l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

nonlinear resonance in systems of conservation laws the riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed the inhomogeneity is modeled by a linearly degenerate field the solution of the riemann problem determines the nature of wave interactions and thus the riemann problem serves as a canonical form for nonlinear systems of conservation laws generic conditions on the fluxes are stated and it is proved that under these conditions the solution of the riemann problem exists is unique and has a fixed structure this demonstrates that in the above sense resonant inhomogeneous systems generically have the same canonical form the wave curves for these systems are only lipschitz continuous in a neighborhood of the states where the wave speeds coincide and so in contrast to strictly hyperbolic systems the implicit function theorem cannot be applied directly to obtain existence and uniqueness here we show that existence

global bv entropy solutions and uniqueness for hyperbolic systems of balance laws we consider the cauchy problem for n n strictly hyperbolic systems of nonresonant balance laws

wavelets fractals and radial basis functions wavelets and radial basis functions rbfs lead to two distinct ways of representing signals in terms of shifted basis functions rbfs unlike wavelets are nonlocal and do not involve any scaling which makes them applicable to nonuniform grids despite these fundamental differences we show that the two types of representation are closely linked together through fractals first we identify and characterize the whole class of self similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases conversely we prove that for any compactly supported scaling function spl phi x there exists a one sided central basis function spl rho sub x that spans the same multiresolution subspaces the central property is that the multiresolution bases are generated by simple translation of spl rho sub without any dilation we also present an explicit time domain representation of a scaling function as a sum of harmonic splines the leading term in the decomposition corresponds to the fractional splines a recent continuous order generalization of the polynomial splines

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

wavelets bases adapted to a self similar quasicrystal given any self similar quasicrystal in rn with inflation 1 we construct bases of l2 rn having the following structure nj 2 jx j z where the mother wavelets are smooth and with exponential decay or compact support we also show that wavelets constitute a relatively compact set in some sobolev space and that they depend continuously on when is equipped with an appropriate topology

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

nonlinear resonance for quasilinear hyperbolic equation the purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system for the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source analytical understanding is used to design a numerical scheme based on the random choice method an important physical example is transonic gas flow through a nozzle this analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow including such phenomena as nonlinear stability instability and changing types of waves

solutions a variations bornees pour certains systemes hyperboliques de lois de conservation abstract blake temple trans amer math soc 280 1983 781 795 has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide in this note we prove the global existence of weak solutions for such systems with any bounded variation initial condition the proof is based upon standard numerical schemes as well as upon parabolic regularization the key is that the total variation of the riemann invariants is decreasing in time at least in the case of the initial condition with compact support we prove by using the glimm scheme that the system is decoupling in two conservation laws in one unknown in finite time

equilibrium schemes for scalar conservation laws with stiff sources we consider a simple model case of stiff source terms in hyperbolic conservation laws namely the case of scalar conservation laws with a zeroth order source with low regularity it is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced cfl condition and refined meshes required because of the lack of accuracy on equilibrium states the source term should be taken into account in the upwinding and discretized at the nodes of the grid in order to solve numerically the problem we introduce a so called equilibrium schemes with the properties that i the maximum principle holds true ii discrete entropy inequalities are satisfied iii steady state solutions of the problem are maintained one of the difficulties in studying the convergence is that there are no bv estimates for this problem we therefore introduce a kinetic interpretation of upwinding taking into account the source terms based on the kinetic formulation we give a new convergence proof that only uses property ii in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions the computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that in comparison with an usual upwind scheme the corresponding equilibrium version is far more accurate furthermore numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

likelihood approximation by numerical integration on sparse grids the calculation of likelihood functions of many econometric models requires the evaluation of integrals without analytical solutions approaches for extending gaussian quadrature to multiple dimensions discussed in the literature are either very specific or suffer from exponentially rising computational costs in the number of dimensions we propose an extension that is very general and easily implemented and does not suffer from the curse of dimensionality monte carlo experiments for the mixed logit model indicate the superior performance of the proposed method over simulation techniques

godunov type approximation for a general resonant balance law with large data abstract we consider the cauchy problem for the 2 2 nonstrictly hyperbolic system a t 0 u t f a u x g a u a x 0 a u t 0 a 0 u 0 for possibly large discontinuous and resonant data the generalized solution to the riemann problem is introduced interaction estimates are carried out using an original change of variables and the convergence of godunov approximations is shown uniqueness is addressed relying on a suitable extension of kru kov s techniques

systems of conservation laws with invariant submanifolds systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation multicomponent chromatogra phy as well as in the study of nonlinear motion in elastic strings here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws this coinci dence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension in the case of 2 x 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide introduction systems of conservation laws which have coinciding shock and rarefaction curves arise in the study of oil reservoir simulation nonlinear wave motion in elastic strings as well as in multicomponent chromatography 1 4 5 6 9 11 12 these systems have many interesting features the riemann problem for these equations can be explicitly solved in the large and wave interactions have a simplified structure even in the presence of a nonconvex flux function for this reason these systems represent some of the few examples for which the cauchy problem has been solved for arbitrary data of bounded variation also hyperbolic degeneracies appear in each of these systems in the present paper we are concerned with characterizing the phenomenon of coinciding shock and rarefaction curves this phenomenon turns out to be a special case of the general phenomenon of a submanifold of r being invariant for weak solutions of a given system of n conservation laws invariant manifolds of higher dimension also appear in physical conservation equations in which the conserved quantities represent concentrations i e when the concentrations c of any of the species vanish the system must reduce to a lower order system that expresses the conservation of the remaining species thus c 0 defines a manifold which is invariant in the sense that solutions that start on the manifold remain there for all time moreover such an invariant manifold of dimension n 1 is the boundary of an dimensional invariant region in the sense of chueh conley and smoller 14 in the case of

applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets the objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary nonuniform knots both construction procedures are based on certain optimally local interpolatory fundamental spline functions which are not required to possess any approximation property

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

training with noise is equivalent to tikhonov regularization it is well known that the addition of noise to the input data of a neural network during training can in some circumstances lead to significant improvements in generalization performance previous work has shown that such training with noise is equivalent to a form of regularization in which an extra term is added to the error function however the regularization term which involves second derivatives of the error function is not bounded below and so can lead to difficulties if used directly in a learning algorithm based on error minimization in this paper we show that for the purposes of network training the regularization term can be reduced to a positive semi definite form that involves only first derivatives of the network mapping for a sum of squares error function the regularization term belongs to the class of generalized tikhonov regularizers direct minimization of the regularized error function provides a practical alternative to training with noise

quadratic spline wavelets with arbitrary simple knots on the sphere in this paper we extend the method for fitting functions on the sphere described in lyche and schumaker siam j sci comput 22 2 2000 724 to the case of nonuniform knots we present a multiresolution method leading to functions on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three with arbitrary simple knot sequences we determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces we describe the tensor product decomposition and reconstruction algorithms in matrix forms which are convenient for the compression of surfaces we give the different steps of computer implementation and finally we present a test example by using two knot sequences a uniform one and a sequence of chebyshev points

well posedness for a scalar conservation law with singular nonconservative source abstract we consider the cauchy problem for the 2 2 strictly hyperbolic system a t 0 a u 0 a 0 u 0 u t f a u x g a u a x 0 for possibly large discontinuous data the continuous dependence of the solution with respect to both a0 and u0 is shown moreover the solutions are characterized as unique limits of kru kov s entropic solutions constructed with regularized initial data ae0

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

surface based matching using elastic transformations abstract we introduce a methodology for the alignment of multidimensional data such as brain scans the proposed approach does not require fiducial point correspondence correspondence of surfaces provides sufficient data for registration we extend multidimensional interpolation theory by using a more general form of energy functional which leads to basis functions that have different orders at zero and infinity this allows flexibility in the design of the interpolation solution the problem is transformed into a linear algebra problem two techniques for better conditioning of the system matrix are described experimental results on two and three dimensional alignment of brain data used in neurochemistry research are shown

automatic generation of rbf networks using wavelets abstract learning can be viewed as mapping from an input space to an output space examples of these mappings are used to construct a continuous function that approximates the given data and generalizes for intermediate instances radial basis function rbf networks are used to formulate this approximating function a novel method is introduced that automatically constructs a generalized radial basis function grbf network for a given mapping and error bound this network is shown to be the smallest network within the error bound for the given mapping the integral wavelet transform is used to determine the parameters of the network simple one dimensional examples are used to demonstrate how the network constructed using the transform is superior to that constructed using standard ad hoc optimization techniques the paper concludes with the automatic generation of grbf networks for a multi dimensional problem namely real time 3d object recognition and pose estimation the results of this application are favorable

stability of conservation laws with discontinuous coefficients we prove l1 contractivity of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable furthermore we show that the l1 difference between solutions to conservation laws with different flux functions is bounded by the total variation with respect to the space variable of the difference between the flux functions

on the stability functional for conservation laws this note is devoted to the explicit construction of a functional defined on all pairs of l 1 functions with small total variation which is equivalent to the l 1 distance and non increasing along the trajectories of a given system of conservation laws the present definition of this functional does not need any construction of approximate solutions

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

multiquadric prewavelets on nonequally spaced knots in one dimension we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with nonequally spaced centers or knots although the multiquadric function and its relations are our prime examples the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

electromagnetic uncertainty analysis using stochastic fdfd method this paper presents a novel stochastic finite difference frequency domain s fdfd method that incorporates the statistical variations of electrical properties directly into the fdfd algorithm by using the delta method approximation the method is validated through two examples a 1 d bioelectromagnetic layered model and specific absorption rate in a 2 d slice of a human head model comparisons between the single run s fdfd results and those of the monte carlo method the benchmark in statistical simulations show that the s fdfd method can predict the mean and variance of electromagnetic fields with a fair degree of accuracy in a computationally efficient way our numerical experiment also shows that the accuracy is highly dependent on the approximation for the correlation coefficients of the variance equations

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

wavelet approximations for first kind boundary integral equations on polygons an elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation its galerkin discretization with n degrees of freedom on the boundary with spline wavelets as basis functions is analyzed a truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from o n 2 to o n log n entries the condition numbers are bounded independently of the meshwidth it is proved that the compressed scheme thus obtained yields in o n log n 2 operations approximate solutions with the same asymptotic convergence rates as the full galerkin scheme in the boundary energy norm as well as in interior points numerical examples show the asymptotic error analysis to be valid already for moderate values of n

well posedness for a class of 2 2 conservation laws withl data abstract the cauchy problem for a special class of 2 2 systems of conservation laws with data in l 1 l is considered in the strictly hyperbolic case we prove the existence of a weak solution which depends continuously on the initial data with respect to the l 1 norm this solution can be characterized in terms of a kru kov type entropy condition which is introduced here

application of polynomial chaos to quantify uncertainty in deterministic channel models a non intrusive formulation of the polynomial chaos method is applied to quantify the uncertainties in deterministic models of the indoor radio channel deterministic models based on the finite difference time domain fdtd method and ray tracing are examined various sources of parameter uncertainty are considered including randomness in the material properties building geometry and the spatial location of transmitting and receiving antennas the polynomial chaos results are confirmed against monte carlo simulations and experimental measurements the analysis shows the expected variation in the sector averaged path loss can be considerable for relatively small input parameter uncertainties leading to the conclusion that a single simulation run using nominal values may be insufficient to adequately characterize the indoor radio channel

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed