improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

estimation for the linear model with uncertain covariance matrices we derive a maximum a posteriori estimator for the linear observation model where the signal and noise covariance matrices are both uncertain the uncertainties are treated probabilistically by modeling the covariance matrices with prior inverse wishart distributions the nonconvex problem of jointly estimating the signal of interest and the covariance matrices is tackled by a computationally efficient fixed point iteration as well as an approximate variational bayes solution the statistical performance of estimators is compared numerically to state of the art estimators from the literature and shown to perform favorably

an optimization problem for mass transportation with congested dynamics starting from the work by brenier extended monge kantorovich theory in optimal transportation and applications martina franca 2001 lecture notes in math 1813 springer verlag berlin 2003 pp 91 121 where a dynamic formulation of mass transportation problems was given we consider a more general framework where different kinds of cost functions are allowed this seems relevant in some problems presenting congestion effects as for instance traffic on a highway crowds moving in domains with obstacles and in general in all cases where the transportation does not behave as in the classical monge setting we show some numerical computations obtained by generalizing to our framework the approximation scheme introduced in benamou and brenier a computational fluid mechanics solution to the monge kantorovich mass transfer problem numer math 84 2000 pp 375 393

numerical integration of related hankel transforms by quadrature and continued fraction expansion an algorithm is presented for the accurate evaluation of hankel or bessel transforms of algebraically related kernel functions defined here as the non bessel function portion of the integrand that is more widely applicable than the standard digital filter methods without enormous increases in computational burden the algorithm performs the automatic integration of the product of the kernel and bessel functions between the asymptotic zero crossings of the latter and sums the series of partial integrations using a continued fraction expansion equivalent to an analytic continuation of the series the integrands may be saved to allow the rapid computation of related transforms without recalculating the kernel or bessel functions and the integration steps use interlacing quadrature formulas so that no function evaluations are wasted when it is necessary to increase the order of the quadrature rule the continued fraction algorithm allows very slowly divergent or even formally divergent integrals to be computed quite easily the local error is controlled at each step in the algorithm and accuracy is limited largely by machine resolution the algorithm is written in fortran and is listed in an appendix along with a driver program that illustrates its features the driver program and subroutine are available from the seg business offtce

imaging low frequency and dc electromagnetic fields using a simple linear approximation we consider that all types of electromagnetic measurements represent weighted averages of the subsurface electrical conductivity distribution and that to each type of measurement there corresponds a different weighting function we use this concept for the quantitative interpretation of dc resistivity magnetometric resistivity and low frequency electric and magnetic measurements at low induction numbers in all three cases the corresponding inverse problems are nonlinear because the weighting functions depend on the unknown conductivity distribution we use linear approximations that adapt to the data and do not require reference resistivity values the problem is formulated numerically as a solution of a system of linear equations the unknown conductivity values are obtained by minimizing an objective function that includes the quadratic norm of the residuals as well as the spatial derivatives of the unknowns we also apply constraints through the use of quadratic programming the final product is the flattest model that is compatible with the data under the assumption of the given weighting functions this approximate inversion or imaging technique produces reasonably good results for low and moderate conductivity contrasts we present the results of inverting jointly and individually different data sets using synthetic and field data

minimax mse ratio estimation with signal covariance uncertainties in continuation to an earlier work we further consider the problem of robust estimation of a random vector or signal with an uncertain covariance matrix that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix while in the earlier work we developed and proposed a competitive minimax approach of minimizing the worst case mean squared error mse difference regret criterion here we study in the same spirit the minimum worst case mse ratio regret criterion namely the worst case ratio rather than difference between the mse attainable using a linear estimator ignorant of the exact signal covariance and the minimum mse mmse attainable by optimum linear estimation with a known signal covariance we present the optimal linear estimator under this criterion in two ways the first is as a solution to a certain semidefinite programming sdp problem and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure we then show that the linear minimax ratio regret estimator can also be interpreted as the mmse estimator that minimizes the mse for a certain choice of signal covariance that depends on the uncertainty region we demonstrate that in applications the proposed minimax mse ratio regret approach may outperform the well known minimax mse approach the minimax mse difference regret approach and the plug in approach where in the latter one uses the mmse estimator with an estimated covariance matrix replacing the true unknown covariance

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

a simple sensitivity analysis of time domain and frequency domain electromagnetic measurements electromagnetic responses in both time and frequency domains can be looked upon as weighted spatial averages of the nearby conductivity distribution using this concept i analyze the simplest possible situation a uniform half space in the presence of uniform current sources both transient and harmonic measurements are equally effective in providing conductivity values for the half space while the conductivity values are the same the averaging processes are not to different types of measurements even within the same domain there correspond different types of sensitivity or weighting functions some of these functions emphasize the near surface conductivity while others emphasize the conductivity at depth these and other features of the weighting functions explain the behavior of magnetotelluric sounding curves

embedded trees estimation of gaussian processes on graphs with cycles graphical models provide a powerful general framework for encoding the structure of large scale estimation problems however the graphs describing typical real world phenomena contain many cycles making direct estimation procedures prohibitively costly in this paper we develop an iterative inference algorithm for general gaussian graphical models it operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph when these subproblems are suitably chosen the algorithm converges to the correct conditional means moreover and in contrast to many other iterative methods the tree based procedures we propose can also be used to calculate exact error variances although the conditional mean iteration is effective for quite densely connected graphical models the error variance computation is most efficient for sparser graphs in this context we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree structured counterparts thanks both to the expressive power of these models and to the efficient inference algorithms developed herein the convergence properties of the proposed tree based iterations are characterized both analytically and experimentally in addition by using the basic tree based iteration to precondition the conjugate gradient method we develop an alternative accelerated iteration that is finitely convergent simulation results are presented that demonstrate this algorithm s effectiveness on several inference problems including a prototype distributed sensing application

interior and boundary continuity of weak solutions of degenerate parabolic equations in this paper we consider degenerate parabolic equations of the form bf u t diva x t u u b x t u u d 0 where a and b are respectively vector and scalar valued baire functions defined on u x r1 x r where u is an open subset of rn x t the functions a and b are subject to natural structural inequalities sufficiently general conditions are allowed on the relation b c r1 x rl so that the porus medium equation and the model for the two phase stefan problem can be considered the main result of the paper is that weak solutions of are continuous throughout u in the event that u q x 0 t where q is an open set of r it is also shown that a weak solution is continuous at xo to c aq x 0 t provided xo is a regular point for the laplacian on u 0 introduction in this paper we consider the problem of continuity of weak solutions of parabolic equations of the form at 1 tf3 u diva x t u ux b x t u ux 3 0 where a and b are respectively vector and scalar valued baire functions defined on u x r x rn and u is an open subset of r n x t throughout this paper u will take the form qt q x 0 t where q is an open bounded subset of rn the functions a and b are required to satisfy the following structural inequalities ia x t u p i co p 12 c1 x t i u 12 c2 x t where ao bo and c0 are constants with ao and bo nonnegative and c0 0 the remaining coefficients are nonnegative measurable functions which for simplicity of exposition we will assume to be bounded thus we assume there is a nonnegative constant k such that 3 a x t k b1 x t k ci x t k our results remain valid if the coefficients lie in those appropriate lebesgue classes which have been considered by previous authors e g lsu t throughout we received by the editors march 28 1981 1980 mathematics subject classification primary 35k10 35k15 secondary 35k20 35k65 research supported in part by a grant from the national science foundation d1982 american mathematical society 0002 9947 81 0000 0376 05 00

continuity in two dimensions for a very degenerate elliptic equation an elliptic equation f u f whose ellipticity strongly degenerates for small values of u say f 0 on b 0 1 is considered the aim is to prove regularity for f u the paper proves a continuity result in dimension 2 and presents some applications

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

global l gradient estimates for solutions to a certain degenerate elliptic equation abstract in view of the applications in the study of regularity properties of minimizers for a continuous model of transportation which is a kind of divergence constrained optimization problem we prove a global l gradient estimate for solutions of an elliptic equation whose ellipticity constants become degenerate at every point where u with 0 the exposition is as self contained as possible

optimal transportation with traffic congestion and wardrop equilibria in the classical monge kantorovich problem the transportation cost depends only on the amount of mass sent from sources to destinations and not on the paths followed by this mass thus it does not allow for congestion effects using the notion of traffic intensity we propose a variant taking into account congestion this variant is a continuous version of a well known traffic problem on networks that is studied both in economics and in operational research the interest of this problem is in its relations with traffic equilibria of wardrop type what we prove in the paper is exactly the existence and the variational characterization of equilibria in a continuous space setting

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

quasi linear approximation in 3 d electromagnetic modeling the born approximation in electromagnetic em numerical modeling has limited application for solving 3 d electromagnetic induction problems because in structures with high conductivity contrasts and at high frequencies this approximation is inaccurate in this paper we develop a new and relatively simple approximation for the em field called a quasi linear approximation which is based on the evaluation of the anomalous field ea by a linear transformation of the normal primary field ea en where is called the electrical reflectivity tensor the reflectivity tensor inside inhomogeneities can be approximated by a slowly varying function that can be determined numerically by a simple optimization technique the new approximation gives an accurate estimate of the em response for conductivity contrasts of more than one hundred to one and for a wide range of frequencies it also opens the possibility for fast 3 d electromagnetic inversion

insights into capacity constrained optimal transport a variant of the classical optimal transportation problem is the following among all joint measures with fixed marginals and that are dominated by a given density find the optimal one existence and uniqueness of solutions to this variant were established by korman and mccann in the present article we expose an unexpected symmetry leading to explicit examples in two and more dimensions these are inspired in part by simulations in one dimension that display singularities and topology and in part by two further developments the identification of all extreme points in the feasible set and an approach to uniqueness based on constructing feasible perturbations

direct inversion of time domain electromagnetic data this note describes a simple method for converting transient electromagnetic em sounding data into profiles of conductivity versus depth based on an approximate image representation for the decaying induced ground currents the method can provide one dimensional 1 d inversion for any kind of time domain em sounding data the discussion here is limited to the case of central loop sounding in particular i apply the inversion to some time derivative central loop sounding data and demonstrate that essentially the same interpretation of the geoelectric section is obtained with the simple image method as is obtained using traditional iterative least squares fitting to layered models this simple inverse can be computed much more quickly than an iterative least squares inverse making it possible to estimate the geoelectric section concomitant with data acquisition

the use and misuse of apparent resistivity in electromagnetic methods problems and misunderstandings arise with the concept of apparent resistivity when the analogy between an apparent resistivity computed from geophysical observations and the true resistivity structure of the subsurface is drawn too tightly several definitions of apparent resistivity are available for use in electromagnetic methods however those most commonly used do not always exhibit the best behavior many of the features of the apparent resistivity curve which have been interpreted as physically significant with one definition disappear when alternative definitions are used it is misleading to compare the detection or resolution capabilities of different field systems or configurations solely on the basis of the apparent resistivity curve for the in loop transient electromagnetic tem method apparent resistivity computed from the magnetic field response displays much better behavior than that computed from the induced voltage response a comparison of exact and asymptotic formulas for the te

robust distributed estimation using the embedded subgraphs algorithm we propose a new iterative distributed approach for linear minimum mean square error lmmse estimation in graphical models with cycles the embedded subgraphs algorithm esa decomposes a loopy graphical model into a number of linked embedded subgraphs and applies the classical parallel block jacobi iteration comprising local lmmse estimation in each subgraph involving inversion of a small matrix followed by an information exchange between neighboring nodes and subgraphs our primary application is sensor networks where the model encodes the correlation structure of the sensor measurements which are assumed to be gaussian the resulting lmmse estimation problem involves a large matrix inverse which must be solved in network with distributed computation and minimal intersensor communication by invoking the theory of asynchronous iterations we prove that esa is robust to temporary communication faults such as failing links and sleeping nodes and enjoys guaranteed convergence under relatively mild conditions simulation studies demonstrate that esa compares favorably with other recently proposed algorithms for distributed estimation simulations also indicate that energy consumption for iterative estimation increases substantially as more links fail or nodes sleep thus somewhat surprisingly sensor network energy conservation strategies such as low powered transmission and aggressive sleep schedules could actually prove counterproductive our results can be replicated using matlab code from www dsp rice edu software

first order mean field games with density constraints pressure equals price in this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality a weak solution of the system contains an extra term an additional price imposed on the saturated zones we show that this price corresponds to the pressure field from the models of incompressible euler equations a la brenier by this observation we manage to obtain a minimal regularity which allows us to write optimality conditions at the level of single agent trajectories and to define a weak notion of nash equilibrium for our model

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

3d resistivity mapping of airborne em data a 3d resistivity mapping technique has been developed to provide fast estimates of resistivity distributions in airborne electromagnetic surveys this proposed 3d mapping method consists of an approximate 3d linear inverse operator and a generalized subspace solver the 3d inverse operator can be generated using any forward approximation that is linear in resistivity the generalized subspace method is an alternative to the conjugate gradient method and it reduces the original large linear system of equations to a much smaller but nonlinear one that is solved iteratively the major benefit of using generalized subspace methods is that subspace vectors can be built based upon physical principles such as skin and investigation depths since the 3d mapping is a linear inverse problem no iteration and thus no forward modeling nor sensitivity updating is needed the 3d resistivity mapping technique can be used directly to estimate 3d resistivity distribution or to provide a model update during an intermediate iteration in a nonlinear 3d inversion synthetic and field data examples indicate that the 3d mapping can provide quantitative information about the resistivity and spatial distributions of the 3d targets

regularity via duality in calculus of variations and degenerate elliptic pdes abstract a technique based on duality to obtain h 1 or other sobolev regularity results for solutions of convex variational problems is presented this technique first developed in order to study the regularity of the pressure in the variational formulation of the incompressible euler equation has been recently re employed in mean field games here it is shown how to apply it to classical problems in relation with degenerate elliptic pdes of p laplace type this allows to recover many classical results via a different point of view and to have inspiration for new ones the applications include among others variational models for traffic congestion and more general minimization problems under divergence constraints but the most interesting results are obtained in dynamical problems such as mean field games with density constraints or density penalizations

an automatic technique for presentation of coincident loop impulse response transient electromagnetic data coincident loop tem sounding data are often presented by plotting the half space apparent conductivity as a function of delay time a new algorithm generates an improved presentation that plots the apparent conductivity as a function of depth the resulting data may be further processed to sharpen or spike the smoothly varying apparent conductivity depth curves in an attempt to better represent the rapid changes in conductivity that often exist in the earth the algorithm described involves an approximation but is simple easy to use and computationally efficient a layered conductivity structure is assumed so the algorithm is best for areas where the geology is approximately horizontal however the algorithm can also be used to identify anomalous features that are not infinite horizontal layers the spiked conductivity models derived from synthetic data are consistent with the original layered earth models and show a greater resolution than the apparent conductivity depth curves and sometimes ampl

nonlinear conjugate gradients algorithm for 2 d magnetotelluric inversion we investigate a new algorithm for computing regularized solutions of the 2 d magnetotelluric inverse problem the algorithm employs a nonlinear conjugate gradients nlcg scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity we compare this algorithm theoretically and numerically to two previous algorithms for constructing such minimum structure models the gauss newton method which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics and an algorithm due to mackie and madden which solves a sequence of linearized inverse problems incompletely using a linear conjugate gradients technique numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients nlcg and mackie madden are more efficient than the gauss newton algorithm in terms of both computer memory requirements and cpu time needed to find accurate solutio

application of the modified image method to inversion of synthetic active electromagnetic bathymetric data a modified image method is used to invert synthetic active electromagnetic data generated from shallow ocean models by means of the exact sommerfeld theory the modified image method offers approximate solutions to the sommerfeld problem the response of a layered structure of ohmic conductors to an active electromagnetic source the image plane for the active electromagnetic source is assumed to be at a complex depth given by q m delta 1 exp ipi 4 rad 2 where delta 1 is the skin depth of the first layer and q m is the multilayer correction factor two ad hoc correction factors have been introduced into the modified image method they bring the image field h i of the single image at a complex depth into very good agreement with the secondary field h s of the exact sommerfeld theory of a two layer bathymetric model we demonstrate that these correction factors can be calculated from synthetic data this allows for an accurate algebraic and fast inversion of active electromagnetic data in terms of the parameters of a two layer model the conductivities of the sea and sea bottom and the depth and ohmic skin depth of the sea the beta factor which takes into account the departure of the primary active electromagnetic field from a plane wave form at the air sea interface is incorporated into the complex two layer correction function q m a rescaling factor f brings h i into good agreement with h s in the depth regime where the beta correction factor is ineffective i e where the first layer depth is greater than two skin depths

review on airborne electromagnetic inverse theory and applications airborne electromagnetic aem data processing and inversion has progressed from the early conductivity depth imaging to 1d inversion for a layered earth model and most recently to 2d 3d inversions the new processes have used methods for searching for solutions applying constraints and focusing images such as occam s laterally constrained and holistic inversions for airborne data imaging and 1d inversion some algorithms are fast whereas others have control over data misfit and the vertical or lateral smoothness beyond 1d algorithms 2d 3d inversions are based on currently used methods such as isolated conductor models which are good for highly conductive orebodies in resistive background and techniques that discretized regions that are used for more complex structures and backgrounds in the latter case the computations are slow so research is focusing on time efficient computer algorithms for solving the equations such as gauss newton quasi newton and nonlinear conjugate gradient algorithms for the electromagnetic em problem the solution can be obtained in a more practical time frame if material that has minimal impact is ignored two approaches can be used to speedup the em inversion process the moving footprint and direct solver methods we hope our work will to some extent help stimulate and focus the research in aem inversion

derivatives with respect to metrics and applications subgradient marching algorithm this paper introduces a subgradient descent algorithm to compute a riemannian metric that minimizes an energy involving geodesic distances the heart of the method is the subgradient marching algorithm to compute the derivative of the geodesic distance with respect to the metric the geodesic distance being a concave function of the metric this algorithm computes an element of the subgradient in o n 2 log n operations on a discrete grid of n points it performs a front propagation that computes a subgradient of a discrete geodesic distance we show applications to landscape modeling and to traffic congestion both applications require the maximization of geodesic distances under convex constraints and are solved by subgradient descent computed with our subgradient marching we also show application to the inversion of travel time tomography where the recovered metric is the local minimum of a non convex variational problem involving geodesic distances

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

coincident loop transient electromagnetic master curves for interpretation of two layer earths a set of apparent conductivity master curves has been calculated for the coincident loop transient electromagnetic tem method used over a two layer earth conductivity contrasts range from 0 001 to 1000 loop radius layer depth ratios range from 0 01 to 100 the time range is sufficient to see the entire shape of the curves from the early to the late time asymptotes these curves allow the determination of the parameters of a two layer earth for accurate data over a sufficient time range examples using the curves to interpret multilayered earths are given the curves are also used to show the limitations placed on interpretation by existing tem equipment

conductivity depth imaging of airborne electromagnetic step response data an adaptation of the macnae lamontagne method allows transform of airborne step response electromagnetic em data to a conductivity depth image the algorithm is based on a nonlinear transformation of the amplitude of the measured response at each delay time to an apparent mirror image depth using matrix algebra the set of mirror image depth delay time data pairs can then be used to derive a conductivity section data can be efficiently processed on a personal computer at rates faster or comparable to the rate required for collection stable conductivity fitting as a function of depth is obtained by damping the matrix inversion by specification of the first and second derivative smoothness weights of the fitted conductivity depth sounding damping parameters may be either fixed or varied along the profile their choice can be constrained by geologic control stability of the process is enhanced by accounting for the transmitter and receiver tilts the mirror image depth delay time data can also be use

robust estimation of a random parameter in a gaussian linear model with joint eigenvalue and elementwise covariance uncertainties we consider the estimation of a gaussian random vector x observed through a linear transformation h and corrupted by additive gaussian noise with a known covariance matrix where the covariance matrix of x is known to lie in a given region of uncertainty that is described using bounds on the eigenvalues and on the elements of the covariance matrix recently two criteria for minimax estimation called difference regret dr and ratio regret rr were proposed and their closed form solutions were presented assuming that the eigenvalues of the covariance matrix of x are known to lie in a given region of uncertainty and assuming that the matrices h t cw 1 h and c x are jointly diagonalizable where c w and c x denote the covariance matrices of the additive noise and of x respectively in this work we present a new criterion for the minimax estimation problem which we call the generalized difference regret gdr and derive a new minimax estimator which is based on the gdr criterion where the region of uncertainty is defined not only using upper and lower bounds on the eigenvalues of the parameter s covariance matrix but also using upper and lower bounds on the individual elements of the covariance matrix itself furthermore the new estimator does not require the assumption of joint diagonalizability and it can be obtained efficiently using semidefinite programming we also show that when the joint diagonalizability assumption holds and when there are only eigenvalue uncertainties then the new estimator is identical to the difference regret estimator the experimental results show that we can obtain improved mean squared error mse results compared to the mmse dr and rr estimators

the usefulness of multicomponent time domain airborne electromagnetic measurements time domain airborne electromagnetic aem systems historically measure the inline horizontal x component new versions of the electromagnetic systems are designed to collect two additional components the vertical z component and the lateral horizontal y component to provide greater diagnostic information in areas where the geology is near horizontal the z component response provides greater signal to noise particularly at late delay times this allows the conductivity to be determined to greater depth in a layered environment the symmetry implies that the y component will be zero hence a nonzero y component will indicate a lateral inhomogeneity the three components can be combined to give the energy envelope of the response over a vertical plate the response profile of this envelope has a single positive peak and no side lobes the shape of the energy envelope is dependent on the flight direction but less so than the shape of the x component response profile in the interpretation of

robust training sequence design for correlated mimo channel estimation we study how to design a worst case robust training sequence for multiple input multiple output mimo channel estimation we consider mean squared error of channel estimates as the figure of merit which is a function of second order statistics of the mimo channel i e channel covariance matrix in order to optimize training sequences under a total power constraint in practical applications the channel covariance matrix is not known perfectly thus the main aspect of our design is to improve robustness of the training sequences against possible uncertainties in the available channel covariance matrix using a deterministic uncertainty model we formulate a robust training sequence design as a minimax optimization problem where we take such imperfections into account we investigate the robust design problem assuming the general case of an arbitrarily correlated mimo channel and a non empty compact convex uncertainty set we prove that such a problem admits a globally optimal solution by exploiting the convex concave structure of the objective function and propose numerical algorithms to address the robust training design problem we proceed the analysis by considering multiple input single output miso channels and kronecker structured mimo channels along with unitarily invariant uncertainty sets for these scenarios we show that the problem is diagonalized by the eigenvectors of the nominal covariance matrices so that the robust design is significantly simplified from a complex matrix variable problem to a real vector variable power allocation problem for the miso channel we provide closed form solutions for the robust training sequences with the uncertainty sets defined by the spectral norm and nuclear norm

global lipschitz continuity for minima of degenerate problems we consider the problem of minimizing the lagrangian int f nabla u f u among functions on omega subset mathbb r n with given boundary datum varphi we prove lipschitz regularity up to the boundary for solutions of this problem provided omega is convex and varphi satisfies the bounded slope condition the convex function f is required to satisfy a qualified form of uniform convexity only outside a ball and no growth assumptions are made

imaging quasi layered conductive structures by simple processing of transient electromagnetic data an imaged conductivity section of a layered earth can be obtained by simple transformation of step response electromagnetic data measured in the quasi static zone this method of data transformation is presented as an alternative to conventional apparent conductivity transformations at each delay time the variation of the step response as a function of geometry transmitter and receiver location is transformed to an equivalent reference depth h which can be related to the depth of electromagnetic field diffusion the behavior of h as a function of delay time is nearly independent of the source receiver geometry the slowness dt dh divided by the magnetic permeability is almost exactly proportional to the cumulative conductance measured from the surface down to a depth h thus we can estimate an apparent conductivity which we call the imaged conductivity at depth to be d2t 0dh2 the cost of this transformation is a fraction of the cost of conventional data inversion and it does not require an a

nonlinear filtering using a new proposal distribution and the improved fast gauss transform with tighter performance bounds we propose a new approach to approximating the chapman kolmogorov equation cke for particle based nonlinear filtering algorithms using a new proposal distribution and the improved fast gauss transform ifgt with tighter accuracy bounds the new proposal distribution used to obtain a monte carlo mc approximation of the cke is based on the proposal distribution found in the auxiliary marginal particle filter ampf by using mc integration to approximate the integrals of the ampf proposal distribution as well as the cke we demonstrate significant improvement in terms of both error and computation time we consider the additive state noise case where the evaluation of the cke is equivalent to performing kernel density estimation kde thus fast methods such as the ifgt can be used in practice the ifgt demonstrates performance far better than that which is predicted by current error analysis therefore the existing bounds are not useful for determining the ifgt parameters which in practice have to be chosen experimentally in order to obtain satisfactory compromise between accuracy and speed of the filtering algorithm we provide in this paper much improved performance bounds for the ifgt and which unlike the previous bound are consistent with the expectation that the error decreases as the truncation order of the ifgt increases the new bounds lead to a new definition of the ifgt parameters and give important insight into the effect that these parameters have on the error therefore facilitating the choice of parameters in practice the experimental results show that we can obtain similar error to the sequential importance sampling sis particle filter while using fewer particles furthermore the choice of the new ifgt parameters remains roughly the same for all the examples that we give

some nonlinear equations of the nonstable filtration type and embedding theorems abstract in this article we consider some mixed problems with implicit degenerate equations for their investigation we study certain properties of a class of pn spaces connected with nonlinear boundary value problems further using the results obtained we prove the solvability of some nonlinear boundary value problems with various elliptic parabolic equations and also a boundary problem for an equation of the nonstable filtration type

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

inversion of horizontal loop electromagnetic frequency soundings a procedure has been developed to perform automatic inversion of hlem frequency soundings by the method of backus and gilbert bg using a linearized iterative scheme a layered conductivity is constructed satisfying the available soundings to an accuracy consistent with the observational uncertainties subsequently in order to investigate the resolving power of the data and hence hopefully to determine the broad scale features which are common to all conductivity models which satisfy the data bg average conductivities are computed for the constructed model at a number of depths in examples with both synthetic data and field data these averages proved to be identical except for random error to the corresponding average conductivities computed for a number of different models satisfying the same data the additional models for this analysis were generated by maximizing or minimizing the conductivities of individual layers in the original constructed model using linear programming techniques on the

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem