constructing robust good lattice rules for computational finance the valuation of many financial derivatives leads to high dimensional integrals the constructions of robust or universal good lattice rules for financial applications are both important and challenging an important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low dimensional functions i e functions that depend on only a small number of variables usually just two variables for numerical integration of such functions the quality of the low order i e low dimensional projections of the node set is crucial in this paper we propose methods to construct good lattice points with optimal low order projections the quality of a point set is measured by a new measure called elementary order ell discrepancy which measures the quality of all order ell projections and is more informative than usual measures two constructions namely the korobov and the component by component constructions are studied such that the low order projections are optimized numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order 2 projections that are better than those of the sobol points and random points and with higher order projections that are no worse while the sobol points lost the advantage over random points in order 2 projections on the average the new lattice rules have the potential to improve upon the accuracy for favorable functions while doing no harm for unfavorable ones their applications for pricing path dependent options and american options based on the least square monte carlo method are studied and their high efficiency is demonstrated a nice surprise revealed is the robustness property of such lattice rules the good projection property and the suitability for a large range of problems the potential possibility and limitations of good lattice points in achieving good quality of moderate and high order projections is investigated the reason why classical lattice rules may not be efficient for high dimensional finance problems is also discussed

pricing derivative securities using integrated quasi monte carlo methods with dimension reduction and discontinuity realignment the dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi monte carlo qmc method the first factor implies that the qmc method can have high performance on a problem with low effective dimension even though its nominal dimension can be very large the second factor suggests that the qmc method can be very effective on a problem that is smooth or even on a problem that is discontinuous as long as its discontinuities are parallel to the axes motivated by these findings methods such as the linear transformation lt method of j imai and k s tan j comput finance 10 2006 pp 129 155 and the orthogonal transformation ot method of x wang and k s tan management sci 59 2013 pp 376 389 have been proposed to increase the efficiency of qmc however both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the lt method enhances qmc by effective dimension reduction while the o

imposing boundary conditions in sinc method using highest derivative approximation sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities a recent implementation of the sinc method li c and wu x numerical solution of differential equations using sinc method based on the interpolation of the highest derivatives applied mathematical modeling 31 1 2007 1 9 in which sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired we suggest an alternative approach for specifying the boundary conditions and we compare the numerical results for analysis of a laminated composite timoshenko beam implementing both li and wu s approach and our alternative approach for applying the boundary conditions for the timoshenko beam problem we obtain accurate results using both approaches including transverse shear stress by integration of the 3d equilibrium equations of elasticity the beam results indicate our approach is less dependent on the selection of the sinc mesh size than li and wu s sihd we also apply sihd to analyze a classical laminated composite plate for the plate example we experience difficulty in obtaining a complete system of equations using li and wu s approach for our approach we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge using this technique we obtain accurate results for deflection and stresses including interlaminar stresses by integration of the 3d equilibrium equations of elasticity our results for both the beam and the plate problems indicate that this approach is easily implemented has a high level of accuracy and good convergence properties

advances in the adomian decomposition method for solving two point nonlinear boundary value problems with neumann boundary conditions a new straightforward approach for solving ordinary and partial second order boundary value problems with neumann boundary conditions is introduced in this research this approach depends mainly on the adomian decomposition method with a new definition of the differential operator and its inverse which has been modified for neumann boundary conditions the effectiveness of the proposed approach is verified by several linear and nonlinear examples

principal warps thin plate splines and the decomposition of deformations the decomposition of deformations by principal warps is demonstrated the method is extended to deal with curving edges between landmarks this formulation is related to other applications of splines current in computer vision how they might aid in the extraction of features for analysis comparison and diagnosis of biological and medical images in indicated

integrated local petrov galerkin sinc method for structural mechanics problems an integrated local petrov galerkin sinc method is introduced and applied to static structural mechanics problems the method approximates the highest derivative in the local weak form of the governing equation on a rectangular grid and the lower derivatives and unknown function are found by numerical indenite integration we suggest that the essential boundary conditions may be applied by the traditional penalty method or by a method in which the stiness matrix is reduced to eliminate dependent degrees of freedom we propose and compare the performance of three basis functions in terms of their accuracy and convergence properties for two problems a one dimensional tapered bar and a two dimensional plane stress elasticity problem our results indicate that the present method can provide greater accuracy than the sinc method based on interpolation of highest derivative an integration based collocation method suggested by li and wu li c and wu x numerical solution of dierential equations using sinc method based on

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

pricing and hedging with discontinuous functions quasi monte carlo methods and dimension reduction quasi monte carlo qmc methods are important numerical tools in the pricing and hedging of complex financial instruments the effectiveness of qmc methods crucially depends on the discontinuity and the dimension of the problem this paper shows how the two fundamental limitations can be overcome in some cases we first study how path generation methods pgms affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of qmc methods the insight is that the discontinuities can be qmc friendly i e aligned with the coordinate axes or not depending on the pgm the pgms that offer the best performance in qmc methods are those that make the discontinuities qmc friendly the structure of discontinuities can affect the accuracy of qmc methods more significantly than the effective dimension this insight motivates us to propose a novel way of handling the discontinuities the basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in qmc methods for pricing options and for estimating greeks it also reduces the effective dimension of the problem this paper was accepted by assaf zeevi stochastic models and simulation

spatial low discrepancy sequences spherical cone discrepancy and applications in financial modeling in this paper we introduce a reproducing kernel hilbert space defined on r d 1 as the tensor product of a reproducing kernel defined on the unit sphere s d in r d 1 and a reproducing kernel defined on 0 we extend stolarsky s invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel hilbert space the idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods an extension of the area preserving lambert transform is used to generate points on s d 1 via lifting sobol points in 0 1 d to the sphere the d th component of each sobol point suitably transformed provides the distance information so that the resulting point set is normally distributed in r d numerical tests provide evidence of the usefulness of constructing quasi monte carlo type methods for integration in such spaces we also test this method on examples from financial applications option pricing problems and compare the results with traditional methods for numerical integration in r d

strong tractability of multivariate integration using quasi monte carlo algorithms we study quasi monte carlo algorithms based on low discrepancy sequences for multivariate integration we consider the problem of how the minimal number of function evaluations needed to reduce the worst case error from its initial error by a factor of e depends on e 1 and the dimension s strong tractability means that it does not depend on s and is bounded by a polynomial in e 1 the least possible value of the power of e 1 is called the e exponent of strong tractability sloan and wozniakowski established a necessary and sufficient condition of strong tractability in weighted sobolev spaces and showed that the e exponent of strong tractability is between 1 and 2 however their proof is not constructive in this paper we prove in a constructive way that multivariate integration in some weighted sobolev spaces is strongly tractable with e exponent equal to 1 which is the best possible value under a stronger assumption than sloan and wozniakowski s assumption we show that quasi monte carlo algorithms using niederreiter s t s sequences and sobol sequences achieve the optimal convergence order o n 1 for any 0 independent of the dimension with a worst case deterministic guarantee where n is the number of function evaluations this implies that strong tractability with the best e exponent can be achieved in appropriate weighted sobolev spaces by using niederreiter s t s sequences and sobol sequences

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

quasi monte carlo methods for linear two stage stochastic programming problems quasi monte carlo qmc algorithms are studied for generating scenarios to solve two stage linear stochastic programming problems their integrands are piecewise linear quadratic but do not belong to the function spaces considered for qmc error analysis we show that under some weak geometric condition on the two stage model all terms of their anova decomposition except the one of highest order are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to l 2 l2 this implies that randomly shifted lattice rules may achieve the optimal rate of convergence o n 1 delta o n 1 with delta in 0 frac 1 2 0 12 and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two the geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution is normal we discuss effective dimensions and techniques for dimension reduction numerical experiments for a production planning model with normal inputs show that indeed convergence rates close to the optimal rate are achieved when using randomly shifted lattice rules or scrambled sobol point sets accompanied with principal component analysis for dimension reduction

knowledge directed left ventricular boundary detection in equilibrium radionuclide angiocardiography a method is presented that will automatically find the left ventricular lv boundary in any of several equilibrium radionuclide angiocardiographic erna image views the lower level image analysis involves the use of directional gradient edge operators and an edge point linking scheme the higher level portion of the algorithm uses a knowledge based strategy that incorporates appropriate local and global information in heuristic cost functions to find complete boundaries a detailed example of lv boundary delineation in the end diastolic frame from a left lateral view erna image sequence is tracked through each stage of the proposed methodology the use of this approach for following the lv boundary through entire temporal sequences is also illustrated on 16 frames of both a left lateral view and a left anterior oblique view erna study

constructing embedded lattice rules for multivariate integration lattice rules are a family of equal weight cubature formulae for approximating high dimensional integrals by now it is well established that good generating vectors for lattice rules having n points can be constructed component by component for integrands belonging to certain weighted function spaces and that they can achieve the optimal rate of convergence although the lattice rules constructed this way are extensible in dimension they are not extensible in n thus when n is changed the generating vector needs to be constructed anew in this paper we introduce a new algorithm for constructing good generating vectors for embedded lattice rules which can be used for a range of n while still being extensible in dimension by using an adaptation of the fast component by component construction algorithm which makes use of fast fourier transforms we are able to obtain good generating vectors for thousands of dimensions and millions of points under both product weight and order dependent weight settings at the cost of calo d n log n 2 operations with a sufficiently large number of points and good overall quality these embedded lattice rules can be used for practical purposes in the same way as a low discrepancy sequence we show for a range of weight settings in an unanchored sobolev space that our embedded lattice rules achieve the same optimal rate of convergence calo n 1 delta delta 0 as those constructed for a fixed number of points and that the implied constant gains only a factor of 1 30 to 1 55

quasi monte carlo methods in financial engineering an equivalence principle and dimension reduction quasi monte carlo qmc methods are playing an increasingly important role in the pricing of complex financial derivatives for models in which the prices of the underlying assets are driven by brownian motions the performance of qmc methods is known to depend crucially on the construction of brownian motions this paper focuses on the impact of various constructions although the brownian bridge bb construction often yields very good results as papageorgiou pointed out there are financial derivatives for which the bb construction performs badly papageorgiou a 2002 the brownian bridge does not offer a consistent advantage in quasi monte carlo integration j complexity18 1 171 186 in this paper we first extend papageorgiou s analysis to establish an equivalence principle if the bb construction or any other construction is the preferred construction for a particular financial derivative then for any other construction there is another financial derivative for which the latter construction is the preferred one in this sense all methods of construction are equivalent and no method is consistently superior to others it all depends on the particular financial derivative we then show how to find a good construction for a particular class of financial derivatives in practice our strategy is to find a good construction for an easy problem and then apply it to more complicated problems related to the easy one this strategy is applied to the arithmetic asian options including bermudan asian options based on the weighted average of the stock prices we do this by studying a simpler problem namely the geometric asian option for which the best construction is easily available and applying it to the arithmetic asian option numerical experiments confirm the success of this strategy whereas in qmc all the common methods the standard method bb and principal component analysis may lose their power in some situations the new method behaves very well in all cases further large variance reduction can be achieved in combination with a control variate the new method can be interpreted as a practical way of reducing the effective dimension for some class of functions

sequential quasi monte carlo this is a written discussion of the paper sequential quasi monte carlo sampling by m gerber and n chopin following the presentation given before the royal statistical society in london on december 10th 2014

left ventricle automated detection method in gated isotopic ventriculography using fuzzy clustering a method that uses the fuzzy isodata clustering algorithm and fourier analysis is proposed for automated detection of heart left ventricle contours this operation is used for quantitative analysis of cardiac function the computation begins by finding the phase image the fuzzy isodata algorithm is first applied to this image to generate a number of clusters that correspond to the organ substructures ventricles atria second the ventricles cluster is isolated and the intensities of its points are replaced by the corresponding ones from the original end diastolic frame finally a reduced image representing the ventricular region is obtained and an additional clustering is performed to find the left ventricular boundary automatically this algorithm is tested by application of 105 sets of 16 images each these results are compared with the measurements obtained with two semi automatic methods used respectively on the philips and the sopha medical gamma cameras

quasi monte carlo methods for two stage stochastic mixed integer programs we consider randomized qmc methods for approximating the expected recourse in two stage stochastic optimization problems containing mixed integer decisions in the second stage it is known that the second stage optimal value function is piecewise linear quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets this structure is exploited to provide conditions implying that first and higher order terms of the integrand s anova decomposition math comp 79 2010 953 966 have mixed weak first order partial derivatives this leads to a good smooth approximation of the integrand and hence to good convergence rates of randomized qmc methods if the effective superposition dimension is low

space time adaptive finite difference method for european multi asset options the multi dimensional black scholes equation is solved numerically for a european call basket option using a priori a posteriori error estimates the equation is discretized by a finite difference method on a cartesian grid the grid is adjusted dynamically in space and time to satisfy a bound on the global error the discretization errors in each time step are estimated and weighted by the solution of the adjoint problem bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations comparisons are made with monte carlo and quasi monte carlo methods in one dimension and the performance of the method is illustrated by examples in one two and three dimensions

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

quasi monte carlo methods in numerical finance this paper introduces and illustrates a new version of the monte carlo method that has attractive properties for the numerical valuation of derivatives the traditional monte carlo method has proven to be a powerful and flexible tool for many types of derivatives calculations under the conventional approach pseudo random numbers are used to evaluate the expression of interest unfortunately the use of pseudo random numbers yields an error bound that is probabilistic which can be a disadvantage another drawback of the standard approach is that many simulations may be required to obtain a high level of accuracy there are several ways to improve the convergence of the standard method this paper suggests a new approach which promises to be very useful for applications in finance quasi monte carlo methods use sequences that are deterministic instead of random these sequences improve convergence and give rise to deterministic error bounds the method is explained and illustrated with several examples these examples include complex derivatives such as basket options asian options and energy swaps

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

matching two perspective views a computational approach to image matching is described it uses multiple attributes associated with each image point to yield a generally overdetermined system of constraints taking into account possible structural discontinuities and occlusions in the algorithm implemented intensity edgeness and cornerness attributes are used in conjunction with the constraints arising from intraregional smoothness field continuity and discontinuity and occlusions to compute dense displacement fields and occlusion maps along the pixel grids the intensity edgeness and cornerness are invariant under rigid motion in the image plane in order to cope with large disparities a multiresolution multigrid structure is employed coarser level edgeness and cornerness measures are obtained by blurring the finer level measures the algorithm has been tested on real world scenes with depth discontinuities and occlusions a special case of two view matching is stereo matching where the motion between two images is known the algorithm can be easily specialized to perform stereo matching using the epipolar constraint

path generation for quasi monte carlo simulation of mortgage backed securities monte carlo simulation is playing an increasingly important role in the pricing and hedging of complex path dependent financial instruments low discrepancy simulation methods offer the potential to provide faster rates of convergence than those of standard monte carlo methods however in high dimensional problems special methods are required to ensure that the faster convergence rates hold indeed ninomiya and tezuka 1996 have shown highdimensional examples in which low discrepancy methods perform worse than monte carlo methods the principal component construction introduced by acworth et al 1998 provides one solution to this problem however the computational effort required to generate each path grows quadratically with the dimension of the problem this article presents two new methods that offer accuracy equivalent in terms of explained variability to the principal components construction with computational requirements that are linearly related to the problem dimension one method is to take into account knowledge about the payoff function which makes it more flexible than the brownian bridge construction numerical results are presented that show the benefits of such adjustments the different methods are compared for the case of pricing mortgage backed securities using the hull white term structure model

wavelet galerkin solutions for one dimensional partial differential equations in this paper we describe how wavelets may be used to solve partial differential equations these problems are currently solved by techniques such as finite differences finite elements and multi grid the wavelet method however offers several advantages over traditional methods wavelets have the ability to represent functions at different levels of resolution thereby providing a logical means of developing a hierarchy of solutions furthermore compactly supported wavelets such as those due to daubechies1 are localized in space which means that the solution can be refined in regions of high gradient e g stress concentrations without having to regenerate the mesh for the entire problem in order to demonstrate the wavelet technique we consider the one dimensional counterpart of helmholtz s equation by comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed dirichlet boundary conditions are then imposed using the capacitance matrix method described by proskurowski and widlund2 and others the convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method at least for problems with simple geometries

general multilevel monte carlo methods for pricing discretely monitored asian options abstract we describe general multilevel monte carlo methods that estimate the price of an asian option monitored at m fixed dates for a variety of processes that can be simulated exactly we prove that for the same computational cost our method yields an unbiased estimator with variance lower than the variance of the standard monte carlo estimator by a factor of order m we show how to combine our approach with the milstein scheme for processes driven by scalar stochastic differential equations and with the euler scheme for processes driven by multidimensional stochastic differential equations numerical experiments confirm that our method outperforms the conventional monte carlo algorithm by a factor proportional to m

solving second kind integral equations by galerkin methods with continuous orthogonal wavelets abstract in this paper we use the continuous wavelets on the interval constructed by cohen et al appl comput harm anal 1 1993 54 81 to solve the second kind integral equations to this end we give the decomposition and reconstruction algorithm for these wavelets and construct the quadrature formulae for the calculation of inner products of any functions and the scaling functions which are required in the wavelet galerkin methods for integral equations in this method the integral kernels are represented in these wavelet bases as sparse matrices to high precision thus we present an efficient algorithm for numerical solution of second kind integral equations

low discrepancy sequences in high dimensions how well are their projections distributed quasi monte carlo qmc methods have been successfully used to compute high dimensional integrals arising in many applications especially in finance to understand the success and the potential limitation of qmc this paper focuses on quality measures of point sets in high dimensions we introduce the email protected superposition and truncation discrepancies which measure the quality of selected projections of a point set on lower dimensional spaces these measures are more informative than the classical ones we study their relationships with the integration errors and study the tractability issues we present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the sobol nets with that of the sets of latin hypercube sampling and random points numerical results show that in high dimensions the superiority of the sobol nets mainly derives from the one dimensional projections and the projections associated with the earlier dimensions for order 2 and higher order projections all these point sets have similar behavior on the average in weighted cases with fast decaying weights the sobol nets have a better performance than the other two point sets the investigation enables us to better understand the properties of qmc and throws new light on when and why qmc can have a better or no better performance than monte carlo for multivariate integration in high dimensions

solution of nonlinear initial boundary value problems by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the application refers to some direct and inverse type problems

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

analytic solution of nonlinear boundary value problems in several dimensions by decomposition abstract nonlinear partial differential equations are systematically solved by the decomposition method of g adomian for general boundary conditions described by boundary operator equations linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure

on the effects of dimension reduction techniques on some high dimensional problems in finance many problems in finance can be formulated as high dimensional integrals which are often attacked by quasi monte carlo qmc algorithms to enhance qmc algorithms dimension reduction techniques such as the brownian bridge bb and principal component analysis pca are used to reduce the effective dimension this paper explores in depth the effects of these techniques on the dimension structure of some typical high dimensional problems from finance the pricing of path dependent options and bond valuation according to term structure models by deriving explicit expressions for the underlying integrands and the associated weights that control the relative importance of different variables and by investigating the variance ratios the effective dimensions the mean dimension and their limiting behavior as the nominal dimension tends to infinity we show theoretically and empirically how and to what extent the bb and pca algorithms change the dimension structure including the degree of additivity of the underlying functions they change the functions to be strongly weighted and substantially reduce the effective dimensions and the mean dimension and they enhance the degree of additivity which is particularly important for qmc moreover the resulting functions are of low effective dimension not only in the superposition sense but also in the truncation sense the variance ratios the effective dimensions and the mean dimension associated with these techniques are very insensitive to the nominal dimension they are essentially constant which highlights the possibility of removing the curse of dimensionality when dimension reduction techniques are used in combination with qmc a counterexample is also shown for which the bb and pca may increase the effective dimension the investigation provides further insight into the effects of dimension reduction techniques

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

reclaiming quasimonte carlo efficiency in portfolio value at risk simulation through fourier transform quasimonte carlo methods overcome the problem of sample clustering in regular monte carlo simulation and have been shown to improve simulation efficiency in the derivatives pricing literature when the price is expressed as a multidimensional integration and the integrand is suitably smooth for portfolio value at risk var problems the distribution of portfolio value change is based on the expectation of an indicator function hence the integrand is discontinuous the purpose of this paper is to smooth the expectation estimation of an indicator function via fourier transform so that the faster convergence rate of quasimonte carlo methods can be reclaimed theoretically under fairly mild assumptions the simulation of portfolio value at risk is fast and accurate numerical examples elucidate the advantage of the proposed approach over regular monte carlo and quasimonte carlo methods

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

generating quasi random paths for stochastic processes the need to simulate stochastic processes numerically arises in many fields frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness this paper addresses the question of how to use quasi monte carlo methods to improve this simulation special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps is required two such techniques the generalized brownian bridge and particle reordering are described here these methods are applied to a problem from finance the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate when expressed as an integral this problem is nominally 360 dimensional the analysis of the integrand presented here explains the effectiveness of the quasi random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi random sequences

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

efficient weighted lattice rules with applications to finance good lattice rules are an important type of quasi monte carlo algorithm they are known to have good theoretical properties in the sense that they can achieve an error bound or optimal error bound that is independent of the dimension for weighted function spaces with suitably decaying weights to use the theory of weighted function spaces for practical applications one has to determine what weights should be used after explaining that lattice rules based on figures of merit with classical weights classical lattice rules may not give good results for high dimensional problems we propose a matching strategy which chooses the weights in such a way that the typical functions of a weighted korobov space are similar in the sense of similar relative sensitivity indices to those of the given function the matching strategy relates the weights of the spaces to the sensitivity indices of the given function we apply the korobov construction and the component by component construction of lattice rules with the suitably chosen weights to the valuation of high dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules moreover they are competitive with other types of low discrepancy sequences we also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly we show that an acceptance rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy

the error bounds and tractability of quasi monte carlo algorithms in infinite dimension dimensionally unbounded problems are frequently encountered in practice such as in simulations of stochastic processes in particle and light transport problems and in the problems of mathematical finance this paper considers quasi monte carlo integration algorithms for weighted classes of functions of infinitely many variables in which the dependence of functions on successive variables is increasingly limited the dependence is modeled by a sequence of weights the integrands belong to rather general reproducing kernel hilbert spaces that can be decomposed as the direct sum of a series of their subspaces each subspace containing functions of only a finite number of variables the theory of reproducing kernels is used to derive a quadrature error bound which is the product of two terms the generalized discrepancy and the generalized variation tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by ce p for some exponent p and some positive constant c the e exponent of tractability is defined as the smallest power of e 1 in these bounds it is shown by using monte carlo quadrature that the e exponent is no greater than 2 for these weighted classes of integrands under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the halton sequence that the e exponent of tractability is 1 which implies that infinite dimensional integration is no harder than one dimensional integration

handling discontinuities in financial engineering good path simulation and smoothing discontinuities are common in the pricing and hedging of complex financial derivatives quasi monte carlo qmc methods for high dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality interestingly path simulation method psm may affect both factors implying its significance in qmc methods what defines a good psm for problems with discontinuities the ability to align the discontinuities with the coordinate axes is a desirable property for a psm we show that for an arbitrary psm there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis parallel discontinuities for which good qmc performance can be expected in this sense any psm can be good in qmc methods for a specific class of problems we analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness up to a permutation of the standard co

comparison of the solutions obtained by adomian decomposition and wavelet galerkin methods of boundary value problems abstract in this paper we will compare the performance of adomian decomposition method and the wavelet galerkin method applied to the solution of boundary value problems involving non homogeneous heat and wave equations it is shown that the adomian decomposition method in many instances gives better results in the wavelet galerkin solutions daubechies six wavelets are used because they give better results than those of lower degree wavelets the results are then compared with those obtained using the adomian decomposition method although the adomian decomposition solution required slightly more computational effort than the wavelet galerkin solution it resulted in more accurate results than the wavelet galerkin method

a multiresolution image segmentation technique based on pyramidal segmentation and fuzzy clustering in this paper an unsupervised image segmentation technique is presented which combines pyramidal image segmentation with the fuzzy c means clustering algorithm each layer of the pyramid is split into a number of regions by a root labeling technique and then fuzzy c means is used to merge the regions of the layer with the highest image resolution a cluster validity functional is used to find the optimal number of objects automatically segmentation of a number of synthetic as well as clinical images is illustrated and two fully automatic segmentation approaches are evaluated which determine the left ventricular volume lv in 140 cardiovascular magnetic resonance mr images first fuzzy c means is applied without pyramids in the second approach the regions generated by pyramidal segmentation are merged by fuzzy c means the correlation coefficients of manually and automatically defined lv lumen of all 140 and 20 end diastolic images were equal to 0 86 and 0 79 respectively when images were segmented with fuzzy c means alone these coefficients increased to 0 90 and 0 93 when the pyramidal segmentation was combined with fuzzy c means this method can be applied to any dimensional representation and at any resolution level of an image series the evaluation study shows good performance in detecting lv lumen in mr images

a comparison between adomian decomposition method and taylor series method in the series solutions in this work we will compare the performance of the adomian decomposition method and the taylor series method applied to the solution of linear and nonlinear ordinary differential equations the comparison shows that the decomposition method is reliable efficient and easy to use from a computational viewpoint although the two methods provide the solution in an infinite series the decomposition method provides a fast convergent series of easily computable components and eliminates cumbersome computational work needed by taylor series method

efficient monte carlo and quasi monte carlo option pricing under the variance gamma model we develop and study efficient monte carlo algorithms for pricing path dependent options with the variance gamma model the key ingredient is difference of gamma bridge sampling based on the representation of a variance gamma process as the difference of two increasing gamma processes for typical payoffs we obtain a pair of estimators named low and high with expectations that 1 are monotone along any such bridge sampler and 2 contain the continuous time price these estimators provide pathwise bounds on unbiased estimators that would be more expensive infinitely expensive in some situations to compute by using these bounds with extrapolation techniques we obtain significant efficiency improvements by work reduction we then combine the gamma bridge sampling with randomized quasi monte carlo to reduce the variance and thus further improve the efficiency we illustrate the large efficiency improvements on numerical examples for asian lookback and barrier options

a markov random field model based approach to image interpretation an image is segmented into a collection of disjoint regions that form the nodes of an adjacency graph and image interpretation is achieved through assigning object labels or interpretations to the segmented regions or nodes using domain knowledge extracted feature measurements and spatial relationships between the various regions the interpretation labels are modeled as a markov random field mrf on the corresponding adjacency graph and the image interpretation problem is then formulated as a maximum a posteriori map estimation rule given domain knowledge and region based measurements simulated annealing is used to find this best realization or optimal map interpretation this approach also provides a systematic method for organizing and representing domain knowledge through appropriate design of the clique functions describing the gibbs distribution representing the pdf of the underlying mrf a general methodology is provided for the design of the clique functions results of image interpretation experiments on synthetic and real world images are described

random weights robust lattice rules and the geometry of the cbcrc algorithm in this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube 0 1 s from a weighted reproducing kernel hilbert space we assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications i it is usually not known in practice how to choose the weights thus by assuming that the weights are random variables we obtain robust constructions with respect to the weights of lattice rules this to some extend removes the necessity to carefully choose the weights ii in practice it is convenient to use the same lattice rule for many different integrands the best choice of weights for each integrand may vary to some degree hence considering the weights random variables does justice to how lattice rules are used in applications in this paper the worst case error is therefore a random variable depending on random weights we show how one can construct lattice rules which perform well for weights taken from a set with large measure such lattice rules are therefore robust with respect to certain changes in the weights the construction algorithm uses the component by component cbc idea based on two criteria one using the mean of the worst case error and the second criterion using a bound on the variance of the worst case error we call the new algorithm the cbc2c component by component with 2 constraints algorithm we also study a generalized version which uses r constraints which we call the cbcrc component by component with r constraints algorithm we show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights 1 r thus in applications instead of finding one set of weights it is enough to find a convex polytope in which the optimal weights lie the price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights

why are high dimensional finance problems often of low effective dimension many problems in mathematical finance can be formulated as high dimensional integrals where the large number of dimensions arises from small time steps in time discretization and or a large number of state variables quasi monte carlo qmc methods have been successfully used for approximating such integrals to understand this success this paper focuses on investigating the special features of some typical high dimensional finance problems namely option pricing and bond valuation we provide new insight into the connection between the effective dimension and the efficiency of qmc and we present methods to analyze the dimension structure of a function we confirm the observation of caflisch morokoff and owen that functions from finance are often of low effective dimension in the sense that they can be well approximated by their low order anova analysis of variance terms usually just the order 1 and order 2 terms we explore why the effective dimension is small for many integrals from finance by deriving explicit forms of the anova terms in simple cases we find that the importance of each dimension is naturally weighted by certain hidden weights these weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher order anova terms we study the variance ratios captured by low order anova terms and their asymptotic properties as the dimension tends to infinity and we show that with the increase of dimension the lower order terms continue to play a significant role and the higher order terms tend to be negligible this provides some insight into high dimensional problems from finance and explains why qmc algorithms are efficient for problems of this kind

randomized dimension reduction for monte carlo simulations we present a new unbiased algorithm that estimates the expected value of f u via monte carlo simulation where u is a vector of d independent random variables and f is a function of d variables

an auto realignment method in quasi monte carlo for pricing financial derivatives with jump structures abstract discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi monte carlo qmc method while if the discontinuities are parallel to the axes good efficiency of the qmc method can still be expected by realigning the discontinuities to be axes parallel wang tan 2013 succeeded in recovering the high efficiency of the qmc method for a special class of functions motivated by this work we propose an auto realignment method to deal with more general discontinuous functions the k means clustering algorithm a classical algorithm of machine learning is used to select the most representative normal vectors of the discontinuity surface by applying this new method the discontinuities of the resulting function are realigned to be friendly for the qmc method numerical experiments demonstrate that the proposed method significantly improves the performance of the qmc method

the computation of wavelet galerkin approximation on a bounded interval this paper describes exact evaluations of various finite integrals whose integrands involve products of daubechies compactly supported wavelets and their derivatives and or integrals these finite integrals play an essential role in the wavelet galerkin approximation of differential or integral equations on a bounded interval

cluster validity for the fuzzy c means clustering algorithrm the uniform data function is a function which assigns to the output of the fuzzy c means fc m or fuzzy isodata algorithm a number which measures the quality or validity of the clustering produced by the algorithm for the preselected number of cluster c the fc m algorithm produces c vectors in the space in which the data lie called cluster centers which represent points about which the data are concentrated it also produces for each data point c membership values numbers between zero and one which measure the similarity of the data points to each of the cluster centers it is these membership values which indicate how the point is classified they also indicate how well the point has been classified in that values close to one indicate that the point is close to a particular center but uniformly low memberships indicate that the point has not been classified clearly the uniform data functional udf combines the memberships in such a way as to indicate how well the data have been classified and is computed as follows for each data point compute the ratio of its smallest membership to its largest and then compute the probability that one could obtain a smaller ratio indicating better classification from a clustering of a standard data set in which there is no cluster structure these probabilities are then averaged over the data set to obtain the values of the udf

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

quasi random sequences and their discrepancies quasi random also called low discrepancy sequences are a deterministic alternative to random sequences for use in monte carlo methods such as integration and particle simulations of transport processes the error in uniformity for such a sequence of n points in the s dimensional unit cube is measured by its discrepancy which is of size log n s n 1 for large n as opposed to discrepancy of size log log n 1 2 n 1 2 for a random sequence i e for almost any randomly chosen sequence several types of discrepancies one of which is new are defined and analyzed a critical discussion of the theoretical bounds on these discrepancies is presented computations of discrepancies are presented for a wide choice of dimension s number of points n and different quasi random sequences in particular for moderate or large s there is an intermediate regime in which the discrepancy of a quasi random sequence is almost exactly the same as that of a randomly chosen sequence a simplified p

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

constructing sobol sequences with better two dimensional projections direction numbers for generating sobol sequences that satisfy the so called property a in up to 1111 dimensions have previously been given in joe and kuo acm trans math software 29 2003 pp 49 57 however these sobol sequences may have poor two dimensional projections here we provide a new set of direction numbers alleviating this problem these are obtained by treating sobol sequences in d dimensions as t d sequences and then optimizing the t values of the two dimensional projections our target dimension is 21201

acceleration of quasi monte carlo approximations with applications in mathematical finance a procedure to accelerate the convergence of quasi monte carlo approximations is described and investigated the procedure applies a weighted least squares smoothing method to improve the approximations that were obtained using low discrepancy sequences the acceleration procedure is tested computationally on several numerical integration problems including two problems of interest in mathematical finance the results indicate that when halton low discrepancy points are used in the quasi monte carlo method the accelerated sequence often converges a factor of five or more faster than the original quasi monte carlo approximations

efficient algorithms for computing the l 2 discrepancy the l 2 discrepancy is a quantitative measure of precision for multivariate quadrature rules it can be computed explicitly previously known algorithms needed o m 2 operations where m is the number of nodes in this paper we present algorithms which require o m log m d operations

on the step by step construction of quasi monte carlo integration rules that achieve strong tractability error bounds in weighted sobolev spaces we develop and justify an algorithm for the construction of quasi monte carlo qmc rules for integration in weighted sobolev spaces the rules so constructed are shifted rank 1 lattice rules the parameters characterising the shifted lattice rule are found component by component the d 1 th component of the generator vector and the shift are obtained by successive 1 dimensional searches with the previous d components kept unchanged the rules constructed in this way are shown to achieve a strong tractability error bound in weighted sobolev spaces a search for n point rules with n prime and all dimensions 1 to d requires a total cost of o n3d2 operations this may be reduced to o n3d operations at the expense of o n2 storage numerical values of parameters and worst ease errors are given for dimensions up to 40 and n up to a few thousand the worst case errors for these rules are found to be much smaller than the theoretical bounds

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

fast convergence of quasi monte carlo for a class of isotropic integrals we consider the approximation ofd dimensional weighted integrals of certain isotropic functions we are mainly interested in cases where d is large we show that the convergence rate of quasi monte carlo for the approximation of these integrals iso p logn n since this is a worst case result compared to the expected convergence rate o n 1 2 of monte carlo it shows the superiority of quasi monte carlo for this type of integral this is much faster than the worst case convergence o log d n n of quasi monte carlo

galerkin wavelet methods for two point boundary value problems anti derivatives of wavelets are used for the numerical solution of differential equations optimal error estimates are obtained in the applications to two point boundary value problems of second order the orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems numerical examples are given

pricing asian options via compound gamma and orthogonal polynomials we develop two approximations cg3 and cgn for pricing arithmetic asian options approximation cg3 utilizes a compound gamma distribution of the price average it is calibrated by analytically matching the first three moments approximation cg3 outperforms many other existing approximations approximation cgn expands cg3 by utilizing the concept of orthogonal polynomials and matches n 3 moments it produces near exact results within 1 s when volatility and maturity are not too high two useful insights of our work are i demonstrating that orthogonal polynomials can be used effectively to improve accuracy and ii showing that matching higher moments is beneficial