gridding with continuous curvature splines in tension a gridding method commonly called minimum curvature is widely used in the earth sciences the method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature the minimum curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points minimum curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used these extraneous inflection points can be eliminated by adding tension to the elastic plate flexure equation it is straightforward to generalize minimum curvature gridding algorithms to include a tension parameter the same system of equations must be solved in either case and only the relative weights of the coefficients change therefore solutions under tension require no more computational effort than minimum curvature solutions and any algorithm which can solve the minimum curvature equations can solve the more general system we give common geologic examples where minimum curvature gridding produces erroneous results but gridding with tension yields a good solution we also outline how to improve the convergence of an iterative method of solution for the gridding equations

error estimates for the fast multipole method i the two dimensional case the fast multipole method is a new and interesting method for computing long range interactions in particle systems although the method already has been implemented and compared to traditional and other methods with respect to accuracy and speed no accurate error estimates of the fast multipole method have been given in this paper we develop an explicit though complicated form for the error in the three dimensional case and derive an estimate for it the estimate has a simple analytic form which will allow its use in tuning the method for best efficiency and for comparison of the method with other methods at the same accuracy

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

global existence for systems of parabolic conservation laws in several space variables abstract we prove the global existence of solutions of the cauchy problem for certain systems of conservation laws with artificial viscosity terms added the system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix and the initial data is assumed to be close to a constant in l2 l in particular our result applies to the equations of compressible fluid flow in two and three space variables

a scattered equivalent source method for interpolation and gridding of potential field data in three dimensions potential field geophysical data observed at scattered discrete points in three dimensions can be interpolated gridded for example onto a level surface by relating the point data to a continuous function of equivalent discrete point sources the function used here is the inverse distance newtonian potential the sources located beneath some of the data points at a depth proportional to distance to the nearest neighboring data point are determined iteratively areas of no data are filled by minimum curvature for two dimensional 2 d data all data points at the same elevation grids calculated by minimum curvature and by equivalent sources are similar but the equivalent source method can be tuned to reduce aliasing gravity data in an area of high topographic relief in southwest u s a were gridded by minimum curvature a 2 d algorithm and also by equivalent sources 3 d the minimum curvature grid shows strong correlation with topography as expected because variation in gravity effect due to

generalized sampling a variational approach i theory we consider the problem of reconstructing a multidimensional vector function f sub in spl ropf sup m spl rarr spl ropf sup n from a finite set of linear measures these can be irregularly sampled responses of several linear filters traditional approaches reconstruct in an a priori given space e g the space of bandlimited functions instead we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion j first we present the solution of the generalized interpolation problem later we also consider the approximation problem and we show that both lead to the same class of solutions imposing generally desirable properties on the reconstruction largely limits the choice of the criterion j linearity leads to a quadratic criterion based on bilinear forms specifically we show that the requirements of translation rotation and scale invariance restrict the form of the criterion to essentially a one parameter family we show that the solution can be obtained as a linear combination of generating functions we provide analytical techniques to find these functions and the solution itself

orthogonality from disjoint support in reproducing kernel hilbert spaces we investigate reproducing kernel hilbert spaces rkhs where two functions are orthogonal whenever they have disjoint support necessary and sufficient conditions in terms of feature maps for the reproducing kernel are established we also present concrete examples of finite dimensional rkhs and rkhs with a translation invariant reproducing kernel in particular it is shown that a sobolev space has the orthogonality from disjoint support property if and only if it is of integer index

a unified approach to optimal image interpolation problems based on linear partial differential equation models the unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal spatial the unknown image is reconstructed from a finite set of sampled data in such a way that a mean square error is minimized by first expressing the solution in terms of the reproducing kernel of a related hilbert space and then constructing this kernel using the fundamental solution of an induced linear partial differential equation or the green s function of the corresponding self adjoint operator it is proved that in most cases closed form fundamental solutions or green s functions for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first or second order linear partial differential operator an efficient method for obtaining the corresponding closed form fundamental solutions or green s functions of the operators is presented a computer simulation demonstrates the reconstruction procedure

existence and uniqueness of solutions for some hyperbolic systems of conservation laws we study the cauchy problem for systems of conservation laws which belong to the temple class the compensated compactness theory is used to prove existence of solutions some uniqueness results are established by means of holmgren s principle

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

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

interpolation of potential field data by equivalent layer and minimum curvature a comparative analysis interpolation using only the observations at discrete points is an ill posed problem because it admits infinite solutions usually to reduce ambiguity a priori information about the sample function is introduced current interpolation methods in mineral exploration introduce only the constraints of continuity and smoothness of the interpolating function in interpolating potential field anomalies the constraint that the sampled function is harmonic may be introduced by the equivalent layer method elm we compare the performance of the elm and the minimum curvature method mcm in interpolating potential field anomalies by applying these methods to synthetic magnetic data simulating an aeromagnetic survey in the case the anomaly flanks and peak are undersampled the elm performs better than the mcm in recovering the anomaly gradients and peak in the case of elongated linear anomalies the elm recovers the exact linear pattern but the mcm introduces spurious oscillations in the linear pattern also

reproducing kernel hilbert space method for optimal interpolation of potential field data the rkhs based optimal image interpolation method presented by chen and de figueiredo 1993 is applied to scattered potential field measurements the rkhs which admits only interpolants consistent with laplace s equation is defined and its kernel derived the algorithm is compared to bicubic spline interpolation and is found to yield vastly superior results

stability of godunov s method for a class of 2 2 systems of conservation laws on demontre la stabilite et la convergence du schema de godunov pour une classe particuliere de systemes 2 2 non lineaires de lois de conservation

a new approach for rapid evaluation of the potential field in three dimensions efficient evaluation of the potential field is an essential requirement for simulation of large ensembles of particles in many applications including astrophysics plasma physics molecular dynamics very large scale integration systems and micro electro mechanical systems current methods use multipole expansion of spherical harmonics for the potential field which is computationally expensive in terms of running time and memory requirements when a high degree of accuracy is desired in this paper a new approach is presented for efficient and rapid evaluation of the potential field in three dimensions the mathematical background for the proposed approach stems from an exponential integral representation of green s function 1 r and an approximation to the integral by using gauss quadratures which distinctively differs from the theory of spherical harmonics the translations are simple in structure error free and independent of the approximation which enables the overall accuracy and computational performance to be controlled externally via the approximation in addition the gradient of the potential can be readily retrieved as a by product of the computational process more importantly the memory requirement is independent of the desired degree of accuracy the technique presented here opens new possibilities for efficient distributed computing and parallel processing of large scale simulation of particle systems

asymptotic stability of riemann solutions for a class of multidimensional systems of conservation laws with viscosity we prove the asymptotic stability of two state nonplanar riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added the class considered here is those systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes in particular we obtain the uniqueness of the self similar l entropy solution of the two state nonplanar riemann problem the asymptotic stability to which the main result refers is in the sense of the convergence as t in lloc1 of the space of directions x t that is the solution u t x of the perturbed problem satisfies u t t r as t in lloc1 n where r is the self similar entropy solution of the corresponding two state nonplanar riemann problem

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

convergence of an implicit spacetime godunov finite volume method for a class of hyperbolic systems we study an implicit spacetime godunov method for a class of hyperbolic conservation laws known as temple class systems we establish the well posedness of this method a discrete entropy inequality a property analogous to the total variation diminishing property of certain numerical schemes for scalar conservation laws and as a consequence the convergence of the numerical method

pdl g splines defined by partial differential operators with initial and boundary value conditions as a natural generalization of l g splines and thin plate splines text pdl g splines are introduced in this paper a text pdl g spline is defined as a solution of the optimal scattered data interpolation problem described by a general higher order separable linear partial differential operator which may or may not incorporate time with initial terminal and boundary value conditions over the time domain and an arbitrary bounded spatial domain with a continuous and piecewise smooth boundary a closed form expression for the text pdl g spline is obtained by means of the reproducing kernel of a related hilbert space a feasible constructive method for finding the reproducing kernel via a fundamental solution or green s function of an induced linear and self adjoint partial differential operator is also included this method provides a way to take care of the initial terminal and boundary value conditions moreover an explicit formulation for finding the best approximation i

oleinik type estimates and uniqueness for n n conservation laws abstract let ut f u x 0 be a strictly hyperbolic n n system of conservation laws in one space dimension relying on the existence of a semigroup of solutions we first establish the uniqueness of entropy admissible weak solutions to the cauchy problem under a mild assumption on the local oscillation of u in a forward neighborhood of each point in the t x plane in turn this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families thus extending a classical result proved by oleinik in the scalar case

large time behavior of entropy solutions of conservation laws abstract we are concerned with the large time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws we present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions these approaches allow the solutions of arbitrarily large oscillation without a priori assumption on the ways from which the solutions come the relation between the large time behavior of entropy solutions and the uniqueness of riemann solutions leads to an extensive study of the uniqueness problem we use a direct method to show the large time behavior of large l solutions for a class of m m systems including a model in multicomponent chromatography we employ the uniqueness of riemann solutions and the convergence of self similar scaling sequence of solutions to show the asymptotic behavior of large bv solutions for the 3 3 system of euler equations in thermoelasticity these results indicate that the riemann solution is the unique attractor of large discontinuous entropy solutions whose initial data are l l 1 or bv l 1 perturbation of the riemann data for these systems these approaches also work for proving the large time behavior of approximate solutions to hyperbolic conservation laws

the equivalent data concept applied to the interpolation of potential field data the equivalent layer calculation becomes more efficient by first converting the observed potential data set to a much smaller equivalent data set thus saving considerable cpu time this makes the equivalent source method of data interpolation very competitive with other traditional gridding techniques that ignore the fact that potential anomalies are harmonic functions the equivalent data set is obtained by using a least squares iterative algorithm at each iteration that solves an underdetermined system fitting all observations selected from previous iterations and the observation with the greatest residual in the preceding iteration the residuals are obtained by computing a set of predicted observations using the estimated parameters at the current iteration and subtracting them from the observations the use of cholesky s decomposition to implement the algorithm leads to an efficient solution update everytime a new datum is processed in addition when applied to interpolation problems using equiva

generalized sampling a variational approach ii applications for pt i see ibid vol 50 no 8 p 1965 76 2000 the variational reconstruction theory from a companion paper finds a solution consistent with some linear constraints and minimizing a quadratic plausibility criterion it is suitable for treating vector and multidimensional signals here we apply the theory to a generalized sampling system consisting of a multichannel filterbank followed by a nonuniform sampling we provide ready made formulas which should permit application of the technique directly to problems at hand we comment on the practical aspects of the method such as numerical stability and speed we show the reconstruction formula and apply it to several practical examples including new variational formulation of derivative sampling landmark warping and tomographic reconstruction