also investigate its asymptotic properties finally we discuss briefly the structure of a neural net which approximates the optimal decision rule

computing the hilbert transform and its inverse we construct a new method for approximating hilbert transforms and their inverse throughout the complex plane both problems can be formulated as riemann hilbert problems via plemelj s lemma using this framework we re derive existing approaches for computing hilbert transforms over the real line and unit interval with the added benefit that we can compute the hilbert transform in the complex plane we then demonstrate the power of this approach by generalizing to the half line combining two half lines we can compute the hilbert transform of a more general class of functions on the real line than is possible with existing methods

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

the newton method for solving the theodorsen integral equation the newton method for the solution of the theodorsen integral equation in conformal mapping is studied one step of this method consists of solving a linear integral equation the solution of which is given explicitly as the result of a riemann hilbert problem quadratic convergence of the newton method is established under certain assumptions whereas in other methods a so called condition with 1 is required to hold our method converges also for 1 we will also present a numerical implementation in which the result of one step of the newton method is approximately by a vector in r2n which can be computed with 2n log n o n multiplications in comparison one step of the newton method for the discrete theodorsen equation requires o n3 multiplications

extremal distance harmonic measure and numerical conformal mapping abstract delillo t k and j a pfaltzgraff extremal distance harmonic measure and numerical conformal mapping journal of computational and applied mathematics 46 1993 103 113 estimates of extremal distance and harmonic measure are used to show how the geometric properties of a simply connected domain influence the boundary distortion of a conformal map from the unit disk to the domain numerical examples and remarks on the conditioning of numerical conformal mapping methods are included a sharp estimate is given of the exponential ill conditioning known as the crowding phenomenon which occurs for slender regions

conformal mapping by the method of alternating projections the functional analytic principle of alternating projections is used to construct an iterative method for numerical conformal mapping of the unit disc onto regions with smooth boundaries the result is a simple method which requires in each iterative step only two complex fourier transforms local convergence can be proved using a theorem of ostrowski convergence is linear the asymptotic convergence factor is equal to the spectral radius of a certain operator a version with overrelaxation as well as a discretized version are discussed along the same lines for regions which are close to the unit disc convergence is fast for some familiar regions the convergence factors can be calculated explicitly finally the method is compared with theodorsen s

iterative methods for discrete nonlinear riemann hilbert problems abstract let d denote the complex unit disk t d and let f t r 2 r be a given function the paper presents iterative methods for the computation of functions w u iv which are holomorphic in d and satisfy the boundary relation f t u t v t 0 for all t t it is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle t the proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution error estimates for the iterative solutions are given in the scale of the sobolev spaces wk2

the numerical solution of theodorsen integral equation abstract in this paper a numerical solution of the theodorsen integral equation is studied using an adequate quadrature formula which eliminates the singularity of the integral part of the theodorsen equation we obtain a system of nonlinear algebraical equations this system may be solved using a jacobi type method and the procedure can be easily implemented using a programming language with parallel facilities examples are given using ada eval and parallaxis a convergence result is established 1999 elsevier science ltd all rights reserved

a charge simulation method for the numerical conformal mapping of interior exterior and doubly connected domains abstract a simple numerical method is described for computing the following three conformal maps a from a domain interior to a closed jordan curve onto the interior of the unit disk b from a domain exterior to a closed jordan curve onto the exterior of the unit disk and c from a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials without integration typical examples show that numerical results of high accuracy can be obtained by the simple method if the domains have no concave corners

a spectral galerkin method for a boundary integral equation on considere une equation integrale liee au probleme bidimensionnel de dirichlet resolu par la methode du potentiel de simple couche on analyse une methode de galerkin spectrale adaptee au cas des donnees aux limites et d un domaine reguliers on accelere la convergence dans les espaces de sobolev en utilisant des polynomes trigonometriques

flow around simply and multiply connected bodies a new iterative scheme for conformal mapping discussion des techniques de representation conforme numerique des profils aerodynamiques des cascades soit pour le calcul des champs d ecoulement irrotationnel soit pour la construction de grilles de calcul numerique presentation d un algorithme iteratif a convergence quadratique

numerical conformal mappings of bounded multiply connected domains by the charge simulation method we present here a simple method of numerical conformal mappings from bounded multiply connected domains onto the bounded canonical slit domains of nehari i e a disk with concentric circular slits and an annulus with concentric circular slits in this method we express the mapping functions in terms of a pair of conjugate harmonic functions and approximate them using the charge simulation method by a linear combination of complex logarithmic functions some numerical examples show the effectiveness of the method

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

on the computation of the conjugate trigonometric rational function and on a related splitting problem we prove several splitting theorems related to the following one every continuous real trigonometric rational function of t with period 2 pi can be written as the real part of a rational function of z e it that is analytic for z leqq 1 as an application we obtain immediately a method for computing the conjugate function of a real trigonometric rational function

numerical conformal mapping methods for simply and doubly connected regions methods are presented and analyzed for approximating the conformal map from the interior exterior of the disk to the interior exterior of a smooth simple closed curve and from an annulus to a bounded doubly connected region with smooth boundaries the methods are newton like methods for computing the boundary correspondences and conformal moduli similar to fornberg s method for the interior of the disk we show that the linear systems are discretizations of the identity plus a compact operator and hence that the conjugate gradient method converges superlinearly

numerical conformal mapping of multiply connected regions onto the fifth category of koebe s canonical slit regions abstract this paper presents a boundary integral method for approximating the conformal mapping from bounded multiply connected regions onto the fifth category of koebe s classical canonical slit regions the method is based on a uniquely solvable boundary integral equation with generalized neumann kernel the results of some test calculations illustrate the performance of the presented method

classification by pairwise coupling we discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes and then coupling the estimates together the coupling model is similar to the bradley terry method for paired comparisons we study the nature of the class probability estimates that arise and examine the performance of the procedure in real and simulated data sets classifiers used include linear discriminants nearest neighbors adaptive nonlinear methods and the support vector machine

notions of maximality for integral lattice free polyhedra the case of dimension three lattice free sets and their applications for cutting plane methods in mixed integer optimization have been studied in recent literature the family of all integral lattice free polyhedra that are not properly contained in another integral lattice free polyhedron has been of particular interest we call these polyhedra d maximal for fixed d the family of d maximal integral lattice free polyhedra is finite up to unimodular equivalence in view of possible applications in cutting plane theory one would like to have a classification of this family this is a challenging task already for small dimensions in contrast the subfamily of all integral lattice free polyhedra that are not properly contained in any other lattice free set which we call d maximal lattice free polyhedra allow a rather simple geometric characterization hence the question was raised for which dimensions the notions of d maximality and d maximality are equivalent this was known to be the case for dimensions one and two on the

equivalence of nystrom s method and fourier methods for the numerical solution of fredholm integral equations on demontre un resultat d equivalence de la methode de nystrom et de la methode de fourier avec des fonctions orthogonales pour la solution numerique des equations integrales de fredholm on en deduit des estimations d erreur pour la methode de nystrom on decrit exactement la regle de quadrature pour l integration numerique

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

numerical conformal mapping using cross ratios and delaunay triangulation we propose a new algorithm for computing the riemann mapping of the unit disk to a polygon also known as the schwarz christoffel transformation the new algorithm crdt for cross ratios of the delaunay triangulation is based on cross ratios of the prevertices and also on cross ratios of quadrilaterals in a delaunay triangulation of the polygon the crdt algorithm produces an accurate representation of the riemann mapping even in the presence of arbitrary long thin regions in the polygon unlike any previous conformal mapping algorithm we believe that crdt solves all difficulties with crowding and global convergence although these facts depend on conjectures that we have so far not been able to prove we demonstrate convergence with computational experiments the riemann mapping has applications in two dimensional potential theory and mesh generation we demonstrate crdt on problems in long thin regions in which no other known algorithm can perform comparably

discretized versions of newton type iterative methods for conformal mapping abstract it is shown that for nearly circular regions discretized versions of the newton type methods of wegmann 1978 and hubner 1986 converge locally to fixed points convergence is linear the rates can be determined approximately for several standard regions the dominant operator acts only on a subspace of high order harmonics therefore under conditions of smoothness and or symmetry convergence can be much faster the fixed points satisfy a discrete version of lavrentev s variational principle therefore the resulting approximations for conformal mapping are essentially as accurate as wittich s approximation for the conjugation operation

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

median radial basis function neural network radial basis functions rbfs consist of a two layer neural network where each hidden unit implements a kernel function each kernel is associated with an activation region from the input space and its output is fed to an output unit in order to find the parameters of a neural network which embeds this structure we take into consideration two different statistical approaches the first approach uses classical estimation in the learning stage and it is based on the learning vector quantization algorithm and its second order statistics extension after the presentation of this approach we introduce the median radial basis function mrbf algorithm based on robust estimation of the hidden unit parameters the proposed algorithm employs the marginal median for kernel location estimation and the median of the absolute deviations for the scale parameter estimation a histogram based fast implementation is provided for the mrbf algorithm the theoretical performance of the two training algorithms is comparatively evaluated when estimating the network weights the network is applied in pattern classification problems and in optical flow segmentation

lattice free sets multi branch split disjunctions and mixed integer programming in this paper we study the relationship between valid inequalities for mixed integer sets lattice free sets associated with these inequalities and the multi branch split cuts introduced by li and richard discret optim 5 724 734 2008 by analyzing n dimensional lattice free sets we prove that for every integer n there exists a positive integer t such that every facet defining inequality of the convex hull of a mixed integer polyhedral set with n integer variables is a t branch split cut we use this result to give a finite cutting plane algorithm to solve mixed integer programs we also show that the minimum value t for which all facets of polyhedral mixed integer sets with n integer variables can be generated as t branch split cuts grows exponentially with n in particular when n 3 we observe that not all facet defining inequalities are 6 branch split cuts

on fornberg s numerical method for conformal mapping it is shown that fornberg s method 1 is a discretized version of an iterative process which is essentially the same we introduced and studied in 7 8 the main difference consists in the determination of the shift u k in each iterative step while we give an explicit representation of u k fornberg determines u k from a linear equation which involves a certain integral operator r we prove some properties and estimates for this operator which explain several experimental results reported in 1 especially the fact that the conjugate gradient method is so efficient the results of 8 concerning the convergence can be applied also for fornberg s method finally we give estimates for the numerical error

the treatment of corner and pole type singularities in numerical conformal mapping techniques abstract this paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques the paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration

fast conformal mapping of multiply connected regions abstract an iterative method is presented which constructs for an unbounded region g with m holes and sufficiently smooth boundary a circular region h and a conformal mapping from h to g with the usual normalization both h and are uniquely determined by g with a few modifications the method can also be applied to a bounded region g with m holes the canonical region h is then the unit disc with m circular holes the proposed method also determines the centers and radii of the boundary circles of h and requires at each iterative step the solution of a riemann hilbert rh problem which has a unique solution numerically the rh problem can be treated efficiently by the method of successive conjugation using the fast fourier transform fft the iteration for the solution of the rh problem converges linearly the conformal mapping method converges quadratically the results of some test calculations exemplify the performance of the method

on some relations among numerical conformal mapping methods abstract gaier and gutknecht have shown that many numerical methods for producing the conformal map from the unit disk to a simply connected region share a common theoretical basis as solutions of nonlinear integral equations arising from the hilbert transform of a function of the boundary correspondence we give a brief presentation of this classification and extend it somewhat to include some equations for the inverse correspondence such as those of menikoff and zemach noble and schwarz christoffel the use of explicit maps and the method of bisshopp are also brought into this framework an example illustrating the use of explicit maps is given

reducing multiclass to binary a unifying approach for margin classifiers we present a unifying framework for studying the solution of multiclass categorization problems by reducing them to multiple binary problems that are then solved using a margin based binary learning algorithm the proposed framework unifies some of the most popular approaches in which each class is compared against all others or in which all pairs of classes are compared to each other or in which output codes with error correcting properties are used we propose a general method for combining the classifiers generated on the binary problems and we prove a general empirical multiclass loss bound given the empirical loss of the individual binary learning algorithms the scheme and the corresponding bounds apply to many popular classification learning algorithms including support vector machines adaboost regression logistic regression and decision tree algorithms we also give a multiclass generalization error analysis for general output codes with adaboost as the binary learner experimental results with svm and adaboost show that our scheme provides a viable alternative to the most commonly used multiclass algorithms

a simplified fornberg like method for the conformal mapping of multiply connected regions comparisons and crowding we present a new fornberg like method for the numerical conformal mapping of multiply connected regions exterior to circles to multiply connected regions exterior to smooth curves the method is based on new symmetric conditions for analytic extension of functions given on circular boundaries we also briefly discuss a similar method due to wegmann and compare some computations with both methods some examples of regions which exhibit crowding of the circles are also presented

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

an iterative method for solving nonlinear riemann hilbert problems abstract a geometrically motivated linearization is used to construct an iterative method for solving nonlinear boundary value problems for holomorphic functions the nondiscrete method is quadratically convergent results of some test calculations are reported

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

a fornberg like conformal mapping method for slender regions abstract a method is presented for approximating the conformal map from the interior of an ellipse to the interior of a simply connected target region the map is represented as a truncated chebyshev series conditions that the mapping function be conformal are transplanted from the ellipse to an annulus with the joukowski map the resulting conditions on the laurent coefficients then give a system of equations for the newton update of the approximate boundary correspondence this system is a generalization of fornberg s system for maps from the disk and is solved similarly in o n log n operations by the conjugate gradient method our numerical experiments demonstrate that the maps from the ellipse to a slender target region of similar aspect ratio can be constructed with far fewer mesh points than are required for maps from the disk thus circumventing the ill conditioning due to crowding in these cases

approximate solution of periodic riemann boundary value problem for analytic functions a direct method for the approximate solution of periodic riemann boundary value problems for analytic functions is given through the approximation by complex splines by the cardinal splines of the first degree we get the approximation of the canonical function based on the approximate result of singular integrals with hilbert kernel furthermore we obtain the approximate solution which may be sufficiently close to the exact solution to any degree when the partition is sufficiently fine

nonparametric estimation via empirical risk minimization a general notion of universal consistency of nonparametric estimators is introduced that applies to regression estimation conditional median estimation curve fitting pattern recognition and learning concepts general methods for proving consistency of estimators based on minimizing the empirical error are shown in particular distribution free almost sure consistency of neural network estimates and generalized linear estimators is established

an iterative method for the conformal mapping of doubly connected regions an iterative method is described for the determination of the conformal mapping of a circular annulus rq z q z 1 onto a doubly connected region g with smooth boundary curves the method is analogous to the method of 7 for simply connected regions it requires in each step the solution of a riemann hilbert problem on rq this problem can be solved explicitly in terms of a generalized conjugation operator k a degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step if the boundary curves of g admit parametrizations by functions with second derivatives which are holder continuous with exponent 0 1 then the method converges locally in the sobolev space w of 2 periodic absolutely continuous functions with square integrable derivatives the order of convergence is at least 1 the convergence is quadratic if the have lipschitz continuous second derivatives for the numerical implementation of the method the operator k can be approximated by a wittich type method which can be performed very effectively using fft a calculation with n 2m grid points on the computer requires storage of the order o n and computing time o mn the results of some test calculations are reported and compared with results of calculations with the method of theodorsen garrick

recursive numerical algorithm for conformal mapping in two dimensional hydrodynamics an original recursive numerical method rt algorithm for conformal mapping is presented based on the lavrentiev variational principle it is a powerful means for the solution of various two dimensional boundary value problems the rt algorithm can be used to solve internal and external hydrodynamics problems occurring in spacecraft research namely the calculation of fluid motions with local vorticity regions for external problems conformal mapping of the fluid flow area onto the outside of the unit circle is constructed for internal problems the area is mapped onto one of standard domains the unit circle or circular annulus fluid flows in a cylindrical tank with inner radial damping ribs and nearby an air brake with sharp edges with or without free eddies are discussed as examples

fast potential theory ii layer potentials and discrete sums we present three new families of fast algorithms for classical potential theory based on ewald summation and fast transforms of gaussians and fourier series ewald summation separates the green function for a cube into a high frequency localized part and a rapidly converging fourier series each part can then be evaluated efficiently with appropriate fast transform algorithms our algorithms are naturally suited to the use of green functions with boundary conditions imposed on the boundary of a cube rather than free space green functions our first algorithm evaluates classical layer potentials on the boundary of a d dimensional domain with d equal to two or three the quadrature error is o hm e where h is the mesh size on the boundary and m is the order of quadrature used the algorithm evaluates the discretized potential using n elements at o n points in o n log n arithmetic operations the constant in o n log n depends logarithmically on the desired error tolerance our second scheme evaluates a layer potential on the domain itself with the same accuracy it produces md values using n boundary elements in o n md log m arithmetic operations our third method evaluates a discrete sum of values of the green function of the type which occur in particle methods it attains error at a cost o na log n where a 2 1 dd and d is the hausdorff dimension of the set where the sources concentrate in the limit n thus it is o n log n when the sources do not cluster too much and close to o n log n in the important practical case when the points are uniformly distributed over a hypersurface we also sketch an o n log n algorithm based on special functions two dimensional numerical results are presented for all three algorithms layer potentials are evaluated to second order accuracy in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation discrete sum calculations are speeded up astronomically our algorithm reduces the cpu time required for a calculation with 40 000 points from six months to one hour

constructive solution of a certain class of riemann hilbert problems on multiply connected circular regions abstract a class of riemann hilbert rh problems on multiply connected circular regions is treated iteratively by a method of successive conjugation sc which solves the problem on a single boundary circle each time running through all circles in each iterative step it is shown that the method converges convergence is linear with a rate which can be calculated with an eigenvalue problem the rh problems considered have negative index in order to fulfill the solvability conditions free parameters are introduced in form of polynomials the proof of convergence of the sc method shows at the same time that the rh problems with this specification of the free parameters have unique solutions rh problems of this kind are of practical use in conformal mapping problems of multiply connected regions to circular regions they play a role in the mapping of nearly circular regions lavrentev s principle there is also a close relationship to koebe s iterative method for the calculation of conformal mappings the properties of rh problems are different for unbounded and bounded regions the sc iteration must be adapted accordingly the performance of the sc iteration is illustrated with several examples

rapid methods for the conformal mapping of multiply connected regions abstract we present fast methods for the conformal mapping of simply doubly and multiply connected regions onto certain canonical regions in the plane our mapping procedure consists of two parts first we solve an integral equation on the boundary of the region we wish to map the solution of this integral equation is needed to determine the boundary correspondence we have chosen to use the integral equation formulation of mikhlin although it is not widely used this formulation has the advantage that it leads to integral equations of the second kind with unique solutions and bounded kernels the solutions are also periodic allowing for effective use of the trapezoid rule once we have solved the integral equation we use a rapid method we have previously developed to determine the mapping function in the interior of the region this method makes use of fast poisson solvers and thereby circumvents the difficulties associated with computing integrals at points near the boundary of the region and avoids the expense of computing many integrals we also provide results of numerical experiments

orthogonal grid generation in a 2d domain via the boundary integral technique a new numerical scheme is proposed for the generation of an orthogonal coordinate grid in an arbitrary simply connected two dimensional domain the scheme is robust and non iterative and is based on the conjunction of the familiar boundary integral technique with the covariant laplace equation method for mapping in the proposed scheme two types of problems are considered 1 boundary correspondence is specified on two adjacent sides of the boundary or 2 the distortion factor is specified in the product form f x h x q h

a conformal map formula for difficult cases abstract a simple approximate formula is derived for the boundary map function of a conformal map which provides a remarkably good fit for mappings to regions with highly distorted boundaries the approximation follows from the reduction of a nonlocal integral equation for the map function to a local equation based on the nature of the crowding associated with distorted regions numerical examples are given

convergence proofs and error estimates for an iterative method for conformal mapping the iterative method as introduced in 8 and 9 for the determination of the conformal mapping of the unit disc onto a domaing is here described explicitly in terms of the operatork which assigns to a periodic functionu its periodic conjugate functionk u it is shown that whenever the boundary curve ofg is parametrized by a function with lipschitz continuous derivative dot eta ne 0 then the method converges locally in the sobolev spacew of 2 periodic absolutely continuous functions with square integrable derivative if is in a holder classc 2 the order of convergence is at least 1 if is inc l 1 withl 1 0 1 then the iteration converges inc l for analytic boundary curves the convergence takes place in a space of analytic functions for the numerical implementation of the method the operatork can be approximated by wittich s method which can be applied very effectively using fast fourier transform the sobolev norm of the numerical error can be estimated in terms of the numbern of grid points it iso n 1 l if is inc l 1 ando exp n 2 if is an analytic curve the number in the latter formula is bounded by logr wherer is the radius of the largest circle into which can be extended analytically such that z 0 for z r the results of some test calculations are reported

the riemann hilbert problem and the generalized neumann kernel this paper presents two new fredholm integral equations associated to the interior and the exterior riemann hilbert problems in simply connected regions with smooth boundaries the kernel of these integral equations is the generalized neumann kernel the solvability of the integral equations depends on whether l 1 are eigenvalues of the kernel which in turn depends on the index of the riemann hilbert problem the complete discussion of the solvability of the integral equations with the generalized neumann kernel is presented the integral equations can be used effectively to solve numerically the riemann hilbert problems the case of non uniquely solvable riemann hilbert problems is treated by imposing additional constraints to get a uniquely solvable problem fredholm integral equations with generalized neumann kernels are also derived for the problem of the interior and the exterior harmonic conjugate functions as applications we study the problem of conformal mapping to a nearby region and extend wegmann s iterative method to general regions numerical examples reveal that the present method offers an effective solution technique for the riemann hilbert problems when the boundaries are sufficiently smooth

numerical conformal mapping via the szego kernel abstract a new method to compute the riemann mapping function is numerically implemented and tested on examples the method expresses the szego kernel as the solution of a second kind integral equation the equation has its origin in an earlier non numerical work of stein and one of the authors 8 the experimental results show the algorithm to be effective and stable

a domain decomposition method for approximating the conformal modules of long quadrilaterals this paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals the method has been studied already by us and also by d gaier and w k hayman but only in connection with a special class of quadrilaterals viz quadrilaterals where a the defining domain is bounded by two parallel straight lines and two jordan arcs and b the four specified boundary points are the four corners where the arcs meet the straight lines our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above

on the fast solution of a linear system arising in numerical conformal mapping abstract one step of the newton method for the discretized theodorsen equation in conformal mapping requires the solution of a certain 2n 2n system application of the gaussian algorithm costs o n3 arithmetic operations a o we present an algorithm which reduces the problem to the solution of three n n linear toeplitz systems these systems can be solved in o n log2n a o this is also the amount of work required by the whole algorithm

fast conformal mapping of an ellipse to a simply connected region the iterative method of wegmann 1978 for conformal mapping of a region e to a region g is reformulated in terms of a conjugation operator if the boundary of g is sufficiently smooth the method converges quadratically in a sobolev space w to overcome the crowding an elongated canonical region e should be chosen if the image region g is elongated for ellipses e the operator of conjugation can be expressed in a very simple way in terms of fourier series numerically this can be performed very efficiently by fast fourier transform fft if the calculation is done on a grid with n points and if only the first m fourier terms are retained then for fixed m this smoothed numerical method converges if the initial approximation is sufficiently close to the solution and n is large enough the limit function approximates the parameter mapping function an error estimate is given for regions with analytic boundaries the error is bounded by cpm for all m with some constant p 1 these results increase the flexibility and applicability of the method appreciably several examples are discussed in detail and some numerical experiments are reported

pattern recognition algorithms based on space filling curves and orthogonal expansions the classical problem of constructing a multidimensional pattern classifier in the bayesian framework is considered preprocessing of the learning sequence by a quasi inverse of a space filling curve is proposed and properties of space filling curves which are necessary to obtain bayes risk consistency are indicated the learning sequence transformed into the unit interval is used to estimate the coefficients in an orthogonal expansion of the bayes decision rule to transform a new observation into the unit interval requires o d elementary operations where d is the dimension of the observation space strong bayes risk consistency of the classifiers is proved when distributions of the random pair of the observation vector and its class are absolutely continuous with respect to the lebesgue measure attainable convergence rate of the bayes risk is discussed details of the classification algorithm based on the haar series and its properties are presented distribution free consistency of the classifier is established the performance of such a classifier is tested both on simulated data and on the standard benchmarks providing misclassification errors comparable to or even better than the k nearest neighbors k nn method

discrete riemann hilbert problems interpolation of simply closed curves and numerical conformal mapping abstract let g be a simply connected region with 0 g and with a twice lipschitz continuously differentiable boundary curve and let z 1 n be an even number of n 2 n equidistant grid points on the unit circle sfnc z sfnc 1 with z 1 1 then there exists for all sufficiently large n a polynomial p n of degree n 1 normalized by the condition that the coefficient p 0 0 and the coefficients p 1 and p n 1 are real such that p n satisfies the interpolation condition p n z for all 1 n in a neighbourhood of the normalized conformal mapping function there is exactly one such interpolating polynomial the sequence of these p n converges to the conformal mapping function as n if is three times lipschitz continuously differentiable then the p n are also conformal mappings of the unit circle onto regions which approximate g an important tool for the theoretical investigation is a discrete analogon of the riemann hilbert problem we present two fast procedures for the numerical solution of the discrete riemann hilbert problem a conjugate gradient method with computational cost o n log n and a toeplitz matrix method with cost o n log 2 n using this one can calculate the interpolating polynomials by a newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function

numerical conformal mapping onto a rectangle with applications to the solution of laplacian problems abstract let f be the function which maps conformally a simply connected domain onto a rectangle r so that four specified points on are mapped respectively onto the four vertices of r in this paper we consider the problem of approximating the conformal map f and present a survey of the available numerical methods we also illustrate the practical significance of the conformal map by presenting a number of applications involving the solution of laplacian boundary value problems

numerical conformal mapping of multiply connected regions by fornberg like methods we develop a new algorithm for computing conformal maps from regions exterior to non overlapping disks to unbounded multiply connected regions exterior to non overlapping smoothly bounded jordan regions the method is an extension of fornberg s original newton like method for mapping of the disk to simply connected regions a fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case numerical examples are reported

on the construction and training of reformulated radial basis function neural networks presents a systematic approach for constructing reformulated radial basis function rbf neural networks which was developed to facilitate their training by supervised learning algorithms based on gradient descent this approach reduces the construction of radial basis function models to the selection of admissible generator functions the selection of generator functions relies on the concept of the blind spot which is introduced in the paper the paper also introduces a new family of reformulated radial basis function neural networks which are referred to as cosine radial basis functions cosine radial basis functions are constructed by linear generator functions of a special form and their use as similarity measures in radial basis function models is justified by their geometric interpretation a set of experiments on a variety of datasets indicate that cosine radial basis functions outperform considerably conventional radial basis function neural networks with gaussian radial basis functions cosine radial basis functions are also strong competitors to existing reformulated radial basis function models trained by gradient descent and feedforward neural networks with sigmoid hidden units

a fast method for solving certain integral equations of the first kind with application to conforma mapping abstract we describe a fast iterative method for solving fredholm integral equations of the first kind whose kernels have a logarithmic principal part the method is a fourier galerkin method and due to the singularity of the kernel the linear system of simultaneous equations is block diagonally dominant and can be solved rapidly by an iterative method we present examples of the computation of conformal mappings of simply and multiply connected domains

inviscid analysis of two dimensional airfoils in unsteady motion using conformal mapping using a conformal mapping technique a procedure for calculating the irrotational flow about a two dimensional airfoil of arbitrary shape in unsteady motion is developed the two dimensional form with a sharp trailing edge is transformed into a unit circle by two successive transformations the latter of the two is a modified version of the gershgorin integral equation which yields solutions much more accurate than those obtained with panel methods the change in circulation around the airfoil due to unsteadiness is modelled by discrete vortices which are shed from the trailing edge and allowed to move freely with the local stream the strength of these vortices is determined by the kutta condition that at each time step the velocity be zero at the trailing edge in the circle plane the procedure is efficient since the integral equation is solved only once numerical examples are presented for a sinusoidal heaving motion and for an impulsively started airfoil

blowing up convex sets in the plane abstract let k be a convex set in r 2 such that every line in r 2 meets k z 2 we prove that k z 2 r 2 for 1 2 3 3 and that this bound is best possible thus solving a problem of kannan and lovasz

on polynomial approximation in the complex plane with application to conformal mapping we consider the selection of polynomial bases for polynomial approximation of analytic functions on bounded simply connected regions in the complex plane while a monomial basis may be very ill conditioned we show that a basis of lagrange polynomials with fejer points as nodes is well conditioned numerical examples where we compute polynomial approximations of conformal mappings conclude the paper

an integral equation method for the numerical conformal mapping of interior exterior and doubly connected domains a numerical method based on the integral equation formulation of symm is described for computing approximations to the mapping functions which accomplish the following conformal maps a the mapping of a domain interior to a closed jordan curve onto the interior of the unit disc b the mapping of a domain exterior to a closed jordan curve onto the exterior of the unit disc c the mapping of a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on approximating the unknown source density by cubic splines and singular functions and is particularly suited for the mapping of difficult domains having sharp corners

on the determination of boundary collocation points for solving some problems for the laplace operator abstract boundary collocation is a method for obtaining approximate solutions of boundary problems for linear partial differential equations for which complete families of particular solutions are explicitly known the method contains various decisions which are important for its performance such as choice of solution subspace choice of basis for the subspace and choice of collocation points using a model problem some particular strategies for the determination of collocation points are investigated

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions