the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

the collocation method for first kind boundary integral equations on polygonal regions in this paper the collocation method for first kind boundary integral equations by using piecewise constant trial functions with uniform mesh is shown to be equivalent to a projection method for second kind fredholm equations in a certain sense this projection is an interpolation projection by introducing this technique of analysis we particularly consider the case of polygonal boundaries we give asymptotic error estimates in l 2 norm on the boundaries and some superconvergence results for the single layer potential

boundary element collocation methods using splines with multiple knots we extend the theory of boundary element collocation methods by allowing reduced inter element smoothness or in other words by allowing trial functions that are splines with multiple knots our convergence analysis is based on a recurrence relation for the fourier coefficients of the numerical solution and so is restricted to uniform grids on smooth closed curves superconvergence is possible with special choices of the collocation points numerical experiments with a model problem confirm the convergence rates predicted by our theory

spline collocation for strongly elliptic equations on the torus we prove convergence and error estimates in sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus shaped domains in three or more dimensions for odd degree splines we prove convergence of nodal collocation for any strongly elliptic operator for even degree splines and midpoint collocation we find an additional condition for the convergence which is satisfied for the classical boundary integral operators our analysis is a generalization to higher dimensions of the corresponding analysis of arnold and wendland 4

bubble tower radial solutions in the slightly supercritical brezis nirenberg problem abstract in this paper we consider the brezis nirenberg problem in dimension n 4 in the supercritical case we prove that if the exponent gets close to n 2 n 2 and if simultaneously the bifurcation parameter tends to zero at the appropriate rate then there are radial solutions which behave like a superposition of bubbles namely solutions of the form j 1 k 1 1 m j 4 n 2 y 2 n 2 2 m j 1 o 1 n n 2 n 2 4 where mj and mj o mj 1 for all j these solutions lie close to turning points to the right of the associated bifurcation diagram

a piecewise constant collocation method using cosine mesh grading for symm s equation here we study the piecewise constant collocation method using mesh grading to solve symm s integral equation on 1 1 we give a mesh grading for which this method achieves the optimal order of convergence even though the piecewise constant galerkin method with the same mesh grading does not some numerical results are given

some discrete methods for boundary integral equations on smooth closed curves in this paper discrete trigonometric methods as well as quadrature methods for solution of boundary integral equations on smooth closed curves are presented and analyzed the analysis applies the general approach based on the consistency stability and convergence properties the trigonometric methods including e g variants based on product integration turn out to have high convergence even exponential convergence the family of quadrature methods covers operators with the even symbol and of order beta 0 they can be considered as fully discretized petrov galerkin methods that employ approximations both for the integral operator and the inner product the operator is approximated by using the trapezoidal rule either directly or after subtraction of the kernel singularity for instance for symm s equation beta 1 a method with the rate o h 5 of convergence is presented furthermore the quadrature solution of the cauchy singular integral equation beta 0 and the hypersingular

proof number search abstract proof number search pn search is designed for finding the game theoretical value in game trees it is based on ideas derived from conspiracy number search and its variants such as applied cn search and cn search while in cn search the purpose is to continue searching until it is unlikely that the minimax value of the root will change pn search aims at proving the true value of the root therefore pn search does not consider interim minimax values pn search selects the next node to be expanded using two criteria the potential range of subtree values and the number of nodes which must conspire to prove or disprove that range of potential values these two criteria enable pn search to treat efficiently game trees with a non uniform branching factor it is shown that in non uniform trees pn search outperforms other types of search such as iterative deepening search even when enhanced with transposition tables move ordering for the full principal variation etc pn search has been used to establish the game theoretical values of connect four qubic and go moku there pn search was able to find a forced win for the player to move first the experiments described here are in the domain of awari a game which has not yet been solved the experiments are repeatable for other games with a non uniform branching factor this article describes the underlying principles of pn search presents an appropriate implementation and provides an analysis of its strengths and weaknesses

best first fixed depth minimax algorithms abstract this article has three main contributions to our understanding of minimax search first a new formulation for stockman s sss algorithm based on alpha beta is presented it solves all the perceived drawbacks of sss finally transforming it into a practical algorithm in effect we show that sss alpha beta transposition tables the crucial step is the realization that transposition tables contain so called solution trees structures that are used in best first search algorithms like sss having created a practical version we present performance measurements with tournament game playing programs for three different minimax games yielding results that contradict a number of publications second based on the insights gained in our attempts at understanding sss we present a framework that facilitates the construction of several best first fixed depth game tree search algorithms known and new the framework is based on depth first null window alpha beta search enhanced with storage to allow for the refining of previous search results it focuses attention on the essential differences between algorithms third a new instance of this framework is presented it performs better than algorithms that are currently used in most state of the art game playing programs we provide experimental evidence to explain why this new algorithm mtd f performs better than other fixed depth minimax algorithms

properties of certain trigonometric series arising in numerical analysis h o f kdlsinn6 n l where 0 is real tx 0 and i 1 series of this kind have obvious connec tions with the riemann zeta function and with bernoulli polynomials and are important for fractional integration 6 xii sect 83 they have also arisen recently 2 4 in the study of certain numerical methods for bound ary integral equations on smooth curves in the plane for the applications in 2 we are interested firstly in knowing that g has a unique zero in the interval 0 rc so that a quadrature rule employing this zero is well defined and for later applications we need to know that this zero increases monotonically with c in the case where o 3 1 these questions were addressed in l as were the corresponding trickier questions about the partial sums for o b 1 we write g e p gol e b g e 4 me d fvbut fi f h e mi and for 0 p d f define k 4 b f p g 0 pi of crucial importance to the applications in 2 are the facts proved in theorem 2 below that h is nonnegative and h non positive and that k are both positive except in special cases in essence these properties ensure that the numerical method treated in 2 is stable the rate of convergence of the numerical methods of 2 is determined by the behaviour of g f i for small values of p the salient properties follow from 2 below in section 5 of the paper we indicate some extensions of our results to other classical orthogonal polynomials 2

an analysis of the conspiracy numbers algorithm abstract mcallester s conspiracy numbers algorithm is a minimax search procedure that builds game trees to variable depths without application dependent knowledge the algorithm gathers information to determine how likely it is that the search of a sub tree will produce a useful result likeliness is measured by the conspiracy numbers the minimum number of leaf nodes that must change their value by being searched deeper to cause the minimax value of a sub tree to change the search is controlled by the conspiracy threshold ct the minimum number of conspirators beyond which it is considered unlikely that a sub tree s value can be changed this paper analyzes the best case performance for the algorithm mcallester s original algorithm is shown to build a tree of size o w ct w 1 where w is the branching factor of the tree a new improvement to the algorithm is shown to reduce this complexity to o ct 2 hence for a given fixed w the algorithm s best case can be improved from exponential to quadratic growth although the minimal tree case is not necessarily representative of the search trees built in practice experimental results are presented that demonstrate the improvement appears to be significant in the expected case as well

nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains let be a bounded domain in rn with n 3 and assume p 2n n 2 we study problems of the type u u p 2u 0 in u 0 on we prove that for suitable p 2n n 2 there exists some bounded domain with nontrivial topology in the sense of bahri and coron such that the above problem has only the solution u 0

on the super approximation property of galerkin s method with finite elements for galerkin s method with finite elements as trial functions for strongly elliptic operator equations in the hilbert scaleh t the super approximation property and the optimal convergence rate are obtained by using the aubin nitsche lemma this applies in particular to spline collocation methods for a wide class of pseudodifferential equations

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

game tree searching by min max approximation abstract we present an iterative method for searching min max game trees based on the idea of approximating the min and max operators by generalized mean valued operators this approximation is used to guide the selection of the next leaf node to expand since the approximations allow one to select efficiently that leaf node upon whose value the approximate value at the root most highly depends experimental results from almost 1 000 games of connect four 1 suggest that our scheme is superior to minimax search with alpha beta pruning for the same number of calls to the move routine however our scheme has higher overhead so that further work is needed before it becomes competitive when cpu time per turn is the limiting resource

double layer potential scheme for dirichlet problems on smooth open arcs abstract in this paper we present a new numerical scheme for the dirichlet problem on the smooth open arcs in the plane by using the double layer potential to construct the solution of the problem we induce a fredholm integral equation of the second kind existence and uniqueness of the solution of this integral equation are proved in the case when the boundary arc is convex a numerical example is given to show the validity of the present scheme

spline collocation for convolutional parabolic boundary integral equations summary we consider spline collocation methods for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover for example the case of the single layer heat operator equation when the spatial domain is a disc

on the numerical evaluation of legendre s chi function legendre s chi function formula is reexpanded in a power series in powers of formula the expansion obtained is well suited for the computation of formula in the two cases of real z close to 1 and formula for n 2 and n 3 the present computational procedure is shown to be superior to the procedure recently proposed by dempsey liu and dempsey which uses plana s summation formula

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

solutions to a singularly perturbed supercritical elliptic equation on a riemannian manifold concentrating at a submanifold abstract given a smooth riemannian manifold m g we investigate the existence of positive solutions to the equation e 2 g u u u p 1 on m which concentrate at some submanifold of m as e 0 for supercritical nonlinearities we obtain a positive answer for some manifolds which include warped products using one of the projections of the warped product or some harmonic morphism we reduce this problem to a problem of the form e 2 div h c x h u a x u b x u p 1 with the same exponent p on a riemannian manifold m h of smaller dimension so that p turns out to be subcritical for this last problem then applying lyapunov schmidt reduction we establish existence of a solution to the last problem which concentrates at a point as e 0

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

spline qualocation methods for variable coefficient elliptic equations on curves here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type the analysis exploits localization in the form of frozen coefficients pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders numerical examples show the predicted high orders of convergence

boundary towers of layers for some supercritical problems we consider the supercritical problem u u p 1uin d u 0on d where d is a bounded smooth domain in rn and p is smaller than the th critical sobolev exponent 2n n 2n 2 with 1 n 3 we show that in some suitable torus like domains d there exists an arbitrary large number of sign changing solutions with alternate positive and negative layers which concentrate at different rates along a dimensional submanifold of d as p approaches 2n from below

conspiracy numbers for min max search abstract a new procedure is presented for growing min max game trees in large games such as chess decisions must be based on incomplete search trees the new tree growth procedure is based on conspiracy numbers as a measure of the accuracy of the root minimax value of an incomplete tree conspiracy numbers measure the number of leaf nodes whose value must change in order to change the minimax root value by a given amount trees are grown in a way that maximizes the conspiracy required to change the root value the trees grown by this procedure are often deep and narrow however if all static values in a game are the same this new procedure reduces to d ply search with pruning unlike b search nonuniform growth is achieved without any modification of the static board evaluator

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

a generalised quiescence search algorithm abstract this paper describes how the concept of a null move may be used to define a generalised quiescence search applicable to any minimax problem experimental results in the domain of chess tactics show major gains in cost effectiveness over full width searches and it is suggested that null move quiescence may be almost as widely useful as the alpha beta mechanism the essence of the mechanism is that null moves give rise to bounds on position values which are more reliable than evaluations when opposing bounds touch they create a single value which is more reliable than ordinary evaluations and the search is terminated at that point these terminations are prior to any alpha beta cutoffs and can lead to self terminating searches

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

extrapolation methods for spline collocation solutions of pseudodifferential equations on curves in this article we consider extrapolation methods in approximating smooth linear functionals by means of collocation solutions of boundary integral equations we are able to derive an asymptotic expansion of the error in main applications this expansion covers arbitrary high powers of the discretization parameter if the boundary solution is smooth the expansion gives rise to richardson type extrapolation schemes which require only simple postprocessing of calculated numbers the results are applicable for example in calculating pointwise values in the space domain when solving boundary value problems by means of integral equations in such examples the extrapolation rapidly improves the original rate of the convergence numerical experiments conform our results

orthogonal collocation in the nonconforming boundary element method this paper outlines the use of non conforming discontinuous elements in the collocation boundary element method for solving two dimensional potential and poisson type problems the roots of an orthogonal polynomial shifted jacobi polynomial are used as the collocation points this results in increased accuracy due to the least square minimization property of the orthogonal polynomials the advantage of using non conforming elements is realized when the method is applied i to problems with singularities both due to geometry and boundary conditions and ii in conjunction with domain decomposition techniques also the collocation points can be relocated within an element by changing two user defined parameters in the shifted jacobi polynomial thus providing an error indicator which can be used for mesh refinement purposes this technique called the rh method is discussed and illustrated the results obtained by using non conforming boundary elements for standard test problems are shown to be accurate

bubble towers for supercritical semilinear elliptic equations we construct positive solutions of the semilinear elliptic problem u u up 0 with dirichet boundary conditions in a bounded smooth domain rn n 4 when the exponent p is supercritical and close enough to n 2n 2 and the parameter r is small enough as p n 2n 2 the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves green s function our result extends the result of del pino et al j differential equations 193 2 2003 280 when is a ball and the solutions are radially symmetric

quadrature methods for strongly elliptic equations of negative order on smooth closed curves in this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order the results cover all the classical pseudodifferential equations of this kind and hence the boundary integral equations appearing in applications the quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh and on varepsilon collocation they are proved to have the stability and the convergence with the maximal rate o h beta of convergence in general and o h beta 1 for some special values of varepsilon here beta is the order of the operator the method is applicable to systems of boundary integral equations as well in this connection the authors have conducted some numerical experiments which confirm their theoretical results

strategies anticipating a difference in search depth using opponent model search in this contribution we propose a class of strategies which focus on the game as well as on the opponent preference is given to the thoughts of the opponent so that the strategy under investigation might be speculative we describe a generalization of om search called d d om search where d stands for the depth of search by the player and d for the opponent s depth of search a known difference in search depth can be exploited by purposely choosing a suboptimal variation with the aim to gain a larger advantage than when playing the objectively best move the difference in search depth may have the result that the opponent does not see the variation in sufficiently deep detail we then give a pruning alternative for d d om search denoted by 2 pruning a best case analysis shows that 2 prunes very efficiently comparable to the efficiency of with regard to minimax the effectiveness of the proposed strategy is confirmed by simulations using a game tree model including an opponent model and by experiments in the domain of othello

superapproximation for projections on spline spaces in this paper general conditions are given for the superapproximation of projections on non uniform mesh multiple knot splines in lp spaces various known results are contained as special cases

boundary clustered layers near the higher critical exponents abstract we consider the supercritical problem u u p 2 u in u 0 on where is a bounded smooth domain in r n and p smaller than the critical exponent 2 n k 2 n k n k 2 for the sobolev embedding of h 1 r n k in l q r n k 1 k n 3 we show that in some suitable domains there are positive and sign changing solutions with positive and negative layers which concentrate along one or several k dimensional submanifolds of as p approaches 2 n k from below

the spline collocation method for parabolic boundary integral equations on smooth curves we consider the spline collocation method for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators for the localization we need also some special spline approximation results in anisotropic sobolev spaces

inversion formulas for the discretized hilbert transform on the unit circle a discrete version of the hilbert transform on the unit circle is considered its moore penrose inverse with respect to suitable scalar products is derived for different side conditions furthermore stability of the pseudo inverse is studied these results allow the efficient computation of approximate solutions of singular integral equations with hilbert kernel furthermore the stability analysis of such methods becomes much easier even for graded meshes which are useful for weakly singular solutions

fast isogeometric boundary element method based on independent field approximation abstract an isogeometric boundary element method for problems in elasticity is presented which is based on an independent approximation for the geometry traction and displacement field this enables a flexible choice of refinement strategies permits an efficient evaluation of geometry related information a mixed collocation scheme which deals with discontinuous tractions along non smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions all these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations the system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis for the required geometrical bisection of the domain a strategy for the evaluation of bounding boxes containing the supports of nurbs basis functions is presented the versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions

spline collocation for a boundary integral equation on polygons with cuts in this paper we construct collocation methods for a modified integral equation of the second kind on the boundary of a polygon with a finite number of cuts cuts are often met in practice and are not treated by the classical methods the double layer potential which degenerates in this situation is modified by the addition of two kinds of terms on each cut they take into account the highly singular functions generated by the cut and lead to a bijective boundary operator this operator remains bijective between spaces of sobolev types with high orders of regularity this allows the construction of collocation methods with quasi optimal estimates and high order of convergence with respect to strong norms they use smoothest splines of odd degree numerical examples of the methods are presented

midpoint collocation for cauchy singular integral equations a cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid even if the grid is non uniform suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations the same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

lp estimates of boundary integral equations for some nonlinear boundary value problems recently galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions and stability results and error estimates in theh 1 2 norm have been proved ruotsalainen and wendland and ruotsalainen and saranen we show that these results extend tol p setting without any extra conditions these extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators and then considering the nonlinear equations the fact that inh 1 2 setting the nonlinear operator is a homeomorphism with lipschitz continuous inverse plays a crucial role optimal error estimates for the galerkin and collocation method inl p space then follow

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

numerical evaluation of some trigonometric series we present a method for the accurate numerical evaluation of a family of trigonometric series arising in the design of special purpose quadrature rules for boundary element methods the series converge rather slowly but can be expressed in terms of fourier chebyshev series that converge rapidly let r be a positive integer and define the functions gr and hr by gr t 2 e r cos 2rmt m 1

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis