cubic pencils of lines and bivariate interpolation cubic pencils of lines are classified up to projectivities explicit formulae for the addition of lines on the set of nonsingular lines of the pencils are given these formulae can be used for constructing planar generalized principal lattices which are sets of points giving rise to simple lagrange formulae in bivariate interpolation special attention is paid to the irreducible nonsingular case where elliptic functions are used in order to express the addition in a natural form

on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

message passing for maximum weight independent set in this paper we investigate the use of message passing algorithms for the problem of finding the max weight independent set mwis in a graph first we study the performance of the classical loopy max product belief propagation we show that each fixed point estimate of max product can be mapped in a natural way to an extreme point of the linear programming lp polytope associated with the mwis problem however this extreme point may not be the one that maximizes the value of node weights the particular extreme point at final convergence depends on the initialization of max product we then show that if max product is started from the natural initialization of uninformative messages it always solves the correct lp if it converges this result is obtained via a direct analysis of the iterative algorithm and cannot be obtained by looking only at fixed points the tightness of the lp relaxation is thus necessary for max product optimality but it is not sufficient motivated by this observation we show that a simple modification of max product becomes gradient descent on a smoothed version of the dual of the lp and converges to the dual optimum we also develop a message passing algorithm that recovers the primal mwis solution from the output of the descent algorithm we show that the mwis estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the mwis is unique finally we show that any problem of maximum a posteriori map estimation for probability distributions over finite domains can be reduced to an mwis problem we believe this reduction will yield new insights and algorithms for map estimation

properly posed sets of nodes for multivariate lagrange interpolation in c s in this paper we apply techniques from the theory of ideals and varieties in algebraic geometry to study the geometric structure of a properly posed set of nodes or ppsn for short for multivariate lagrange interpolation along an algebraic hypersurface we provide a hyperplane superposition process to construct the ppsn for interpolation along an algebraic hypersurface and as a result we offer a clear understanding of the geometric structure of the ppsn for multivariate lagrange interpolation in cs

a regularization parameter for nonsmooth tikhonov regularization in this paper we develop a novel rule for choosing regularization parameters in nonsmooth tikhonov functionals it is solely based on the value function and applicable to a broad range of nonsmooth models and it extends one known criterion a posteriori error estimates of the approximations are derived an efficient numerical algorithm for computing the minimizer is developed and its convergence properties are discussed numerical results for several common nonsmooth models are presented including deblurring natural images the numerical results indicate the rule can yield results comparable with those achieved with the discrepancy principle and the optimal choice and the algorithm merits a fast and steady convergence

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

message passing algorithms for inference and optimization belief propagation and divide and concur message passing algorithms can solve a wide variety of optimization inference and constraint satisfaction problems the algorithms operate on factor graphs that visually represent and specify the structure of the problems after describing some of their applications i survey the family of belief propagation bp algorithms beginning with a detailed description of the min sum algorithm and its exactness on tree factor graphs and then turning to a variety of more sophisticated bp algorithms including free energy based bp algorithms splitting bp algorithms that generalize tree reweighted bp and the various bp algorithms that have been proposed to deal with problems with continuous variables

lattices on simplicial partitions in this paper d 1 pencil lattices on simplicial partitions in r d are studied the barycentric approach naturally extends the lattice from a simplex to a simplicial partition providing a continuous piecewise polynomial interpolant over the extended lattice the number of degrees of freedom is equal to the number of vertices of the simplicial partition the constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice

a regularization parameter in discrete ill posed problems the tikhonov regularization method for discrete ill posed problems is considered for the practical choice of the regularization parameter alpha some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all alpha considered this paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the l corner of the plot considered in the logarithmic scale moreover a new criterion for choosing alpha is introduced independent of the shape of the plot which gives a new interpretation of the corner criterion mentioned above the existence of l corner is discussed

on lagrange and hermite interpolation in rk a method for the construction of a set of data of interpolation in several variables is given the resulting data which are either function values or directional derivatives values give rise to a space of polynomials in such a way that unisolvence is guaranteed the interpolating polynomial is calculated using a procedure which generalizes the newton divided differences formula for a single variable

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

uniform bivariate hermite interpolation i coordinate degree on considere des schemas d interpolation d hermite uniforme bivariable a partir d espaces de polynomes de degre coordonne m n on se pose la question de la presque regularite de tels schemas c est a dire celle de l existence d ensembles de n uds pour lesquels avec une constellation de derivees donnees l interpolation est possible on montre que dans le cas affirmatif le schema en question peut etre utilise pour presque tous les ensembles de n uds l etude est basee sur des techniques de reduction

on computing first and second order derivative spectra enhancing resolution in spectral response and an ability to differentiate spectral mixing in delineating the endmembers from the spectral response are central to the spectral data analysis first and higher order derivatives analysis of absorbance and reflectance spectral data is commonly used techniques in differentiating the spectral mixing but high sensitivity of derivative to the noise in data is a major problem in the robust estimation of derivative of spectral data an algorithm of robust estimation of first and second order derivative spectra from evenly spaced noisy normal spectral data is proposed the algorithm is formalized in the framework of an inverse problem where based on the fundamental theorem of calculus a matrix equation is formed using a volterra type integral equation of first kind a regularization technique where the balancing principle is used in selecting a posteriori optimal regularization parameter is designed to solve the inverse problem for robust estimation of first order derivative spectra the higher order derivative spectra are obtained while using the algorithm in sequel the algorithm is tested successfully with synthetically generated spectral data contaminated with additive white gaussian noise and also with real absorbance and reflectance spectral data for fresh and sea water respectively robust and efficient algorithm of computing first and second orders derivative spectra is designed the algorithm is based on the premise of inverse problem where a robust solution is achieved via regularization it couples discrepancy and balancing principles to determine regularization parameter and implement stopping rule respectively numerical experiments on noisy data suggest superiority of the method over finite difference and fourier transform based techniques the proposed method is applied on absorbance and reflectance spectral data for fresh and sea water

message passing algorithms and improved lp decoding linear programming lp decoding for low density parity check codes and related domains such as compressed sensing has received increased attention over recent years because of its practical performance coming close to that of iterative decoding algorithms and its amenability to finite blocklength analysis several works starting with the work of feldman showed how to analyze lp decoding using properties of expander graphs this line of analysis works for only low error rates about a couple of orders of magnitude lower than the empirically observed performance it is possible to do better for the case of random noise as shown by daskalakis and koetter and vontobel building on work of koetter and vontobel we obtain a novel understanding of lp decoding which allows us to establish a 0 05 fraction of correctable errors for rate codes this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for lp decoding our analysis exploits an explicit connection between lp decoding and message passing algorithms and unlike other techniques directly works with the primal linear program an interesting byproduct of our method is a notion of a locally optimal solution that we show to always be globally optimal i e it is the nearest codeword such a solution can in fact be found in near linear time by a reweighted version of the min sum algorithm obviating the need for lp our analysis implies in particular that this reweighted version of the min sum decoder corrects up to a 0 05 fraction of errors

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

lattices on simplicial partitions which are not simply connected in this paper d 1 pencil lattices on simplicial partitions in r d which are not simply connected are studied it is shown how the fact that a partition is not simply connected can be used to increase the flexibility of a lattice a local modification algorithm is developed also to deal with slight partition topology changes that may appear afterwards a lattice has already been constructed

interpolation lattices in several variables principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions a principal lattice can be described as the set of intersection points of n 1 pencils of parallel hyperplanes using a projective point of view lee and phillips extended this situation to n 1 linear pencils of hyperplanes in two recent papers two of us have introduced generalized principal lattices in the plane using cubic pencils in this paper we analyze the problem in n dimensions considering polynomial exponential and trigonometric pencils which can be combined in different ways to obtain generalized principal lattices we also consider the case of coincident pencils an error formula for generalized principal lattices is discussed

interpreting potential field anomaly of an isolated source of regular geometry revisited abstracti have developed an improved practical method for interpreting a symmetrical shaped potential field anomaly due to an isolated source body of regular geometric configuration the method uses the first order horizontal derivative of the logarithmically transformed absolute value of the anomaly in estimating the source body parameters such as the location depth of burial and shape factor to tackle noise in data a regularization technique is designed which ensures a robust estimate of the first order derivative of logarithmically transformed data the regularization technique uses an optimal value of regularization parameter that although noise dependent requires no a priori knowledge of the noise level in the data a graphical method is designed to determine an optimal value of the regularization parameter from the position of the local minimum of a specially defined functional with respect to the regularization parameters numerical tests have been conducted on the noise contaminated synthe

polynomial interpolation of minimal degree minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which lagrange interpolation with respect to the given points is uniquely solvable and degree reducing this is a generalization of the concept of least interpolation introduced by de boor and ron this paper investigates the behavior of lagrange interpolation with respect to these spaces giving a newton interpolation method and a remainder formula for the error of interpolation moreover a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view

newton cotes cubature rules over d 1 pencil lattices in this paper newton cotes cubature rules are extended to d 1 pencil lattices over simplices and simplicial partitions the closed form of the cubature rules as well as the error term are determined further the basic cubature rules can be combined with an adaptive algorithm over simplicial partitions the key point of the algorithm is a subdivision step that refines a d 1 pencil lattice over a simplex to its subsimplices if the number of function evaluations is crucial the additional freedom provided by d 1 pencil lattices may be used to decrease it significantly

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

an analysis of polynomial chaos approximations for modeling single fluid phase flow in porous medium systems we examine a variety of polynomial chaos motivated approximations to a stochastic form of a steady state groundwater flow model we consider approaches for truncating the infinite dimensional problem and producing decoupled systems we discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature it would be necessary to find new multivariate cubature rules finally we use the acceleration of monte carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than monte carlo for the relatively small domains considered in this work a curse of dimensionality in the series expansion of the log normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work other than the monte carlo method

on robust estimation of discrete hilbert transform of noisy data abstractwe developed a novel technique of robust estimation of the discrete hilbert transform dht of noisy geophysical data the technique used the sinc method in which the data were transformed via conformal mapping and the sinc bases were determined by solving a linear matrix equation a transformation rule was presented for selecting a suitable conformal mapping function that would transform the class of geophysical data set in an appropriate interval range a novel regularization technique was designed to obtain a robust solution of sinc bases when the data contained noise in which an optimal regularization parameter was obtained in an automated way using a 1d optimization scheme the technique of selecting the optimal value of the regularization parameter required no a priori knowledge about the level of noise contamination in the data numerical experiments were conducted on synthetically generated and published field data sets with a varying level of noise contamination to test the performance

a scheme for stable numerical differentiation a method for stable numerical differentiation of noisy data is proposed the method requires solving a volterra integral equation of the second kind this equation is solved analytically in the examples considered its solution is computed analytically some numerical results of its application are presented these examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods in particular than variational regularization

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

on the history of multivariate polynomial interpolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the first 75 years of this century including a pioneering paper by kronecker in the 19th century

bivariate lagrange interpolation at the padua points the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order o

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

analysis of the min sum algorithm for packing and covering problems via linear programming message passing algorithms based on belief propagation bp are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems the bp based algorithms operate over graph representations called factor graphs that are used to model the input although in many cases the bp based algorithms exhibit impressive empirical results not much has been proved when the factor graphs have cycles this paper deals with packing and covering integer programs in which the constraint matrix is zero one the constraint vector is integral and the variables are subject to box constraints we study the performance of the min sum algorithm when applied to the corresponding factor graph models of packing and covering linear programmings lps we compare the solutions computed by the min sum algorithm for packing and covering problems to the optimal solutions of the corresponding lp relaxations in particular we prove that if the lp has an optimal fractional solution then for each fractional component the min sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction this implies that the min sum algorithm computes the optimal integral solution only if the lp has a unique optimal solution that is integral the converse is not true in general for a special case of packing and covering problems we prove that if the lp has a unique optimal solution that is integral and on the boundary of the box constraints then the min sum algorithm computes the optimal solution in pseudopolynomial time our results unify and extend recent results for the maximum weight matching problem and for the maximum weight independent set problem

on decoding irregular tanner codes with local optimality guarantees we consider decoding of binary linear tanner codes using message passing iterative decoding and linear programming lp decoding in memoryless binary input output symmetric mbios channels we present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular tanner codes with respect to any mbios channel this characterization is a generalization of arora proc acm symp theory of computing 2009 and vontobel proc inf theory and appl workshop 2010 and is based on a conical combination of normalized weighted subtrees in the computation trees of the tanner graph these subtrees may have any finite height h even equal or greater than half of the girth of the tanner graph in addition the degrees of local code nodes in these subtrees are not restricted to two i e these subtrees are not restricted to skinny trees we prove that local optimality in this new characterization implies maximum likelihood ml optimality and lp optimality and show that a certificate can be computed efficiently we also present a new message passing iterative decoding algorithm called normalized weighted min sum nwms nwms decoding is a belief propagation bp type algorithm that applies to any irregular binary tanner code with single parity check local codes e g low density and high density parity check codes we prove that if a locally optimal codeword with respect to height parameter h exists whereby notably h is not limited by the girth of the tanner graph then nwms decoding finds this codeword in h iterations the decoding guarantee of the nwms decoding algorithm applies whenever there exists a locally optimal codeword because local optimality of a codeword implies that it is the unique ml codeword the decoding guarantee also provides an ml certificate for this codeword finally we apply the new local optimality characterization to regular tanner codes and prove lower bounds on the noise thresholds of lp decoding in mbios channels when the noise is below these lower bounds the probability that lp decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the tanner graph

newton basis for multivariate birkhoff interpolation multivariate birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far in this paper we introduce a special new type of multivariate birkhoff interpolation and present a newton paradigm for it using the algorithms proposed in this paper we can construct a hermite system for any interpolation problem of this type and then obtain a newton basis for the problem w r t the hermite system

belief propagation and lp relaxation for weighted matching in general graphs loopy belief propagation has been employed in a wide variety of applications with great empirical success but it comes with few theoretical guarantees in this paper we analyze the performance of the max product form of belief propagation for the weighted matching problem on general graphs we show that the performance of max product is exactly characterized by the natural linear programming lp relaxation of the problem in particular we first show that if the lp relaxation has no fractional optima then max product always converges to the correct answer this establishes the extension of the recent result by bayati shah and sharma which considered bipartite graphs to general graphs perhaps more interestingly we also establish a tight converse namely that the presence of any fractional lp optimum implies that max product will fail to yield useful estimates on some of the edges we extend our results to the weighted b matching and r edge cover problems we also demonstrate how to simplify the max product message update equations for weighted matching making it easily deployable in distributed settings like wireless or sensor networks