symplectic lie algebras with degenerate center abstract every symplectic lie algebra with degenerate including non abelian nilpotent symplectic lie algebras has the structure of a quadratic extension we give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets finally we give a scheme to classify the isomorphism classes of symplectic lie algebras with degenerate center we also give a classification scheme for nilpotent symplectic lie algebras and compute a complete list of all 6 dimensional nilpotent symplectic lie algebras which removes inaccuracies in an older list

convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints in this paper we discuss the sample average approximation saa method for a class of stochastic programs with nonsmooth equality constraints we derive a uniform strong law of large numbers for random compact set valued mappings and use it to investigate the convergence of karush kuhn tucker points of saa programs as the sample size increases we also study the exponential convergence of global minimizers of the saa problems to their counterparts of the true problem the convergence analysis is extended to a smoothed saa program finally we apply the established results to a class of stochastic mathematical programs with complementarity constraints and report some preliminary numerical test results

solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization in this paper we consider the stochastic mathematical programs with linear complementarity constraints which include two kinds of models called here and now and lower level wait and see problems we present a combined smoothing implicit programming and penalty method for the problems with a finite sample space then we suggest a quasi monte carlo approximation method for solving a problem with continuous random variables a comprehensive convergence theory is included as well we further report numerical results with the so called picnic vender decision problem

efficient gravity inversion of discontinuous basement relief abstractwe have developed a gravity inversion method to estimate a discontinuous basement relief based on an extended version of bott s method that allows variable density contrasts between sediments and basement optimizes the modulus of the solution correction at each iteration and provides for solution stabilization initially we obtain a linear approximation stabilized by the total variation functional that correctly maps the horizontal positions of the existing high angle faults but produced poor estimates of the basin depths subsequent iterations update the depth estimates toward the correct values at the same time preserving the correct fault horizontal positions additionally we stabilize each solution correction by the smoothness constraint without inverting any matrix the method was substantially more efficient than the nonlinear method which solves a system of linear equations by the conjugate gradient method at each iteration for 3000 parameters it is almost four times faster than the

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

a level set method for imaging salt structures using gravity data abstractwe have developed a level set method for the inverse gravimetry problem of imaging salt structures with density contrast reversal under such a circumstance a part of the salt structure contributes two completely opposite anomalies that counteract with each other making it unobservable to the gravity data as a consequence this amplifies the inherent nonuniqueness of the inverse gravimetry problem so that it is much more challenging to recover the whole salt structure from the gravity data to alleviate the severe nonuniqueness it is reasonable to assume that the density contrast between the salt structure and the surrounding sedimentary host depends upon the depth only and is known a priori consequently the original inverse gravity problem reduces to a domain inverse problem where the supporting domain of the salt body becomes the only unknown we have used a level set function to parametrize the boundary of the salt body so that we reformulated the domain inverse problem into a nonlinear

stochastic mathematical programs with hybrid equilibrium constraints this paper considers a stochastic mathematical program with hybrid equilibrium constraints smphec which includes either here and now or wait and see type complementarity constraints an example is given to describe the necessity to study smphec in order to solve the problem the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints limiting behaviors of the proposed approach are discussed preliminary numerical experiments show that the proposed approach is applicable

approximating stationary points of stochastic optimization problems in banach space in this paper we present a uniform strong law of large numbers for random set valued mappings in separable banach space and apply it to analyze the sample average approximation of clarke stationary points of a nonsmooth one stage stochastic minimization problem in separable banach space moreover under hausdorff continuity we show that with probability approaching one exponentially fast with the increase of sample size the sample average of a convex compact set valued mapping converges to its expected value uniformly the result is used to establish exponential convergence of stationary sequence under some metric regularity conditions

optimality functions in stochastic programming optimality functions define stationarity in nonlinear programming semi infinite optimization and optimal control in some sense in this paper we consider optimality functions for stochastic programs with nonlinear possibly nonconvex expected value objective and constraint functions we show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer we construct confidence intervals for the value of the optimality function at a candidate point and hence provide a quantitative measure of solution quality based on sample average approximations we develop an algorithm for classes of stochastic programs that include cvar problems and utilize optimality functions to select sample sizes

an approximation scheme for a class of risk averse stochastic equilibrium problems we consider two models for stochastic equilibrium one based on the variational equilibrium of a generalized nash game and the other on the mixed complementarity formulation each agent in the market solves a single stage risk averse optimization problem with both here and now investment variables and production wait and see variables a shared constraint couples almost surely the wait and see decisions of all the agents an important characteristic of our approach is that the agents hedge risk in the objective functions on costs or profits of their optimization problems which has a clear economic interpretation this feature is obviously desirable but in the risk averse case it leads to variational inequalities with set valued operators a class of problems for which no established software is currently available to overcome this difficulty we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents problems such as average or conditional value at risk the smoothed variational inequalities can be tackled by the path solver for example the approximation scheme is shown to converge including the case when smoothed problems are solved approximately an interesting by product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem to assess the proposed technique numerical results are presented the first set of experiments is on randomly generated equilibrium problems for which we compare the proposed methodology with the standard smooth reformulation of average value at risk minimization using additional variables to rewrite the associated max function the second set of experiments deals with a part of the real life european gas network for which dantzig wolfe decomposition can be combined with the smoothing approach

flat affine or projective geometries on lie groups abstract this paper deals essentially with affine or projective transformations of lie groups endowed with a flat left invariant affine or projective structure these groups are called flat affine or flat projective lie groups we give necessary and sufficient conditions for the existence of flat left invariant projective structures on lie groups we also determine lie groups admitting flat bi invariant affine or projective structures these groups could play an essential role in the study of homogeneous spaces m g h having a flat affine or flat projective structures invariant under the natural action of g on m a medina asked several years ago if the group of affine transformations of a flat affine lie group is a flat projective lie group in this work we provide a partial positive answer to this question

a multiple level set method for 3d inversion of magnetic data abstractwe have developed a multiple level set method for inverting magnetic data produced by weak induced magnetization only the method is designed to deal with a specific class of 3d magnetic inverse problems in which the magnetic susceptibility is known and the objective of the inversion is to find the boundary or geometric shape of the causative bodies we adopt the conceptual representation of the subsurface geologic structure by a set of magnetic bodies each having a uniform magnetic susceptibility embedded in a nonmagnetic background this representation enables us to reformulate the magnetic inverse problem into a domain inverse problem for those unknown domains defining the supports of the magnetic causative bodies because each body may take on a variety of shapes and we may not know the number of bodies a priori either we use multiple level set functions to parameterize these domains so that the domain inverse problem can be further reduced to an optimization problem of multiple level set f

a regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints we investigate a class of two stage stochastic programs where the second stage problem is subject to nonsmooth equality constraints parameterized by the first stage variant and a random vector we consider the case when the parametric equality constraints have more than one solution a regularization method is proposed to deal with the multiple solution problem and a sample average approximation method is proposed to solve the regularized problem we then investigate the convergence of stationary points of the regularized sample average approximation programs as the sample size increases the established results are applied to stochastic mathematical programs with p 0 variational inequality constraints preliminary numerical results are reported

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

a local level set method for 3d inversion of gravity gradient data abstractwe have developed a local level set method for inverting 3d gravity gradient data to alleviate the inherent nonuniqueness of the inverse gradiometry problem we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain d so that we may convert the original inverse problem into a domain inverse problem because the unknown domain d may take a variety of shapes we parametrized the domain d by a level set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level set function because the convergence of the level set algorithm relied heavily on initializing the level set function to enclose the gravity center of a source body we applied a weighted l1 regularization method to locate such a gravity center so that the level set function can be properly initialized to rapidly compute the gradient of the nonlinear functional arising in the level

efficient 3d inversion of magnetic data via octree mesh discretization space filling curves and wavelets abstractairborne magnetic survey data sets can contain from hundreds of thousands to millions of observations and typically cover large areas the large number of measurements combined with a model mesh to accommodate the survey extent can render an inversion of these data intractable faced with this challenge we have developed a three step procedure to locally optimize the degree of model discretization and to compress the corresponding sensitivity matrix for the inversion of magnetic data the mesh optimization is achieved through the use of adaptive octree discretization the compression is achieved by first reordering the model cells using the hilbert space filling curve and then applying the one dimensional wavelet transform to the corresponding sensitivities the fractal property of the hilbert curve groups the spatially adjacent cells into algebraically adjacent positions in the reordered model mesh and thereby maximizes the number of zero or near zero coefficients in the one dimensional wavelet

lipschitz stability for stochastic programs with complete recourse this paper investigates the stability of optimal solution sets to stochastic programs with complete recourse where the underlying probability measure is understood as a parameter varying in some space of probability measures in math programming 67 1994 pp 99 108 sha piro proved lipschitz upper semicontinuity of the solution set mapping inspired by this result we introduce a subgradient distance for probability distributions and establish the persistence of optimal solutions for a subclass of recourse models we show that the solution set mapping is hausdorff lipschitz continuous with respect to the subgradient distance moreover the subgradient distance is estimated above by the kolmogorov smirnov distance of certain distribution functions related to the recourse model the lipschitz continuity result is illustrated by verifiable sufficient conditions for stochastic programs to belong to the mentioned subclass and by examples showing its validity and limitations finally the lipschitz continuity result is used to derive some new results on the asymptotic behavior ofoptimal solutions when the probability measure underlying the recourse model is estimated via empirical measures law of iterated logarithm large deviation estimate estimate for asymptotic distribution

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

a saa nonlinear regularization method for a stochastic extended vertical linear complementarity problem in this paper we propose a new stochastic equilibrium model a stochastic extended vertical linear complementarity problem which arises in the stochastic generalized bimatrix games and includes stochastic linear complementarity problem as its special case based on the log exponential function a sample average approximation saa regularization method is proposed for solving this problem the analysis of this regularization method is carried out in two steps first under some mild conditions the existence and convergence results to the proposed method are provided second under conditions on row representative of matrices the exponential convergence rate of this method is established at last the regularization method proposed is applied to finding a generalized nash equilibrium pair for a stochastic generalized bimatrix game

reconciliation of the discrete and integral fourier transforms in the application of harmonic analysis to potential field geophysical studies relationships derived in terms of the continuous fourier integral transform are evaluated in terms of the discrete fourier transform the discrete transform obtained by transforming a finite number of equispaced samples of the actual aperiodic continuous function is too low at the dc level and increasingly too high in the high frequencies compared with the theoretical integral transform as a consequence overly restrictive limitations must be placed on high frequency amplifying operators such as differentiation and downward continuation also a spurious and troublesome azimuthal distortion occurs in the discrete fourier analysis of three dimensional 3 d map data represented as grids the discrete transform can be made essentially equivalent to the integral transform if before sampling the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi

integration of geophysical constraints for multilayer geometry refinements in 2 5d gravity inversion abstractwe have explored the feasibility of estimating depths of multiple interfaces from gravity data the strata are simulated by an aggregate of 3d rectangular prisms whose bottom depths are parameters to be estimated in the inversion process we have integrated geophysical constraints including the borehole information and the sharp condition described by the total variation function the iterative residual function is also introduced to adjust the weighting of the estimated parameters so that layers of different depths have nearly equal likelihood for deviation the inversion is processed by minimizing the tikhonov parametric functional by the reweighted regularized conjugate gradient method inequality constraints are adopted to deal with the coupling of the interfaces synthetic tests show that such integration is conducive to restoring the multilayer depth distribution real data applications in mariana confirm that the inversion method is effective in complex geologic settings in practice we ha

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications inspired by a recent work by alexander et al j bank finance 30 583 605 2006 which proposes a smoothing method to deal with nonsmoothness in a conditional value at risk problem we consider a smoothing scheme for a general class of nonsmooth stochastic problems assuming that a smoothed problem is solved by a sample average approximation method we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w p 1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem moreover under some metric regularity conditions we obtain an error bound on approximate stationary points the convergence result is applied to a conditional value at risk problem and an inventory control problem

noniterative nonlinear gravity inversion to obtain the shape of a homogeneous body from its gravity anomalies the inverse of parker s formula is expressed here by means of a power series expansion in the reciprocal of the density contrast the calculation of the nth coefficient of the series needs the evaluation of n 1 fourier transforms of products and sums of the preceding coefficients and a sufficient storage to contain all of them computer time is modest for the first several terms but grows rapidly as the number of terms increases fortunately most inversions require a small number of terms the method reproduces oldenburg s iteration results exactly and faster whenever it converges and it can also converge in other cases nevertheless its main advantage comes from the fact that once the series coefficients have been computed the inversion can be performed immediately for a variety of density contrasts allowing the choice of best values to match constraints such as seismic or borehole data

a linearization method for nonsmooth stochastic programming problems for a stochastic programming problem with a nonsmooth objective a recursive stochastic subgradient method is proposed in which successive directions are obtained from quadratic programming subproblems the subproblems result from linearization of original constrains and from approximation of the objective by a quadratic function involving stochastic subgradient estimates constructed in the course of computation a special lyapunov function technique is used to show that the method is convergent with probability 1 to stationary points and recovers the subgradient and lagrange multipliers that appear in necessary conditions of optimality

double extension symplectique d un groupe de lie symplectique resume lie groups carrying a left invariant symplectic form symplectic groups are described in terms of semi direct product of lie groups or symplectic reduction and principal fiber bundles with affine fiber we give a generalization of medina and revoy s symplectic double extension which realizes a symplectic group as the reduction of another symplectic group we show that every group obtained by this process carries an invariant lagrangian foliation such that the affine structure defined by the simplectic form over each leaf is complete

fast gravity inversion of basement relief abstractgravity interpretation of large sedimentary basins requires a large number of observations and of parameters estimations so the development of very efficient gravity inversion methods applicable to such environments is of utmost importance we found that the highly efficient bott s method can be cast in the framework of gauss newton s method for minimizing nonlinear functions also we extended bott s method to 1 be applicable to sedimentary basins with density contrast between the sediments and the basement decreasing with depth according to different laws 2 optimize the modulus of the parameter correction vector 3 stabilize the solution by imposing that it be smooth and 4 quit the iteration according to an objective and effective criterion to keep up the efficiency of bott s method stable solutions were obtained by applying a moving average to the usually unstable solution obtained at the last iteration solutions as efficacious as those obtained by the usual nonlinear method were

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

smooth approximation of potential fields from noisy scattered data geophysical potential field data generally are observed at scattered sampling points and are contaminated by measurement and preprocessing errors analysis and further treatment require a representation of the underlying potential fields on a regular grid in the presence of noise an approximation of the data is more appropriate than an exact interpolation analysis of the smoothness of potential fields indicates an approximation by band limited functions the resulting least squares problem can be formulated as a linear system of equations in which the system matrix is of block toeplitz type this special structure allows a computationally attractive and robust solution via the fast fourier transform and the conjugate gradient algorithm on this basis we present a gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data experimental results for synthetic and real world gravity data demonstrate good performance for highly sc

adaptive learning 3d gravity inversion for salt body imaging we have developed an iterative scheme for inverting gravity data produced by salt bodies with density contrasts relative to the sediments varying from positive to negative crossing in this way the nil zone our inversion method estimates a 3d density contrast distribution through a piecewise constant function defined on a user specified grid of cells it consists of two nested iterative loops the outer loop uses an adaptive learning strategy that starts with a coarse grid of cells a set of first guess geometric elements axes and points and the corresponding assigned density contrasts from the second iteration on this strategy refines the grid and automatically creates a new set of geometric elements points only and associated density contrasts each geometric element operates as the first guess skeletal outline of a section of the salt body to be imaged the inner loop estimates the 3d density contrast distribution for the grid of cells and for the set of geometric elements defined in the oute

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

sample path optimization of convex stochastic performance functions in this paper we propose a method for optimizing convex performance functions in stochastic systems these functions can include expected performance in static systems and steady state performance in discrete event dynamic systems they may be nonsmooth the method is closely related to retrospective simulation optimization it appears to overcome some limitations of stochastic approximation which is often applied to such problems we explain the method and give computational results for two classes of problems tandem production lines with up to 50 machines and stochastic pert program evaluation and review technique problems with up to 70 nodes and 110 arcs

symplectic structures on quadratic lie algebras abstract we study quadratic lie algebras over a field k of null characteristic which admit at the same time a symplectic structure we see that if k is algebraically closed every such lie algebra may be constructed as the t extension of a nilpotent lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic lie algebra by the one dimensional lie algebra finally we prove that every symplectic quadratic lie algebra is a special symplectic manin algebra and we give an inductive description in terms of symplectic quadratic double extensions

smooth fitting of geophysical data using continuous global surfaces continuous global surfaces cgs are a general framework for interpolation and smoothing of geophysical data the first of two smoothing techniques we consider in this paper is generalized cross validation gcv which is a bootstrap measure of the predictive error of a surface that requires no prior knowledge of noise levels the second smoothing technique is to define the cgs surface with fewer centers than data points and compute the fit by least squares lsqr the noise levels are implicitly estimated by the number and placement of the centers relative to the data points we show that both smoothing methods can be implemented using extensions to the existing fast framework for interpolation so that it is now possible to construct realistic smooth fits to the very large data sets typically collected in geophysics thin plate spline and kriging surfaces with gcv smoothing appear to produce realistic fits to noisy radiometric data the resulting surfaces are similar yet the thin plate spline required

single and multi period optimal inventory control models with risk averse constraints this paper presents some convex stochastic programming models for single and multi period inventory control problems where the market demand is random and order quantities need to be decided before demand is realized both models minimize the expected losses subject to risk aversion constraints expressed through value at risk var and conditional value at risk cvar as risk measures a sample average approximation method is proposed for solving the models and convergence analysis of optimal solutions of the sample average approximation problem is presented finally some numerical examples are given to illustrate the convergence of the algorithm

imaging geologic surfaces by inverting gravity gradient data with depth horizons the nonuniqueness problem that occurs when inverting potential field data is well known it can however be surmounted by jointly inverting these data with independent data sets incorporating depth information and regularizing the solution the goal is to produce a geologic model that is compatible with all measured quantities does not exceed any prescribed limits and is geologically plausible to achieve this we have developed a spatially based surface inversion algorithm that solves for the geometric interface between geologic bodies the bodies are constructed from grids of rectangular prisms that have their bottom depths adjusted by the algorithm to form the inverted surface to solve large scale inversions approximations are used in the potential field calculations that allow internal matrices to be stored in sparse format with minimal loss of accuracy the impetus for the work came from the need to combine airborne gravity gradient data with depth horizons estimated from interpreted 2d seismic

uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming abstract sample average approximation saa is one of the most popular methods for solving stochastic optimization and equilibrium problems research on saa has been mostly focused on the case when sampling is independent and identically distributed iid with exceptions dai et al 2000 9 homem de mello 2008 16 in this paper we study saa with general sampling including iid sampling and non iid sampling for solving nonsmooth stochastic optimization problems stochastic nash equilibrium problems and stochastic generalized equations to this end we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem exponential convergence of estimators of both optimal solutions and m stationary points characterized by mordukhovich limiting subgradients mordukhovich 2006 23 rockafellar and wets 1998 32 are established under mild conditions we also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

efficient gravity inversion of basement relief using a versatile modeling algorithm abstractwe have developed a novel approach to compute in an efficient and versatile way the gravity anomaly produced by an arbitrary discrete 3d distribution of density contrast the method allows adjustable precision and is particularly suited for the interpretation of sedimentary basins because the gravity field decays with the square of the distance we use a discrete green s operator that may be made much smaller than the whole study area for irregularly positioned observations this discrete green s operator must be computed just at the first iteration and because each of its horizontal layers presents a center of symmetry only one eighth of its total elements need to be calculated furthermore for gridded data on a plane this operator develops translation symmetry for the whole region of interest and has to be computed just once for a single arbitrary observation position the gravity anomaly is obtained as the product of this small operator by any arbitrary density contrast distribution in

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

using an equivalent source with positivity for low latitude reduction to the pole without striation we present a reformulation of reduction to the pole rtp of magnetic data at low latitudes and the equator using equivalent sources the proposed method addresses both the theoretical difficulty of low latitude instability and the practical issue of computational cost we prove that a positive equivalent source exists when the magnetic data are produced by normal induced magnetization and we show that the positivity is sufficient to overcome the low latitude instability in the space domain we further apply a regularization term directly to the recovered rtp field to improve the solution the use of equivalent source also naturally enables the processing of data acquired on uneven surface the result is a practical algorithm that is effective at the equatorial region and can process large scale data sets with uneven observation heights

harmonic and biharmonic biases in potential field inversion abstractwe found that minimum l2 norm and smoothness constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic respectively in the case of a discrete distribution the minimum l2 norm and smoothness constrained solutions become biased toward being harmonic or biharmonic respectively as a result the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle which forces the solution maxima and minima to lie on any boundary of the discretized region the above findings were illustrated with 2d numerical examples the harmonic or biharmonic bias is brought forth when the strengths of the minimum l2 norm or the smoothness constraint are enhanced relative to all other constraints by approximating the continuous case a large number of discretizing cells relative to the number of independent observations and or by using a regularizing parameter we discovered that by inspecting th

3 d inversion of gravity data we present two methods for inverting surface gravity data to recover a 3 d distribution of density contrast in the first method we transform the gravity data into pseudomagnetic data via poisson s relation and carry out the inversion using a 3 d magnetic inversion algorithm in the second we invert the gravity data directly to recover a minimum structure model in both approaches the earth is modeled by using a large number of rectangular cells of constant density and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data the model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints we apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information applications of the algorithms to synthetic and field data produce density models representative of true structures our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information

a sample average approximation regularization method for a stochastic mathematical program with general vertical complementarity constraints based on the log exponential function a sample average approximation saa regularization method is proposed for solving a stochastic mathematical program with general vertical complementarity constraints smpvcc considered by birbil et al 2006 detailed convergence analysis of this method is investigated it is demonstrated that under some regularity conditions any accumulation point of the sequence of optimal solutions of saa regularized problem is almost surely an optimal solution of the smpvcc as the parameter tends to zero and the sample size tends to infinity furthermore the optimal value sequence of saa regularized problem converges to the optimal value of smpvcc with exponential convergence rate with probability one and a sequence of stationary points of regularized saa problem converges almost surely to a stationary point of smpvcc finally we show that a stochastic stackelberg game can be formulated as a smpvcc problem and an equilibrium solution can be obtained by the method proposed

on consistency of stationary points of stochastic optimization problems in a banach space abstract recently balaji and xu studied the consistency of stationary points in the sense of the clarke generalized gradient for the sample average approximations to a one stage stochastic optimization problem in a separable banach space with separable dual we present an alternative approach showing that the restrictive assumptions that the dual space is separable and the clarke generalized gradient is a norm upper semicontinuous and compact valued multifunction can be dropped for that purpose we use two results having independent interest a strong law of large numbers and a multivalued komlos theorem in the dual to a separable banach space and a result on the weak closedness of the expectation of a random weak compact convex set

3d magnetic sparse inversion using an interior point method rock susceptibility measurements are sometimes taken on outcrop and borehole rocks and provide valuable information for constraining magnetic data inversion we present two approaches for 3d magnetic sparse inversion that effectively take advantage of the rock susceptibility information both approaches minimize a total objective function subject to bound constraints using an interior point method the first approach directly minimizes an l1 norm of the susceptibility model by keeping the bounds positive in which case the objective function is differentiable in the feasible region the second approach minimizes a more generalized lp like norm 0 p 1 of the susceptibility model by approximating the lp like norm inversion as an iteratively reweighed least squares problem moreover this approach allows the model values to be either positive or negative we also demonstrate the equivalence of the proposed approaches and the binary inversion the recovered models of both approaches are characterized by

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

a unified algebraic approach to the classical yang baxter equation in this paper the different operator forms of the classical yang baxter equation are given in the tensor expression through a unified algebraic method it is closely related to left symmetric algebras which play an important role in many fields in mathematics and mathematical physics by studying the relations between left symmetric algebras and the classical yang baxter equation we can construct left symmetric algebras from certain classical r matrices and conversely there is a natural classical r matrix constructed from a left symmetric algebra which corresponds to a parakahler structure in geometry moreover the former in a special case gives an algebraic interpretation of the left symmetry as a lie bracket left twisted by a classical r matrix

distributionally robust optimization with matrix moment constraints lagrange duality and cutting plane methods a key step in solving minimax distributionally robust optimization dro problems is to reformulate the inner maximization w r t probability measure as a semiinfinite programming problem through lagrange dual slater type conditions have been widely used for strong duality zero dual gap when the ambiguity set is defined through moments in this paper we investigate effective ways for verifying the slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem moreover we propose two discretization schemes for solving the dro with one for the dualized dro and the other directly through the ambiguity set of the dro in the absence of strong duality the discretization scheme via lagrange duality may provide an upper bound for the optimal value of the dro whereas the direct discretization approach provides a lower bound two cutting plane schemes are consequently proposed one for the discretized dualized dro and the other for the minimax dro with discretized ambiguity set convergence analysis is presented for the approximation schemes in terms of the optimal value optimal solutions and stationary points comparative numerical results are reported for the resulting algorithms

gravity inversion of a discontinuous relief stabilized by weighted smoothness constraints on depth we present a new stable gravity inversion method applied to the mapping of an interface separating two homogeneous media in contrast with previous similar methods it does not impose an overall smoothness on the estimated interface to stabilize the solution the density contrast between the media is assumed to be known the interpretation model for the upper medium consists of rectangular juxtaposed prisms whose thicknesses represent the depths to the interface and are the parameters to be estimated the true interface is assumed to be flat everywhere except at faults to incorporate this attribute into the estimated relief we developed an iterative process in which three kinds of constraints are imposed on parameters 1 proximity between values of adjacent parameters 2 lower and upper bounds to parameters and 3 proximity between the values of parameters and fixed numerical values starting with an initial solution which presents an overall smooth relief the method enhances initially estimated

total variation regularization for depth to basement estimate part 2 physicogeologic meaning and comparisons with previous inversion methods we applied the mathematical basis of the total variation tv regularization to analyze the physicogeologic meaning of the tv method and compared it with previous gravity inversion methods weighted smoothness and entropic regularization to estimate discontinuous basements in the second part we analyze the physicogeologic meaning of the tv method and compare it with previous gravity inversion methods weighted smoothness and entropic regularization to estimate discontinuous basements presenting a mathematical review of these methods we show that minimizing the tv stabilizing function favors discontinuous solutions because a smooth solution to honor the data must oscillate and the presence of these oscillations increases the value of the tv stabilizing function these three methods are applied to synthetic data produced by a simulated 2d graben bordered by step faults tv regularization and weighted smoothness are also applied to the real anomaly of steptoe valley nevada u s a in all applications the three methods perform similarly tv regularization however has the advantage compared with weighted smoothness of requiring no a priori information about the maximum depth of the basin as compared with entropic regularization tv regularization is much simpler to use because it requires in general the tuning of just one regularization parameter

exact penalization level function method and modified cutting plane method for stochastic programs with second order stochastic dominance constraints level function methods and cutting plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance ssd constraints a level function method requires an exact penalization setup because it can only be applied to the objective function not the constraints slater constraint qualification scq is often needed for deriving exact penalization it is well known that ssd usually does not satisfy scq and various relaxation schemes have been proposed so that the relaxed problem satisfies the scq in this paper we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation exact penalization schemes based on l 1 norm and l infty norm are subsequently derived through robinson s error bound on convex systems and clarke s exact penalty function theorem moreover we propose a modified cutting plane method which constructs a cutting plane through t

imaging of location and geometry for extended targets using the response matrix in this paper we study how to image both the location and the shape of extended targets using the response matrix obtained from inter element response of an active array of transducers in particular the time reversal technique is used for efficient initial localization of the target and the level set method is used for shape reconstruction we then show how to couple the location estimation and shape reconstruction in a complementary way to improve accuracy for range estimation resolution analysis for active arrays in remote sensing regime is also presented we illustrate with numerical experiments which show that the method is capable of imaging objects with complicated shapes and that the method is robust with respect to noisy data

a class of smoothing saa methods for a stochastic mathematical program with complementarity constraints abstract a class of smoothing sample average approximation saa methods is proposed for solving the stochastic mathematical program with complementarity constraints smpcc considered by birbil et al s i birbil g gurkan o listes solving stochastic mathematical programs with complementarity constraints using simulation math oper res 31 2006 739 760 the almost sure convergence of optimal solutions of the smoothed saa problem to that of the true problem is established by the notion of epi convergence in variational analysis it is demonstrated that under suitable conditions any accumulation point of karash kuhn tucker points of the smoothed saa problem is almost surely a kind of stationary point of smpcc as the sample size tends to infinity moreover under a strong second order sufficient condition for smpcc the exponential convergence rate of the sequence of karash kuhn tucker points of the smoothed saa problem is investigated through an application of robinson s stability theory some preliminary numerical results are reported to show the efficiency of proposed method

adaptive multinary inversion of gravity and gravity gradiometry data abstractwe have developed a novel approach for inversion of gravity and gravity gradiometry data based on multinary transformation of the model parameters this concept is a generalization of binary density inversion to the models described by any number of discrete model parameters the multinary inversion makes it possible to explicitly exploit the sharp contrasts of the density between the host media and anomalous targets in the inversion of gravity and gravity gradiometry data in the framework of the multinary inversion method we use the given values of density and error functions to transform the density distribution into the desired step function distribution to accommodate a possible deviation of the densities from the fixed discrete values we develop an adaptive technique for selecting the corresponding standard deviations guided by the inversion process the novel adaptive multinary inversion algorithm is demonstrated to be effective in determining the shape location and densities of the a

estimation of derivatives of nonsmooth performance measures in regenerative systems we investigate the problem of estimating derivatives of expected steady state performance measures in parametric systems unlike most of the existing work in the area we allow those functions to benonsmooth and study the estimation ofdirectional derivatives for the class of regenerative markovian systems we provide conditions under which we can obtainconsistent estimators of those directional derivatives an example illustrates that the conditions imposed must be different from those in the differentiable case the result also allows us to derive necessary and sufficient conditions for differentiability of the expected steady state function we then analyze the process formed by thesubdifferentials of the original process and show that the subdifferential set of the expected steady state function can be expressed as an average of integrals of multifunctions which is the approach commonly found in the literature for integrals of sets the latter result can also be viewed as a limit theorem for more general compact convex multivalued processes

practical applications of uniqueness theorems in gravimetry part ii pragmatic incorporation of concrete geologic information we illustrate the importance of establishing solution uniqueness through mathematical restrictions reflecting a source attribute we also illustrate the validity and utility of a guideline derived in an accompanying paper for constructing sound gravity inversion methods for the class of sources presenting either homogeneous or depth independent density distributions the two part guideline is 1 to introduce a priori information favoring uniqueness either by assuming that a nonnull density distribution depending only on x and y is confined to the interior of a horizontal slab with known position or by limiting the class of possible solutions to homogeneous simply connected polygons or polyhedra with known density displaying no fancy shapes and no curling apophyses at their borders and 2 to introduce information favoring solution stability by estimating only the features of the source which may be resolved by the data following the guideline we apply different methods to gravity data using interp

convergence of stationary points of sample average two stage stochastic programs a generalized equation approach this paper presents an asymptotic analysis of a monte carlo method variously known as sample average approximation saa or sample path optimization spo for a general two stage stochastic minimization problem we study the case when the second stage problem may have multiple local optima or stationary points that are not global solutions and saa is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points new optimality conditions are developed for both the true problem and its saa problem to accommodate karush kuhn tucker points because the optimality conditions are essentially stochastic generalized equations the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions for this purpose we analyze piecewise continuous pc0 stochastic mappings to understand when their expectations are piecewise continuous and thereby derive exponential convergence of saa it is shown under moderate conditions that with probability one an accumulation point of the saa stationary points satisfies a relaxed stationary condition for the true problem and further that with probability approaching one exponentially fast with increasing sample size a stationary point of saa converges to the set of relaxed stationary points these results strengthen or complement existing results where the second stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an saa problem rather than the other way around

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

application of cauchy type integrals in developing effective methods for depth to basement inversion of gravity and gravity gradiometry data one of the most important applications of gravity surveys in regional geophysical studies is determining the depth to basement conventional methods of solving this problem are based on the spectrum and or euler deconvolution analysis of the gravity field and on parameterization of the earth s subsurface into prismatic cells we have developed a new method of solving this problem based on 3d cauchy type integral representation of the potential fields traditionally potential fields have been calculated using volume integrals over the domains occupied by anomalous masses subdivided into prismatic cells this discretization can be computationally expensive especially in a 3d case the technique of cauchy type integrals made it possible to represent the gravity field and its gradients as surface integrals in this approach only the density contrast surface between sediment and basement needed to be discretized for the calculation of gravity field this was especially significant in the modeling and inversion of gravity data for determining the depth to the basement another important result was developing a novel method of inversion of gravity data to recover the depth to basement based on the 3d cauchy type integral representation our numerical studies determined that the new method is much faster than conventional volume discretization method to compute the gravity response our synthetic model studies also showed that the developed inversion algorithm based on cauchy type integral is capable of recovering the geometry and depth of the sedimentary basin effectively with a complex density profile in the vertical direction

3d vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3d variable density contrast three dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3d mass bodies with spatially variable density contrasts a 3d vector gravity potential is defined for a 3d rectangular prism with density contrast varying in depth and horizontally the vertical component of the gravity anomaly equals the flux of the 3d vector gravity potential through the enclosed surface of the prism thus the 3d integral for the gravity anomaly is reduced to a 2d surface integral in turn a 2d vector gravity potential is defined the vertical component of the gravity anomaly equals the net circulation of the 2d vector gravity potential along the enclosed contour bounding the surfaces of the prism the 3d integral for the gravity anomaly is reduced to 1d line integrals further analytical or numerical solutions can then be obtained from the line integrals depending on the forms of the density contrast functions if an analytical solution cannot be obtained the line integral method is semianalytical requiring numerical quadratures to be carried out at the final stages singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them then the vector potential line integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast varying simply in one direction and sophisticatedly in three directions the advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast

the subdifferential of measurable composite max integrands and smoothing approximation the subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization it is known that for clarke regular integrands the clarke subdifferential of the expectation equals the expectation of their clarke subdifferential in particular this holds for convex integrands however little is known about the calculation of clarke subgradients for the expectation of non regular integrands the focus of this contribution is to approximate clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand a framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands this class contains non regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning

polynomial equivalent layer we have developed a new cost effective method for processing large potential field data sets via the equivalent layer technique in this approach the equivalent layer is divided into a regular grid of equivalent source windows inside each window the physical property distribution is described by a bivariate polynomial hence the physical property distribution within the equivalent layer is assumed to be a piecewise polynomial function defined on a set of equivalent source windows we perform any linear transformation of a large set of data as follows first we estimate the polynomial coefficients of all equivalentsource windows by using a linear regularized inversion second we transform the estimated polynomial coefficients of all windows into the physical property distribution within the whole equivalent layer finally we premultiply this distribution by the matrix of green s functions associated with the desired transformation to obtain the transformed data the regularized inversion deals with a linear system of equations with dimensions based on the total number of polynomial coefficients within all equivalent source windows this contrasts with the classical approach of directly estimating the physical property distribution within the equivalent layer which leads to a system based on the number of data because the number of data is much larger than the number of polynomial coefficients the proposed polynomial representation of the physical property distribution within an equivalent layer drastically reduces the number of parameters to be estimated by comparing the total number of floating point operations required to estimate an equivalent layer via our method with the classical approach both formulated with cholesky s decomposition we can verify that the computation time required for building the linear system and for solving the linear inverse problem can be reduced by as many as three and four orders of magnitude respectively applications to synthetic and real data show that our method performs the standard linear transformations of potential field data accurately

constraint qualifications for constrained lipschitz optimization problems and applications to a mpcc abstract three constraint qualifications the weak generalized robinson constraint qualification the bounded constraint qualification and the generalized abadie constraint qualification which are weaker than the generalized robinson constraint qualification grcq given by yen 1997 1 are introduced for constrained lipschitz optimization problems relationships between those constraint qualifications and the calmness of the solution mapping are investigated it is demonstrated that the weak generalized robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping whereas the proposed generalized abadie constraint qualification described in terms of graphical derivatives in variational analysis is weaker than the calmness of the solution mapping finally those constraint qualifications are written for a mathematical program with complementarity constraints mpcc and new constraint qualifications ensuring the c stationary point condition of a mpcc are obtained

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

continuation of potential fields between arbitrary surfaces an equivalent source algorithm is described for continuing either one or two dimensional potential fields between arbitrary surfaces in the two dimensional case the dipole surface is approximated as a set of plane faces with constant moments over each face in the one dimensional case the plane faces of the dipole surface reduce to straight line segments application of the algorithm to model and field examples of aeromagnetic data shows the method to be effective and accurate even when the terrain has strong topographic relief and is composed of highly magnetic volcanic rocks

pseudo kahler lie algebras with abelian complex structures we study lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure we prove that each of those lie algebras is completely determined by a pair u h where u is a complex commutative associative algebra and h is a sesquilinear hermitian form on u which verifies certain compatibility conditions with respect to the associative product on u the riemannian and ricci curvatures of the associated pseudo kahler metric are studied and a characterization of those lie algebras which are einstein but not ricci flat is given it is seen that all pseudo kahler lie algebras can be inductively described by a certain method of double extensions applied to the associated complex associative commutative algebras

regularized mathematical programs with stochastic equilibrium constraints estimating structural demand models the article considers a particular class of optimization problems involving set valued stochastic equilibrium constraints we develop a solution procedure that relies on an approxima tion scheme for the equilibrium constraints based on regularization we replace the approximated equilibrium constraints by those involving only single valued lipschitz continuous functions in ad dition sampling has the further effect of replacing the simplified equilibrium constraints by more manageable ones obtained by implicitly discretizing the given probability measure so as to render the problem computationally tractable convergence is obtained by relying in particular on the graphical convergence of the approximated equilibrium constraints the problem of estimating the characteristics of a demand model a widely studied problem in microeconometrics serves both as motivation and illustration of the regularization and sampling procedure

algebres de lie kahleriennes et double extension resume a kahler lie algebra is a real lie algebra carrying a symplectic 2 cocycle and an integrable complex structure j such that x j y is a scalar product we give a process called kahler double extension which realizes a kahler lie algebra as the kahler reduction of another one we show that every kahler algebra is obtained by a sequence of such a process from 0 or a flat kahler algebra it is obtained from 0 iff it contained a lagrangian sub algebra these methods allow us to prove that any completely solvable and unimodular kahler algebra is commutative

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

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

stability analysis of two stage stochastic mathematical programs with complementarity constraints via nlp regularization this paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint first the complementarity constraint is approximated by a parameterized system of inequalities with a well known regularization approach by scholtes siam j optim 11 2001 pp 918 936 in deterministic mathematical programs with equilibrium constraints the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures by treating the approximation problems as a perturbation of the original true problem we carry out a detailed stability analysis of the approximated problems including continuity and local lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points a particular focus is given to the case where the probability distribution is approximated by

uniform laws of large numbers for set valued mappings and subdifferentials of random functions we derive a uniform strong law of large numbers lln for random set valued mappings the result can be viewed as an extension of both a uniform lln for random functions and lln for random sets we apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs

robust polynomial fitting method for regional gravity estimation standard polynomial fitting methods are inconsistent in their formulation the regional field is approximated by a polynomial fitted to the observed field as a result in addition to the nonuniqueness in the definition of the regional field the fitted polynomial is strongly influenced by the residual field observed field minus regional field we present a regional residual separation method for gravity data which uses a robust procedure to determine the coefficients of a polynomial fitted to the observations under the hypothesis that the regional can be modeled correctly by the polynomial surface the proposed method minimizes the influence of the residual field in the fitted surface the proposed method was applied to real gravity data from ceara state brazil and produced information on zones of possible crustal thickening and the occurrence of lower crustal granulitic rocks thrust into the shallow subsurface

convexification in limit laws of random sets in banach spaces a strong law of large numbers for random compact sets in a banach space with the minkowski addition is provided by puri and ralescu 6 and among other results by gine hahn and zinn 2 and hess 3 this follows the finite dimensional version introduced in 1 the latter however allows nonconvex sets while the generalizations to the banach space setting were proved for convex sets a convexification argument stated in the lemma below shows that the convexity assumption can be dropped the proof reveals information about the nature of the convergence since writing this note we learned that f hiai 4 has independently established this result using a different method some notations let b be a banach space with norm 1111 the minkowski addition of two compact subsets k and l is k l x y x e k y e li and ak is the set tax x e k the closed convex hull of k is denoted cok it is a linear operation the hausdorff distance h k l is defined as max maxxekminyel 11 x y 11 maxyelminxek 11 x y y convergence of sets in this paper is meant in the hausdorff metric