appraisal of the generalized likelihood uncertainty estimation glue method 1 recent research documents that the widely accepted generalized likelihood uncertainty estimation glue method for describing forecasting precision and the impact of parameter uncertainty in rainfall runoff watershed models fails to achieve the intended purpose when used with an informal likelihood measure in particular glue generally fails to produce intervals that capture the precision of estimated parameters and the difference between predictions and future observations this paper illustrates these problems with glue using a simple linear rainfall runoff model so that model calibration is a linear regression problem for which exact expressions for prediction precision and parameter uncertainty are well known and understood the simple regression example enables us to clearly and simply illustrate glue deficiencies beven and others have suggested that the choice of the likelihood measure used in a glue computation is subjective and may be selected to reflect the goals of the modeler if an arbitrary likelihood is adopted that does not reasonably reflect the sampling distribution of the model errors then glue generates arbitrary results without statistical validity that should not be used in scientific work the traditional subjective likelihood measures that have been used with glue also fail to reflect the nonnormality heteroscedasticity and serial correlation among the residual errors generally found in real problems and hence are poor metrics for even simple sensitivity analyses and model calibration most previous applications of glue only produce uncertainty intervals for the average model prediction which by construction should not be expected to include future observations with the prescribed probability we show how the glue methodology when properly implemented with a statistically valid likelihood function can provide prediction intervals for future observations which will agree with widely accepted and statistically valid analyses

a recursive generation of local higher order sine gordon equations and their backlund transformation a new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse it is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine gordon and liouville equations in laboratory coordinates a generalization of the well known backlund transformation and nonlinear superposition formula for the sine gordon equation is also obtained

bayesian estimation of uncertainty in runoff prediction and the value of data an application of the glue approach this paper addresses the problem of evaluating the predictive uncertainty of topmodel using the bayesian generalised likelihood uncertainty estimation glue methodology in an application to the small ringelbach research catchment in the vosges france the wide range of parameter sets giving acceptable simulations is demonstrated and uncertainty bands are presented based on different likelihood measures it is shown how the distributions of predicted discharges are non gaussian and vary in shape through time and with discharge updating of the likelihood weights using bayes equation is demonstrated after each year of record and it is shown how the additional data can be evaluated in terms of the way they constrain the uncertainty bands

eikonal equation based front propagation for arbitrary complex configurations this paper presents a front propagation method using the eikonal equation 1 in which represents the smallest euclidean distance field to the front to be propagated the offset capturing approach consists in first calculating the field over a uniform cartesian grid fully covering the front to be propagated and then constructing the iso curves or surfaces as the propagated result the calculation of uses a 3d numerical scheme the fast sweeping scheme validation for accuracy of the method is presented using academic test cases a real 3d industry application draft tube with two piers is successfully propagated and demonstrated using special boundary conditions to cope with inlet and outlet planes during front propagtion this application involves the propagation of a front that exhibits both concave and convex shape regions sharp corners and local tangent plane surface discontinuities as well as a multi connected domain copyright 2007 john wiley sons ltd

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

level set based deformation methods for adaptive grids a new method for generating adaptive moving grids is formulated based on physical quantities level set functions are used to construct the adaptive grids which are solutions of the standard level set evolution equation with the cartesian coordinates as initial values the intersection points of the level sets of the evolving functions form a new grid at each time the velocity vector in the evolution equation is chosen according to a monitor function and is equal to the node velocity a uniform grid is then deformed to a moving grid with desired cell volume distribution at each time the method achieves precise control over the jacobian determinant of the grid mapping as the traditional deformation method does the new method is consistent with the level set approach to dynamic moving interface problems

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

a spectral problem based on so 3 r and its associated commuting soliton equations we present a spectral problem based on the three dimensional real special orthogonal lie algebra so 3 r and construct a hierarchy of commuting bi hamiltonian soliton equations by zero curvature equations associated with the spectral problem an illustrative example of soliton equations is computed together with its associated bi hamiltonian structure

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

field theoretical construction of an infinite set of quantum commuting operators related with soliton equations the quantum version of an infinite set of polynomial conserved quantities of a class of soliton equations is discussed from the point of view of naive continuum field theory by using techniques of two dimensional field theories we show that an infinite set of quantum commuting operators can be constructed explicitly from the knowledge of its classical counterparts the quantum operators are so constructed as to coincide with the classical ones in the 0 limit planck s constant divided by 2 it is expected that the explicit forms of these operators would shed some light on the structure of the infinite dimensional lie algebras which underlie a certain class of quantum integrable systems

comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity the paper appraises two approaches for the treatment of heteroscedasticity and autocorrelation in residual errors of hydrological models both approaches use weighted least squares wls with heteroscedasticity modeled as a linear function of predicted flows and autocorrelation represented using an ar 1 process in the first approach heteroscedasticity and autocorrelation parameters are inferred jointly with hydrological model parameters the second approach is a two stage postprocessor scheme where stage 1 infers the hydrological parameters ignoring autocorrelation and stage 2 conditionally infers the heteroscedasticity and autocorrelation parameters these approaches are compared to a wls scheme that ignores autocorrelation empirical analysis is carried out using daily data from 12 us catchments from the mopex set using two conceptual rainfall runoff models gr4j and hbv under synthetic conditions the postprocessor and joint approaches provide similar predictive performance though the postprocessor approach tends to underestimate parameter uncertainty however the mopex results indicate that the joint approach can be nonrobust in particular when applied to gr4j it often produces poor predictions due to strong multiway interactions between a hydrological water balance parameter and the error model parameters the postprocessor approach is more robust precisely because it ignores these interactions practical benefits of accounting for error autocorrelation are demonstrated by analyzing streamflow predictions aggregated to a monthly scale where ignoring daily scale error autocorrelation leads to significantly underestimated predictive uncertainty and by analyzing one day ahead predictions where accounting for the error autocorrelation produces clearly higher precision and better tracking of observed data including autocorrelation into the residual error model also significantly affects calibrated parameter values and uncertainty estimates the paper concludes with a summary of outstanding challenges in residual error modeling particularly in ephemeral catchments

ensemble evaluation of hydrological model hypotheses it is demonstrated for the first time how model parameter structural and data uncertainties can be accounted for explicitly and simultaneously within the generalized likelihood uncertainty estimation glue methodology as an example application 72 variants of a single soil moisture accounting store are tested as simplified hypotheses of runoff generation at six experimental grassland field scale lysimeters through model rejection and a novel diagnostic scheme the fields designed as replicates exhibit different hydrological behaviors which yield different model performances for fields with low initial discharge levels at the beginning of events the conceptual stores considered reach their limit of applicability conversely one of the fields yielding more discharge than the others but having larger data gaps allows for greater flexibility in the choice of model structures as a model learning exercise the study points to a leaking of the fields not evident from previous field experiments it is discussed how understanding observational uncertainties and incorporating these into model diagnostics can help appreciate the scale of model structural error

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

orthogonal grids around convex bodies using foliations a new technique for the construction of orthogonal grids around convex bodies is presented the method which is analytical or numerical depending on how the body boundary is expressed is based on the development of geometric foliations that follow a prescribed direction for instance the prevailing direction of flow around convex bodies of arbitrary shape the construction of these foliations is straightforward and does not require the solution of any system of algebraic or differential equations nor the use of iterative procedures the method is applicable both to two and three dimensional domains since it is based solely on the concept of local curtvature the lines or surfaces given by the foliations of first and second order together with the complementary orthogonal lines respectively define the orthogonal two or three dimensional grids

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

canonical structure of soliton equations i abstract the canonical structure of the nonlinear evolution equations in 1 1 dimensions solvable in terms of an n n inverse scattering problem is discussed the simplest form of the scattering problems that is those containing the spectral parameter linearly is considered it applies to most of the known soliton equations like the korteweg de vries eq the sine gordon eq and the boussinesq eq discussion of various possible reductions of the number of dependent variables by imposing constraints consistent with the hamiltonian flows is given together with the canonical structure of the reduced systems a direct proof of the involutive character of the infinite number of conserved quantities is given for the general case as well as the reduced case the relation between the conserved quantities and symmetry transformations lie backlund transformations becomes very simple in this framework

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

curvature flow and entropy conditions applied to grid generation we describe a numerical technique to generate logically rectangular body fitted interior and exterior grids the technique is based on solving a hamilton jacobi type equation for a propagating level set function using techniques borrowed from hyperbolic conservation laws coordinate grid lines are kept smooth through curvature terms which regularize the equation of motion and upwind difference schemes which satisfy the correct entropy conditions of front propagation the resulting algorithm can be used to generate two and three dimensional interior and exterior grids around reasonably complex bodies which may contain sharp corners and significant variations in curvature the technique may also be easily extended to problems of boundary fitted moving grids

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

differential operator for the modeling of curvature in structured grid generation elliptic grid generation equations based on the laplacian operator have the well known property of clustering the mesh near convex boundaries and declustering it near concave boundaries the result of this effect is that the mesh spacing near curved boundaries may not reflect the spacing prescribed by the user a number of control functions has been developed over the years to counter this property of the laplacian however these formulations provide only a local improvement of this behavior and often require user intervention in the work presented here a differential operator is derived that retains the strong smoothing properties of the laplacian without the latter s adverse curvature effects the fundamental motivation for this new operator is the premise that the mesh curvature everywhere in the field should reflect the curvature of the domain boundaries without the need for user defined control functions a detailed derivation of the new operator is presented along with a number of applications

on the dissipative evolution equations associated with the zakharov shabat system with a quadratic spectral parameter in this paper we derive some results for the zakharov shabat system of the form dm dx z2 j m zq p m j is diagonal and skewhermitian 8 10 12 following the idea of r beals and r r coifman we estimate the wedge products of the columns of m by l2 norm of the potential q p 4 by this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data q x 0 p x 0 qo po is of schwartz class 1 direct and inverse scattering for the z s system with a quadratic parameter by a potential here we mean a pair of functions q p r m c set of n x n complex matrices q is off diagonal and the diagonal part of p equals the diagonal part of q adj f q and q qx p p e l we consider the following spectral problem given z 0 x z im z o find m z r 4m c with 1 1 am z z 0x z 2 j m x z zq p m x z where j diag id id2 idn d 0 q z im z 0 for a certain set of potentials called generic m has the following properties 8 10 1 3 for any z e x there is a unique matrix v z such that for all x m x z m x z exp xz2j v z exp xz2j where m x z limit of m on i from q received by the editors may 4 1988 1980 mathematics subject classification 1985 revision primary 35q20 35q15

adjoint symmetry constraints of the vector kn equations and their integrable coupling system abstract in this paper we present a soliton hierarchy of vector kn kaup newell equations by using an arbitrary order matrix spectral problem along with its bi hamiltonian formulation then adjoint symmetry constraints are presented for manipulating binary nonlinearization for the associated arbitrary order matrix spectral problem the resulting spatial and temporal constrained flows are shown to provide integrable decompositions of the vector kn equations finally a class of integrable coupling systems of the vector kn soliton equations is obtained through enlarging the spectral problem

computing minimal surfaces via level set curvature flow given a fixed closed curve or curves in r sp3 there is an associated problem of finding a surface with mean curvature zero which has that curve or curves as its boundary in this paper a new approach to numerically solving the minimal surface problem is introduced the surface is represented as a level set of a global function phi r sp3 to r the family of surfaces evolve in time according to mean curvature flow a solution is found when a steady state condition is achieved a new system of interpolatory boundary conditions are used to maintain the connection between the surface and the fixed boundary contour several computer surfaces are shown including examples of topological changes

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise