flux corrected pseudospectral method for scalar hyperbolic conservation laws abstract the pseudospectral method has under used advantages in problems involving shocks and discontinuities these emerge from superior accuracy in phase and group velocities as compared to finite difference schemes of all orders dispersion curves for finite difference schemes suggest that group velocity error typically outranks gibbs error as a cause of numerical oscillation a flux conservative form of the pseudospectral method is derived for compatibility with flux limiters used to preserve monotonicity the resulting scheme gives high quality results in linear advection and shock formation propagation examples

a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

essentially nonoscillatory spectral fourier methods for shock wave calculations in this paper we present an essentially nonoscillatory spectral fourier method for the solution of hyperbolic partial differential equations the method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the fourier method the high accuracy away from the shock is enhanced by using filters numerical results confirm that essentially no oscillations develop in the solution also the accuracy of the spectral solution of the inviscid burgers equation is shown to be higher than a fixed order

recovering exponential accuracy from collocation point values of smooth functions with end point singularities gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity the truncated spectral series has large oscillations near the jump and the overshoot does not decay as the number of terms in the truncated series increases there is therefore no convergence in the maximum norm and convergence in smooth regions away from the discontinuity is also slow in gottlieb and shu 1995 5 a methodology is proposed to completely overcome this difficulty in the context of spectral collocation methods resulting in the recovery of exponential accuracy from collocation point values of a piecewise analytic function in this paper we extend this methodology to handle spectral collocation methods for functions which are analytic in the open interval but have singularities at end points with this extension we are able to obtain exponential accuracy from collocation point values of such functions similar to gottlieb and shu 1995 5 the proof is constructive and uses the gegenbauer polynomials c n l x the result implies that the gibbs phenomenon can be overcome for smooth functions with endpoint singularities

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

an averaging method for the fourier approximation to discontinuous functions both the truncated fourier integral and the truncated fourier series approximations for a discontinuous function bring about the inevitable oscillating error say the gibbs phenomenon most basic filtering methods for the gibbs phenomenon like the fejer averaging method and the lanczos averaging method have a disadvantage that the rise time is very slow even though the filtering effect is prominent away from the discontinuity in this paper we propose a new averaging method of polynomial type which improves the rise time of the existing method the present method can be regarded as a generalization of the traditional lanczos averaging method by several numerical examples we show the efficiency of the present method

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

spectral simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

a fast spectral solver for a 3d helmholtz equation we present a fast solver for the helmholtz equation delta u pm lambda 2 u f in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows us to reduce the errors associated with the gibbs phenomenon and achieve any prescribed rate of convergence the algorithm requires o n3 log n operations where n is the number of grid points in each direction we solve a dirichlet boundary problem for the helmholtz equation we also extend the method to the solution of mixed problems where dirichlet boundary conditions are specified on some faces and neumann boundary conditions are specified on other faces high order accuracy is achieved by a comparatively small number of points for example for the accuracy of 10 8 the resolution of only 16 32 points in each direction is necessary

optimal filter and mollifier for piecewise smooth spectral data we discuss the reconstruction of piecewise smooth data from its pseudo spectral information spectral projections enjoy superior resolution provided the function is globally smooth while the presence of jump discontinuities is responsible for spurious o 1 gibbs oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order classical filters and mollifiers are constructed to have compact support in the fourier frequency and physical time spaces respectively and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region here we construct a noncompactly supported filter and mollifier with optimal joint time frequency localization for a given number of vanishing moments resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains not giving preference to either space allows for a more balanced error decomposition which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters yet obtain true exponential accuracy

approximating the jump discontinuities of a function by its fourier jacobi coefficients in the present paper we generalize eckhoff s method i e the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its fourier chebyshev coefficients a new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function which belongs to a restricted class of the piecewise smooth functions by means of its fourier jacobi coefficients for arbitrary indices approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations it is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities in addition we study the accuracy of the approximations and present some numerical examples

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

an assessment of spectral nonoscillatory schemes a new spectral nonoscillatory interpolation scheme is proposed that achieves spectral accuracy in smooth regions and is nonoscillatory on piecewise discontinuous data the essential idea behind the scheme is to increase the order of an eno scheme in proportion to the number of points wherever possible numerical experiments with the new scheme on interpolation 1 d advection inviscid burgers equation and 1 d gas dynamics confirm the high resolution features of the scheme comparisons with the results of earlier spectral nonoscillatory schemes show that the new scheme is competitive both in efficiency and accuracy

time domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method the acoustical wave propagator awp scheme involves an effective time domain calculation of sound propagation using the combination of chebyshev polynomial expansion and the fourier pseudospectral method the accuracy of this scheme degrades when the media has discontinuities due to the well known gibbs phenomenon in this paper several issues concerning awp are addressed including an analysis of the effect of gibbs phenomenon on the accuracy a mapped pseudospectral method is proposed wherein the grid points are redistributed with the emphasis across the media discontinuities by a pre determined smooth mapping curve then the spatial derivatives are calculated through a modified fourier pseudospectral method using this method the influence of the gibbs phenomenon is effectively alleviated while the computational efficiency of awp is still maintained the superiority of this improved awp scheme is illustrated by three one dimensional 1 d numerical examples

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

gibbs phenomenon for fourier partial sums on zp abstract it is of interest to know whether the gibbs phenomenon occurs on a local field a p adic heaviside function on the group of p adic integers is defined as an analogy of real variable heaviside function and it is also shown that there exists the gibbs phenomenon with an undershoot of at least 1 p 1 as an approximate level as a consequence of the theorem we see that the fourier partial sum for a heaviside function does not converge at the point of discontinuity

an investigation of resolution of 2 variate gibbs phenomenon abstract in this paper we consider an implementation approach of a method for the resolution of the gibbs phenomenon for f x y which has been discussed extensively by jung and shizgal j h jung b d shizgal generalization of the inverse polynomial reconstruction method in the resolution of the gibbs phenomenon j comput appl math 172 2004 131 151 j h jung b d shizgal inverse polynomial reconstruction of two dimensional fourier images j sci comput 25 3 2005 367 399 b d shizgal j h jung towards the resolution of the gibbs phenomenoa j comput appl math 161 2003 41 65 here we go through some details of applying that technique and also we try to work out its application on some time dependent partial differential equations numerical experiments confirm the efficiency of the method

spectral element fct method for the one and two dimensional compressible euler equations abstract a hybrid spectral element fct method is proposed for the solution of the euler equations of gas dynamics in one and two space dimensions it is based on a new conservative formulation starting from standard spectral method expansions and incorporating ideas of modern finite volume schemes a staggered mesh is employed in the discretization consisting of gauss chebyshev and gauss lobatto chebyshev collocation points for the calculation of averaged quantities and fluxes respectively use of limiting is based on the flux corrected transport algorithm the outcome is a geometrically flexible scheme with optimal phase properties both one and two dimensional results are presented in which the gibbs phenomenon is absent while all transition zones are exceptionally sharp

a strategy for choosing gegenbauer reconstruction parameters for numerical stability the gegenbauer reconstruction method was first proposed in 1992 but in early studies no attempts were made to optimize the relevant parameters of this method these parameters were allowed to grow proportionally with the number of nodes which in many cases resulted in exponential convergence for a selected range of the proportionality constants early studies also made clear that very large error bounds could be expected if these key parameters were not chosen carefully subsequent studies then pointed out that although unrelated to the method s analytically predictable domains of poor accuracy round off errors could also sabotage the method s accuracy the challenge of successfully implementing a gegenbauer reconstruction then rests on understanding the performance trade offs we can expect when choosing the key parameters in accordance with different objectives in this study we propose a new strategy for choosing optimal parameters in the chebyshev gegenbauer reconstruction method specifically to achieve numerical stability this strategy is based on asymptotic analysis as well as minimization problems in one and two dimensions the effectiveness of our approach which could also be applied to a wider selection of polynomials is then illustrated with results from numerical experiments

restricting invariants of unitary reflection groups suppose that g is a finite unitary reflection group acting on a complex vector space v and x is the fixed point subspace of an element of g define n to be the setwise stabilizer of x in g z to be the pointwise stabilizer and c n z then restriction defines a homomorphism from the algebra of g invariant polynomial functions on v to the algebra of c invariant functions on x extending earlier work by douglass and rohrle for coxeter groups we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups g in terms of the exponents of g and c and their reflection arrangements a consequence of our main result is that the variety of g orbits in the g saturation of x is smooth if and only if it is normal

resolution properties of the fourier method for discontinuous waves in this paper we discuss the wave resolution properties of the fourier approximations of a wave function with discontinuities it is well known that a minimum of two points per wave is needed to resolve a periodic wave function using fourier expansions for chebyshev approximations of a wave function a minimum of points per wave is needed see numerical analysis of spectral methods theory and applications by d gottlieb and s orszag here we obtain an estimate for the minimum number of points per wave to resolve a discontinuous wave based on its fourier coefficients in our recent work on overcoming the gibbs phenomenon we have shown that the fourier coefficients of a discontinuous function contain enough information to reconstruct with exponential accuracy the coefficients of a rapidly converging gegenbauer expansion we therefore study the resolution properties of a gegenbauer expansion where both the number of terms and the order increase

leave one out approach for matrix completion primal and dual analysis in this paper we introduce a powerful technique based on leave one out analysis to the study of low rank matrix completion problems using this technique we develop a general approach for obtaining fine grained entrywise bounds for iterative stochastic procedures in the presence of probabilistic dependency we demonstrate the power of this approach in analyzing two of the most important algorithms for matrix completion i the non convex approach based on projected gradient descent pgd for a rank constrained formulation also known as the singular value projection algorithm and ii the convex relaxation approach based on nuclear norm minimization nnm using this approach we establish the first convergence guarantee for the original form of pgd without regularization or sample splitting and in particular shows that it converges linearly in the infinity norm for nnm we use this approach to study a fictitious iterative procedure that arises in the dual analysis our results show that nnm recovers an d by d rank r matrix with mathcal o mu r log mu r d log d observed entries this bound has optimal dependence on the matrix dimension and is independent of the condition number to the best of our knowledge none of previous sample complexity results for tractable matrix completion algorithms satisfies these two properties simultaneously

reflection subgroups of finite complex reflection groups abstract all simple extensions of the reflection subgroups of a finite complex reflection group g are determined up to conjugacy as a consequence it is proved that if the rank of g is n and if g can be generated by n reflections then for every set r of n reflections which generate g every subset of r generates a parabolic subgroup of g

the remainder term for analytic functions of gauss radau and gauss lobatto quadrature with multiple end points a study is undertaken of the kernels in the contour integral representations of the remainder terms for gauss radau and gauss lobatto quadrature rules over the interval 1 1 it is assumed that the respective end points in these rules have multiplicity two and that integration is with respect to one of the four chebyshev weight functions of particular interest is the location on the contour where the modulus of the kernel attains its maximum value only elliptic contours are considered having foci at the points 1

iterative methods based on polynomial interpolation filters to detect discontinuities and recover point values from fourier data in previous papers we proposed new filters based on polynomial interpolation to approximate the point values of a piecewise smooth function f on 0 1 from its fourier coefficients and derived error estimates we proved that if all the discontinuity points of f are nodes we can reconstruct point values of f accurately even close to the discontinuities we use the new filters to develop iterative methods for detecting the discontinuity points and therefore accurately approximate the point values of the function from its fourier coefficients

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

on the construction of highly symmetric tight frames and complex polytopes abstract many highly symmetric configurations of vectors in c d such as the vertices of the platonic solids and the regular complex polytopes are equal norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group for nonabelian groups there are uncountably many such tight frames up to unitary equivalence the aim of this paper is to single out those orbits which are particularly nice such as those which are the vertices of a complex polytope this is done by defining a finite class of tight frames of n vectors for c d n and d fixed which we call the highly symmetric tight frames we outline how these frames can be calculated from the representations of abstract groups using a computer algebra package we give numerous examples with a special emphasis on those obtained from the shephard todd finite reflection groups the interrelationships between these frames with complex polytopes harmonic frames equiangular tight frames and heisenberg frames maximal sets of equiangular lines are explored in detail

arrangements defined by unitary reflection groups let v be a complex vector space of dimension i an arrangement in v is a finite set d of hyperplanes all containing the origin let l l d be the set of intersections of elements of r partially order l by reverse inclusion so that l has v as its minimal element and d as its set of atoms the poset l is a finite geometric lattice with rank function r x dim v x xe l without loss of generality we assume that n h h 0 is the maximal element of l and thus l has rank i the characteristic polynomial z l t of l is defined by

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

a spectral method of characteristics for hyperbolic problems a new numerical technique is introduced for first order hyperbolic equations based on combining the spectral method with explicit timestepping along the characteristics which permits the use of large timesteps at no cost in accuracy the method is unconditionally stable and spectrally accurate the theoretical results are illustrated by numerical experiments

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

lagrange interpolation for continuous piecewise smooth functions this note is devoted to lagrange interpolation for continuous piecewise smooth functions a new family of interpolatory functions with explicit approximation error bounds is obtained we apply the theory to the classical lagrange interpolation

on discontinuous solutions of hyperbolic equations abstract a modified fourier method which is applicable to numerical simulations of discontinuous solutions of hyperbolic equations is presented truncated fourier series representations are de truncated at every timestep by utilising step functions in the reconstruction of the discontinuous solutions the de truncated expansions are used to eliminate the numerical dispersion and diffusion which normally ruin the discontinuous solutions in a few timesteps when traditional fourier methods are applied

energy based comparison between the fourier galerkin method and the finite element method abstract the fourier galerkin method in short ffth has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom because the method incorporates the fast fourier transform fft in the linear solver it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method fem here we systematically compare these two methods using the energetic norm of local fields which has the clear physical interpretation as being the error in the homogenised properties this enables the comparison of memory and computational requirements at the same level of approximation accuracy we show that the methods effectiveness relies on the smoothness regularity of the solution and thus on the material coefficients thanks to its approximation properties fem outperforms ffth for problems with jumps in material coefficients while ambivalent results are observed for the case that the material coefficients vary continuously in space ffth profits from a good conditioning of the linear system independent of the number of degrees of freedom but generally needs more degrees of freedom to reach the same approximation accuracy more studies are needed for other fft based schemes non linear problems and dual problems which require special treatment in fem but not in ffth

accuracy of high order and spectral methods for hyperbolic conservation laws with discontinuous solutions higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions on the other hand higher order methods have been applied to hyperbolic problems with less success as higher order approx imations of discontinuous solutions suffer from the gibbs phenomenon we extend past work and show that spectral methods yield spectral convergence of moments even when applied to problems with discontinuous solutions besides spectral fourier methods for periodic domains we also prove high order convergence for adjoint consistent non periodic numerical methods exemplified by the discontinuous galerkin finite element method

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

analysis and application of fourier gegenbauer method to stiff differential equations the fourier gegenbauer fg method introduced by gottlieb shu solomonoff and vandeven icase report 92 4 hampton va 1992 is aimed at removing the gibbs phenomenon that is recovering the point values of a nonperiodic function from its fourier coefficients in this paper we discuss some numerical aspects of the fg method related to its em pseudospectral implementation in particular we analyze the behavior of the gegenbauer series with a moderate several hundred number of terms suitable for computations we also demonstrate the ability of the fg method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles bearing on the previous analysis we suggest a high order spectral fourier method for the solution of nonperiodic differential equations it includes a polynomial subtraction technique to accelerate the convergence of the fourier series and the fg algorithm to evaluate derivatives on the boundaries of nonperiodic functions the present hybrid fourier gegenbauer hfg method possesses better resolution properties than the original fg method the precision of this method is demonstrated by solving stiff elliptic problems with steep solutions

accurate reconstructions of functions of finite regularity from truncated fourier series expansions kowledge of a truncated fourier series expansion for a 2 periodic function of finite regularity which is assumed to be piecewise smooth in each period is used to accurately reconstruct the corresponding function an algebraic equation of degree m is constructed for the m singularity locations in each period for the function in question the m coefficients in this algebraic equation are obtained by solving an algebraic system of m equations determined by the coefficients in the known truncated expansion if discontinuities in the derivatives of the function are considered in addition to discontinuities in the function itself that algebraic system will be nonlinear with respect to the m unknown coefficients the degree of the algebraic system will depend on the desired order of accuracy for the reconstruction i e a higher degree will normally lead to a more accurate determination of the singularity locations by solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations we are able to reconstruct the 2 periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

local approximation on surfaces with discontinuities given limited order fourier coefficients we present a spline approximation method for a piece of a surface where jump discontinuities occur along curves the data for the surface is assumed to be fourier coefficients which are limited in order and possibly contaminated with noise the support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved discontinuities of the surface may occur across the curved side and linear sides adjacent to it the approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve along each line a one dimensional spline approximation is done for a section of the surface over the line this approximation uses two dimensional fourier coefficient data localizing spline functions and a technique which we developed earlier for one dimensional analogues of the problem we use a spline quasi interpolation scheme to create a surface approximation from the section approximations the result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise analytic function we continue the investigation of overcoming gibbs phenomenon i e obtaining exponential accuracy at all points including at the discontinuities themselves from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function we show that if we are given the first n expansion coefficients of an l 2 function f x in terms of either the trigonometrical polynomials or the chebyshev or legendre polynomials we can construct an exponentially convergent approximation to the point values of f x in any sub interval in which it is analytic

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

a stability barrier for reconstructions from fourier samples we prove that any stable method for resolving the gibbs phenomenon that is recovering high order accuracy from the first m fourier coefficients of an analytic and nonperiodic function can converge at best root exponentially fast in m any method with faster convergence must also be unstable and in particular exponential convergence implies exponential ill conditioning this result is analogous to a recent theorem of platte trefethen and kuijlaars concerning recovery from pointwise function values on an equispaced m grid the main step in our proof is an estimate for the maximal behavior of a polynomial of degree n with bounded m term fourier series this estimate is related to a conjecture of hrycak and grochenig in the final part of the paper we discuss the implications of our main theorem to polynomial based interpolation and least squares approaches for overcoming the gibbs phenomenon

an improved estimate of pswf approximation and approximation by mathieu functions abstract in this paper an error estimate of spectral approximations by prolate spheroidal wave functions pswfs with explicit dependence on the bandwidth parameter and optimal order of convergence is derived which improves the existing result in chen et al spectral methods based on prolate spheroidal wave functions for hyperbolic pdes siam j numer anal 43 5 2005 1912 1933 the underlying argument is applied to analyze spectral approximations of periodic functions by mathieu functions which leads to new estimates featured with explicit dependence on the intrinsic parameter

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

the remainder term for analytic functions of gauss radau and gauss lobatto quadrature rules with multiple end points a study is undertaken of the kernels in the contour integral representations of the remainder terms for gauss radau and gauss lobatto quadrature rules over the interval 1 1 it is assumed that

gibbs phenomenon free fourier series for vibration and stability of complex beams the gibbs phenomenon in fourier series has long been recognized as a drawback in its applications in particular when it is used to represent a function having discontinuities one category namely intrinsic discontinuities could be derived from the nature of the physical problems another namely inherent discontinuities is created undesirably through the employment of fourier series to alleviate these drawbacks two techniques are developed the first aims at eliminating the inherent discontinuities the size of the original domain 0 l is first doubled with a virtual function 0 defined over the neighboring domain l 0 the augmented domain l l is then further extended periodically the virtual function 0 y is chosen so that continuities are achieved at y 0 l 2l consequently no gibbs phenomena will occur at the boundaries the second aims at reproducing the intrinsic discontinuities in the representation function by incorporating piecewise cubic polynomials into the fourier base function the enlarged basis function namely the gibbs phenomenon free fourier series function is able to represent accurately any specific boundary conditions and interior intrinsic discontinuous conditions examples of vibration and buckling analysis of complex beams show that the present method is versatile accurate and efficient

detecting the singularities of a function of vp class by its integrated fourier series in the present paper we pursue the general idea suggested in our previous work namely we utilize the truncated fourier series as a tool for the approximation of the points of discontinuities and the magnitudes of jumps of a 2 periodic bounded function earlier we used the derivative of the partial sums while in this work we use integrals first we obtain new identities which determine the jumps of a 2 periodic function of vp 1 p 2 class with a finite number of discontinuities by means of the tails of its integrated fourier series next based on these identities we establish asymptotic expansions for the approximations of the location of the discontinuity and the magnitude of the jump of a 2 periodic piecewise smooth function with one singularity by an appropriate linear combination obtained via integrals of different order we significantly improve the accuracy of the initial approximations then we apply richardson s extrapolation method to enhance the approximation results for a function with multiple discontinuities we use simple formulae which eliminate all discontinuities of the function but one then we treat the function as if it had one singularity finally we give the description of a programmable algorithm for the approximation of the discontinuities investigate the stability of the method study its complexity and present some numerical results

comparison of deterministic and stochastic methods for time dependent wigner simulations recently a monte carlo method based on signed particles for time dependent simulations of the wigner equation has been proposed while it has been thoroughly validated against physical benchmarks no technical study about its numerical accuracy has been performed to this end this paper presents the first step towards the construction of firm mathematical foundations for the signed particle wigner monte carlo method an initial investigation is performed by means of comparisons with a cell average spectral element method which is a highly accurate deterministic method and utilized to provide reference solutions several different numerical tests involving the time dependent evolution of a quantum wave packet are performed and discussed in deep details in particular this allows us to depict a set of crucial criteria for the signed particle wigner monte carlo method to achieve a satisfactory accuracy

optimal rates of convergence for noisy sparse phase retrieval via thresholded wirtinger flow this paper considers the noisy sparse phase retrieval problem recovering a sparse signal mathbf x in mathbb r p from noisy quadratic measurements y j mathbf a j mathbf x 2 varepsilon j j 1 ldots m with independent sub exponential noise varepsilon j the goals are to understand the effect of the sparsity of mathbf x on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates adaptively inspired by the wirtinger flow ieee trans inform theory61 2015 1985 2007 proposed for non sparse and noiseless phase retrieval a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the mathbf a j s are independent standard gaussian random vectors provided that the sample size is sufficiently large compared to the sparsity of mathbf x

nonconvex rectangular matrix completion via gradient descent without l regularization the analysis of nonconvex matrix completion has recently attracted much attention in the community of machine learning thanks to its computational convenience existing analysis on this problem however usually relies on ell 2 infty projection or regularization that involves unknown model parameters although they are observed to be unnecessary in numerical simulations in this paper we extend the analysis of the vanilla gradient descent for positive semidefinite matrix completion in the literature to the rectangular case and more significantly improve the required sampling rate from o mathrm poly kappa mu 3 r 3 log 3 n n to o mu 2 r 2 kappa 14 log n n our technical ideas and contributions are potentially useful in improving the leave one out analysis in other related problems

spectrally accurate solution of nonperiodic differential equations by the fourier gegenbauer method it is well known that the fourier partial sum of an analytic nonperiodic function supported on a finite interval converges slowly inside the interval and exhibits o 1 spurious oscillations near the boundaries the gibbs phenomenon an effective algorithm which allows one to completely overcome the gibbs phenomenon was developed in j comput appl math 43 1992 pp 81 92 the basic concept of this approach consists of the re expansion of the fourier partial sums into the rapidly convergent gegenbauer series in this paper we extend the fourier gegenbauer f g method of j comput appl math 43 1992 pp 81 92 to the evaluation of the spatial derivatives of a piecewise analytic function also we apply this method to the solution of nonperiodic boundary value problems although the derivatives of a discontinuous function are not in l2 the exponential convergence of the truncated gegenbauer series can be proved and the rate of convergence can be estimated the solution of differential equations is accomplished in two steps first a particular solution with arbitrary boundary conditions is constructed using the f g method this particular solution is then corrected to satisfy the prescribed boundary conditions of the problem by adding a proper linear combination of homogeneous solutions for boundary layer problems the intermediate particular solution has steep profiles near the boundaries these steep regions introduce a large error into the final solution which presumably has a smooth profile on the whole interval a method which compensates for this loss of accuracy by using the appropriately constructed homogeneous solutions is proposed

the resolution of the gibbs phenomenon for spliced functions in one and two dimensions abstract in this paper we study approximation methods for analytic functions that have been spliced into nonintersecting subdomains we assume that we are given the first 2n 1 fourier coefficients for the functions in each subdomain the objective is to approximate the spliced function in each subdomain and then to glue the approximations together in order to recover the original function in the full domain the fourier partial sum approximation in each subdomain yields poor results as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain thus once we glue the subdomain approximations back together the approximation for the function in the full domain will exhibit oscillations throughout the entire domain recently methods have been developed that successfully eliminate the gibbs phenomenon for analytic but nonperiodic functions in one dimension these methods are based on the knowledge of the first 2n 1 fourier coefficients and use either the gegenbauer polynomials gottlieb et al or the bernoulli polynomials abarbanel gottlieb cai et al and eckhoff we propose a way to accurately reconstruct a spliced function in a full domain by extending the current methods to eliminate the gibbs phenomenon in each nonintersecting subdomain and then gluing the approximations back together we solve this problem in both one and two dimensions in the one dimensional case we provide two alternative options the bernoulli method and the gegenbauer method as well as a new hybrid method the gegenbauer bernoulli method in the two dimensional case we prove for the very first time exponential convergence of the gegenbauer method and then we apply it to solve the spliced function problem

on robust regression with high dimensional predictors we study regression m estimates in the setting where p the number of covariates and n the number of observations are both large but p n we find an exact stochastic representation for the distribution of argmin p i 1 n y i x i at fixed p and n under various assumptions on the objective function and our statistical model a scalar random variable whose deterministic limit r can be studied when p n 0 plays a central role in this representation we discover a nonlinear system of two deterministic equations that characterizes r interestingly the system shows that r depends on through proximal mappings of as well as various aspects of the statistical model underlying our study several surprising results emerge in particular we show that when p n is large enough least squares becomes preferable to least absolute deviations for double exponential errors

cell averaging chebyshev methods for hyperbolic problems this paper describes a cell averaging method for the chebyshev approximations of first order hyperbolic equations in conservation form we present formulas for transforming between pointwise data at the collocation points and cell averaged quantities and vice versa this step trivial for the finite difference and fourier methods is nontrivial for the global polynomials used in spectral methods we then prove that the cell averaging methods presented are stable for linear scalar hyperbolic equations and present numerical simulations of shock density wave interaction using the new cell averaging chebyshev methods

cuspidal calogero moser and lusztig families for coxeter groups the goal of this paper is to compute the cuspidal calogero moser families for all infinite families of finite coxeter groups at all parameters we do this by first computing the symplectic leaves of the associated calogero moser space and then by classifying certain rigid modules numerical evidence suggests that there is a very close relationship between calogero moser families and lusztig families our classification shows that additionally the cuspidal calogero moser families equal cuspidal lusztig families for the infinite families of coxeter groups

a robust method for accurately representing nonperiodic functions given fourier coefficient information when a function is smooth but not smoothly periodic with a particular period and nonetheless is represented by partial sums of a fourier series calculated using that period the well known gibbs phenomenon defeats uniform convergence of the sums and convergence is slow in recent years several workers have developed methods for recovering accurate and fast converging representations for functions in this situation these efforts have not concentrated on bounds for the operators corresponding to the methods and thus have not explicitly proven robustness in the presence of noise in this paper we present a method for which explicit bounds are established for the operator the method is in effect least squares fitting of the given fourier coefficients by the coefficients of polynomial splines with appropriate discontinuities we obtain bounds by exact calculations of projections in spline spaces using a computer algebra system we give examples of the method and two other published methods working with noisy data

asymptotic analysis for singularly perturbed convection diffusion equations with a turning point turning points occur in many circumstances in fluid mechanics when the viscosity is small very complex phenomena can occur near turning points which are not yet well understood a model problem corresponding to a linear convection diffusion equation e g suitable linearization of the navier stokes or benard convection equations is considered our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the navier stokes or benard convection equations of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems in our case at this stage the full theoretical asymptotic analysis is provided this study is totally new to the best of our knowledge numerical treatment and more complex problems will be considered elsewhere

approximation of the discontinuities of a function by its classical orthogonal polynomial fourier coefficients in the present paper we generalize the method suggested in an earlier paper by the author and overcome its main deficiency first we modify the well known prony method which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its fourier coefficients with respect to any system of the classical orthogonal polynomials next we will show that the method is applicable to a wider class of functions namely to the class of piecewise smooth functions for functions which piecewise belong to c 2 1 1 the locations of discontinuities are approximated to within o 1 n by means of their fourier jacobi coefficients unlike the previous one the generalized method is robust since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial in addition the error estimate is uniform for any c d c 1 1 to the end we discuss the accuracy stability and complexity of the method and present numerical examples

trouble with gegenbauer reconstruction for defeating gibbs phenomenon runge phenomenon in the diagonal limit of gegenbauer polynomial approximations to defeat gibbs phenomenon in fourier and chebyshev series gottlieb et al d gottlieb c w shu a solomonoff h vandeven on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function j comput appl math 43 1992 81 98 developed a gegenbauer reconstruction the partial sums of the fourier or other spectral series are reexpanded as a series of gegenbauer polynomials c n m x recovering spectral accuracy even in the presence of shock waves or other discontinuities to achieve a rate of convergence which is exponential in n however gegenbauer reconstruction requires increasing the order m of the polynomials linearly with the truncation n of the series m bn for some constant b 0 when the order m is fixed it is well known that the gegenbauer series converges as n everywhere on x 1 1 if f x the function being expanded is analytic on the interval but what happens in the diagonal limit in which m n tend to infinity simultaneously we show that singularities of f x off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x 1 1 gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation n this off axis singularity constraint is likely to impair the effectiveness of the reconstruction in some applications

normalisers of parabolic subgroups in finite unitary reflection groups abstract it is well known that the normaliser of a parabolic subgroup of a finite coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots we show that a similar result holds for all finite unitary reflection groups namely the normaliser of a parabolic subgroup is a semidirect product and in many but not all cases the complement can be obtained as the stabiliser of a set of roots in addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup

on one sided filters for spectral fourier approximations of discontinuous functions in this paper the existence of one sided filters is proved for spectral fourier ap proximations of discontinuous functions which can recover spectral accuracy up to the discontinuity from one side a least square procedure is used to construct such a filter and is tested on several discontinuous functions numerically

spectral methods for the euler equations ii chebyshev methods and shock fitting on exprime chaque composante de la vitesse dans les equations d euler par une somme de polynomes de tchebycheff qui constitue une fonction d interpolation on utilise les expressions spectrales ainsi obtenues pour calculer plusieurs ecoulements gazeux bidimensionnels par la methode d ajustement d ondes de choc selon cette methode les conditions initiales correspondent a une onde de choc plane normale a un ecoulement supersonique et on determine la forme veritable de l onde a l aide des formules de rankine hugoniot qui expriment les conditions aux limites

sharp nonasymptotic bounds on the norm of random matrices with independent entries we obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results if xx is the n nn n symmetric matrix with xij n 0 b2ij xij n 0 bij2 we show that this bound is optimal in the sense that a matching lower bound holds under mild assumptions and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum analogous results are obtained for rectangular matrices and for more general sub gaussian or heavy tailed distributions of the entries and we derive tail bounds in addition to bounds on the expected norm the proofs are based on a combination of the moment method and geometric functional analysis techniques as an application we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse wigner matrices we also recover a result of seginer on the norm of rademacher matrices

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

entrywise eigenvector analysis of random matrices with low expected rank recovering low rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning such as in factor analysis community detection ranking matrix completion among others while a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors few results are tight for entrywise analyses which are critical for a number of problems such as community detection this paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low rank which helps settle the conjecture in abbe bandeira and hall 2014 that the spectral algorithm achieves exact recovery in the stochastic block model without any trimming or cleaning steps the key is a first order approximation of eigenvectors under the ell infty norm begin equation u k approx frac au k lambda k end equation where u k and u k are eigenvectors of a random matrix a and its expectation mathbb e a respectively the fact that the approximation is both tight and linear in a facilitates sharp comparisons between u k and u k in particular it allows for comparing the signs of u k and u k even if u k u k infty is large the results are further extended to perturbations of eigenspaces yielding new ell infty type bounds for synchronization mathbb z 2 spiked wigner model and noisy matrix completion

incoherence optimal matrix completion this paper considers the matrix completion problem we show that it is not necessary to assume joint incoherence which is a standard but unintuitive and restrictive condition that is imposed by previous studies this leads to a sample complexity bound that is orderwise optimal with respect to the incoherence parameter as well as to the rank r and the matrix dimension n up to a log factor as a consequence we improve the sample complexity of recovering a semidefinite matrix from o nr 2 log 2 n to o nr log 2 n and the highest allowable rank from theta sqrt n log n to theta n log 2 n the key step in proof is to obtain new bounds in terms of the ell infty 2 norm defined as the maximum of the row and column norms of a matrix to illustrate the applicability of our techniques we discuss extensions to singular value decomposition projection structured matrix completion and semisupervised clustering for which we provide orderwise improvements over existing results finally we turn to the closely related problem of low rank plus sparse matrix decomposition we show that the joint incoherence condition is unavoidable here for polynomial time algorithms conditioned on the planted clique conjecture this means it is intractable in general to separate a rank omega sqrt n positive semidefinite matrix and a sparse matrix interestingly our results show that the standard and joint incoherence conditions are associated respectively with the information statistical and computational aspects of the matrix decomposition problem