gravimagnetic anomaly formulas for polyhedra of spatially linear media formulas for the gravity potential field and field gradient tensor are derived for a polyhedral target body of a spatially linear density medium the formulas also define the magnetic potential and field in the case of a medium of spatially linear magnetization this work generalizes existing solutions for the gravity field of a polyhedral target of linearly varying density the formulas are analyzed for singularities and for numerical error growth error growth with increasing target distance is found to be higher than in the corresponding uniform polyhedral case examination of the error sources reveals that some error reduction is possible on this basis new algorithms of improved error control over existing algorithms are proposed for the linear medium computational results confirm the expected improvement

numerical discretization based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering physics and biomedical sciences parameter estimation of differential equation models is a challenging problem because of high computational cost and high dimensional parameter space in this article we propose a novel class of methods for estimating parameters in ordinary differential equation ode models which is motivated by hiv dynamics modeling the new methods exploit the form of numerical discretization algorithms for an ode solver to formulate estimating equations first a penalized spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ode solver to obtain the ode parameter estimates via a regression approach we consider three different order of discretization methods euler s method trapezoidal rule and runge kutta method a higher order numerical algorithm reduces numerical error in the approximation of the derivative which produces a more accurate estimate but its computational cost is higher to balance the computational cost and estimation accuracy we demonstrate via simulation studies that the trapezoidal discretization based estimate is the best and is recommended for practical use the asymptotic properties for the proposed numerical discretization based estimators are established comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade off between computational cost and estimation accuracy we apply the proposed methods t an hiv study to further illustrate the usefulness of the proposed approaches

a model based initial guess for estimating parameters in systems of ordinary differential equations summary the inverse problem of parameter estimation from noisy observations is a major challenge in statistical inference for dynamical systems parameter estimation is usually carried out by optimizing some criterion function over the parameter space unless the optimization process starts with a good initial guess the estimation may take an unreasonable amount of time and may converge to local solutions if at all in this article we introduce a novel technique for generating good initial guesses that can be used by any estimation method we focus on the fairly general and often applied class of systems linear in the parameters the new methodology bypasses numerical integration and can handle partially observed systems we illustrate the performance of the method using simulations and apply it to real data

generalized ordinary differential equation models existing estimation methods for ordinary differential equation ode models are not applicable to discrete data the generalized ode gode model is therefore proposed and investigated for the first time we develop the likelihood based parameter estimation and inference methods for gode models we propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving odes the simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ode model estimation approach and the extended smoothing based esb method supplementary materials for this article are available online

convergence to diagonal form of block jacobi type methods we provide sufficient conditions for the general sequential block jacobi type method to converge to the diagonal form for cyclic pivot strategies which are weakly equivalent to the column cyclic strategy given a block matrix partition a ij a i j of a square matrix mathbf a a the paper analyzes the iterative process of the form mathbf a k 1 mathbf p k mathbf a k mathbf q k a k 1 p k a k q k k ge 0 k 0 mathbf a 0 mathbf a a 0 a where mathbf p k p k and mathbf q k q k are elementary block matrices which differ from the identity matrix in four blocks two diagonal and the two corresponding off diagonal blocks in our analysis of convergence a promising new tool is used namely the theory of block jacobi operators typical applications lie in proving the global convergence of block jacobi type methods for solving standard and generalized eigenvalue and singular value problems

estimating varying coefficients for partial differential equation models partial differential equations pdes are used to model complex dynamical systems in multiple dimensions and their parameters often have important scientific interpretations in some applications pde parameters are not constant but can change depending on the values of covariates a feature that we call varying coefficients we propose a parameter cascading method to estimate varying coefficients in pde models from noisy data our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed the performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a pde model arising from lidar data

on the singularities of the gravity field of a homogeneous polyhedral body the study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body

on quadratic convergence bounds for thej symmetric jacobi method this paper deals with quadratic convergence estimates for the serialj symmetric jacobi method recently proposed by veselic the method is characterized by the use of orthogonal and hyperbolic plane rotations using a new technique recently introduced by hari we prove sharp quadratic convergence bounds in the general case of multiple eigenvalues

highly accurate symmetric eigenvalue decomposition and hyperbolic svd let g be a m n real matrix with full column rank and let j be a n n diagonal matrix of signs jii 1 1 the hyperbolic singular value decomposition hsvd of the pair g j is defined as g u v 1 where u is orthogonal is positive definite diagonal and v is j orthogonal matrix vtjv j we analyze when it is possible to compute the hsvd with high relative accuracy this essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits even if they have widely varying magnitudes we show that one sided j orthogonal jacobi method method computes the hsvd with high relative accuracy more precisely let b gd 1 where d is diagonal such that the columns of b have unit norms essentially we show that the computed hyperbolic singular values of the pair g j will have log10 e min b correct decimal digits where e is machine precision we give the necessary relative perturbation bounds and error analysis of the algorithm our numerical tests confirmed all theoretical results for the symmetric non singular eigenvalue problem hx x we analyze the two step algorithm which consists of factorization h gjgt followed by the computation of the hsvd of the pair g j here g is square and non singular let b dg where d is diagonal such that the rows of b have unit norms and let b be defined as above essentially we show that the computed eigenvalues of h will have log10 e 2min b e min b correct decimal digits this accuracy can be much higher then the one obtained by the classical qr and jacobi methods applied to h where the accuracy depends on the spectral condition number of h particularly if the matrices b and b are well conditioned and we are interested in the accurate computation of tiny eigenvalues again we give the perturbation and error bounds and our theoretical predictions are confirmed by a series of numerical experiments we also give the corresponding results for eigenvectors and hyperbolic singular vectors

analytic solution of the gravity anomaly of irregular 2d masses with density contrast varying as a 2d polynomial function the analytic solution of the gravity anomaly caused by a 2d irregular mass body with the density contrast varying as a polynomial function in the horizontal and vertical directions is extrapolated from a historical version in which the analytic solution for the gravity anomaly was given only at the origin of the coordinate system to any point for the density function in terms of variables relative to that origin to calculate the gravity anomaly at stations that are not at origins a coordinate transformation is performed in which case the polynomial density contrast function must also be expressed in the transformed coordinates or a transformed solution must be obtained these analytic solutions can be obtained at any station using 1 a solution transformation method in which the density function and boundary of a mass body are kept intact or 2 a coordinate transformation method in which polynomial coefficient and boundary of a mass body are transformed accordingly the issue of singularity and instability of the analytic methods has been related to case studies caution should be exercised in modeling or interpreting the gravity survey data using the analytic methods for large target distance to target size ratios outside the range of numerical stability compared with other published methods the analytic solution results agree very well with other numerical or seminumerical methods indicating the solution is correct and can be applied for any gravity anomaly calculation caused by an irregular 2d mass body with the density contrast approximated as a polynomial function of horizontal position and or vertical position when the observation is within the range of numerical stability

on quadratic convergence bounds of the j symmetric jacobi method this paper deals with quadratic convergence estimates for the serial j symmetric jacobi method recently proposed by veselic the method is characterized by the use of orthogonal and hyperbolic plane rotations using a new technique recently introduced by hari we prove sharp quadratic convergence bounds in the general case of multiple eigenvalues

on the selection of ordinary differential equation models with application to predator prey dynamical models summary we consider model selection and estimation in a context where there are competing ordinary differential equation ode models and all the models are special cases of a full model we propose a computationally inexpensive approach that employs statistical estimation of the full model followed by a combination of a least squares approximation lsa and the adaptive lasso we show the resulting method here called the lsa method to be an asymptotically oracle model selection method the finite sample performance of the proposed lsa method is investigated with monte carlo simulations in which we examine the percentage of selecting true ode models the efficiency of the parameter estimation compared to simply using the full and true models and coverage probabilities of the estimated confidence intervals for ode parameters all of which have satisfactory performances our method is also demonstrated by selecting the best predator prey ode to model a lynx and hare population dynamical system among some well known and biologically interpretable ode models

on the global and cubic convergence of a quasi cyclic jacobi method in this paper we consider the global and the cubic convergence of a quasi cyclic jacobi method for the symmetric eigenvalue problem the method belongs to a class of quasi cyclic methods recently proposed by w mascarenhas mascarenhas showed that the methods from his class asymptotically converge cubically per quasi sweep one quasi sweep is equivalent to 1 25 cyclic sweeps provided the eigenvalues are simple here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues we discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory

gravity anomalies of arbitrary 3d polyhedral bodies with horizontal and vertical mass contrasts up to cubic order abstracta new singularity free analytical formula has been developed for the gravity field of arbitrary 3d polyhedral mass bodies with horizontally and vertically varying density contrast using third order polynomial functions first the observation sites are moved to the origin of the coordinate system then the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies furthermore singularity free closed form solutions are derived for these line integrals over the edges thus the observation sites can be located inside on or outside the 3d distributions a synthetic prismatic mass body is adopted to verify the accuracy and singularity free property of our newly developed analytical expressions excellent agreements are obtained between our solutions and other published closed form solutions with relative errors in the order of 10 12 to 10 13 in a

quick and easy one step parameter estimation in differential equations summary differential equations are customarily used to describe dynamic systems existing methods for estimating unknown parameters in those systems include parameter cascade which is a spline based technique and pseudo least squares which is a local polynomial based two step method parameter cascade is often referred to as a one step method although it in fact involves at least two stages one to choose the tuning parameter and another to select model parameters we propose a class of fast easy to use genuinely one step procedures for estimating unknown parameters in dynamic system models this approach does not need extraneous estimation of the tuning parameter it selects that quantity as well as all the model parameters in a single explicit step and it produces root n consistent estimators of all the model parameters although it is of course not as accurate as more complex methods its speed and ease of use make it particularly attractive for exploratory data analysis

fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast abstractprismatic bodies with variable density contrasts are very useful in modeling and inversion of gravity anomalies for sedimentary basins in which some simple mathematical functions offer better approximations for the density depth relationship of the basin fill than a constant density model we have explored fourier domain modeling of vector and tensor gravity anomalies due to 2d and 3d prismatic bodies following a wide class of variable density functions we placed the emphasis on general polynomial models because they provided a flexible way to approximate arbitrary density functions the recently proposed gauss fast fourier transform method was applied to improve the accuracy of inverse fourier transform the numerical performance of the presented method was examined by comparing with space domain analytical seminumerical or completely numerical solutions singularity and numerical stability of the algorithm were also analyzed with ranges of numerical stability for polynomial density models of

sparse additive ordinary differential equations for dynamic gene regulatory network modeling the gene regulation network grn is a high dimensional complex system which can be represented by various mathematical or statistical models the ordinary differential equation ode model is one of the popular dynamic grn models high dimensional linear ode models have been proposed to identify grns but with a limitation of the linear regulation effect assumption in this article we propose a sparse additive ode sa ode model coupled with ode estimation methods and adaptive group least absolute shrinkage and selection operator lasso techniques to model dynamic grns that could flexibly deal with nonlinear regulation effects the asymptotic properties of the proposed method are established and simulation studies are performed to validate the proposed approach an application example for identifying the nonlinear dynamic grn of t cell activation is used to illustrate the usefulness of the proposed method

gravitational attraction of a rectangular prism with depth dependent density the gravity effect produced by two and three dimensional bodies with nonuniform density contrast has been treated by several authors one of the first attempts in this direction made by cordell 1973 who developed a method to compute the gravity effect due to a two dimensional prism whose density decreases exponentially with depth a different approach was proposed by murthy and rao 1979 they extended the line integral method to obtain the gravity effect for bodies of arbitrary cross sections with density contrast varying linearly with depth chai and hinze 1988 have derived a wavenumber domain approach to compute the gravity effect due to a vertical prism whose density contrast varies exponentially with depth recently rao 1990 has developed a closed expression of the gravity field produced by an asymmetrical trapezoidal body whose density varies with depth following a quadratic polynomial

2 d inversion of gravity data using sources laterally bounded by continuous surfaces and depth dependent density a description is given of numerical methods for 2 d gravity modeling and nonlinear inversion the forward model solution is suitable for calculating the gravity anomaly caused by a 2 d source body with depth dependent density that is laterally bounded by continuous surfaces and can easily accommodate different kinds of geologic structures the weighted and damped discrete nonlinear inverse method addressed here can invert both density and geometry of the source body both modeling and inversion methods are illustrated with several examples using synthetic and two field gravity data sets one over a sulfide ore body and other across a sedimentary basin a sensitivity analysis is carried out for the resulting solutions by means of the resolution covariance and correlation matrices providing insight into the capabilities and limitations of the inversion method the inversion of synthetic data provides meaningful results showing that the method is robust in the presence of noise its sensitivity analysis i

2 d gravity modeling with analytically defined geometry and quadratic polynomial density functions new methods for 2 d modeling of gravity anomaly data are developed following an approach that uses both analytic and numerical methods of integration the forward model solution developed here is suitable to calculate the gravity effect caused by a 2 d source body bounded either laterally or vertically by continuous functions in our models the density contrast is defined by a second order polynomial function of depth and distance along the profile we present several examples to show that our models are capable of accommodating a broad variety of geologic structures

gravity inversion using open reject and shape of anomaly fill criteria a gravity inversion method is developed by iteratively applying open reject and fill o r f criteria within a model space comprising a great many rectangular prisms each prism is assigned a density contrast the modeling procedure consists of filling some prisms while leaving others empty only one element is filled for each pass generally elements are added only to the periphery of the growing model models can be allowed to grow in any combination of directions or in all directions by application of a shape of anomaly fill criterion the model rapidly attains a form which yields gravity approximating the given gravity scaled down by some constant factor as the model continues to grow this scale factor approaches unity the method readily yields inverse models comprising several thousand individual prisms examples presented here give applications to 2 d problems the method is readily applicable to 212 d and 3 d problems as well overhanging elements are obtained by appropriate use of model

block oriented j jacobi methods for hermitian matrices the paper describes a way how one sided jacobi type algorithm of veselic for computing the hyperbolic singular value decomposition of rectangular matrices can be modified to work with blocks the proposed modification preserves the relative accuracy property of the original algorithm and essentially improves its performance special attention is devoted to proving the global convergence of the method under some important classes of block oriented pivot strategies as numerical tests indicate the block oriented j jacobi methods combined with the hermitian indefinite factorization become efficient and accurate eigensolvers for hermitian indefinite matrices

singularity removal a refinement of resistivity modeling techniques modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available a singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth the inability to adequately represent in discrete space a discontinuous function in this case the delta function describing the introduction of current at a point commonly results in numerical error near the source of a modeled singularity inspection of an integrated finite difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin a procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step thus preventing it from contributing to the numerical error using this procedure the average error in apparent resistivity values for a model of a polar dipole traverse over a nonconducting sphere is reduced by 40 percent for a dipole dipole traverse of a two layer model the error decreases by 75 percent and in the case of a wenner profile of a model of a vertically faulted earth the average error is diminished by 90 percent

quadratic convergence estimate of scaled iterates by j symmetric jacobi method abstract this paper estimates the quadratic convergence reduction of scaled iterates by j symmetric jacobi method numer math 64 1993 241 although the method is well defined for a general definite pair h j h ht j diag im in m the paper considers the most important case when h is positive definite in that case the method is an accurate floating point eigensolver for the pair h j as such it is used in a compound algorithm for accurate floating point computation of eigenvalues and eigenvectors of a non singular indefinite symmetric matrix the new result is proved for scaled diagonally dominant matrices in the general case of multiple eigenvalues it uses frobenius norm of the off diagonal part of symmetrically scaled iteration matrix and a relative gap in the spectrum of h j it can be effectively used in connection with stopping criterion of the method especially with its one sided version

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

existence of the hyperbolic singular value decomposition abstract the hyperbolic singular value decomposition is defined on a general n m matrix and an m m signature matrix pair it is employed in finding the eigenstructure of any matrix that is expressed as the difference of two matrix outer products such differences arise in signal processing applications in the context of the covariance differencing the hyperbolic svd applies in problems where the conventional svd cannot be employed the existence of the hyperbolic singular value decomposition is here extended to the most general case where neither the general matrix nor the matrix product is assumed full rank

an analytical expression for the gravity field of a polyhedral body with linearly varying density over the past 24 years the gravity fields of homogeneous polyhedra have been studied extensively paul 1974 barnett 1976 okabe 1979 golizdra 1981 strakhov et al 1986a 1986b gotze and lahmeyer 1988 pohanka 1988 ivan 1990 pohanka 1990 holstein and ketteridge 1996 this intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run time library for most computational programming languages

gravimagnetic similarity in anomaly formulas for uniform polyhedra gravitational and magnetic anomalies of an arbitrary target body are linked through poisson s differential relation for a uniform polyhedral target poisson s relation reduces to an algebraic link between gravity and magnetic anomaly formulas the derivation is given in tensor form it identifies for each target facet edge a vector function in terms of which the gravitational and magnetic potential and field anomaly formulas are similarly expressed as appropriately weighted linear combinations this similarity unifies the theory of uniform polyhedral anomalies it benefits analysis and construction of software that naturally embraces all anomalies in a single code the analysis is exemplified by a discussion of singularities and by the adaptation of three gravity field algorithms to the remaining gravitational and magnetic cases while retaining the respective computational advantages of the former gravity field algorithms

a versatile solution for the gravity anomaly of 3d prism meshed bodies with depth dependent density contrast abstractwe have developed a generalized solution for computing the gravity anomalies of 3d irregular mass bodies with complicated density contrast variation the 3d irregular shaped bodies can be approximated flexibly by a collection of finite juxtaposed right rectangular prisms the complicated density contrast variation of each prism can be well represented by a depth dependent polynomial function a novel analytic solution of gravity anomalies due to a right rectangular prism with an arbitrary order of polynomial density contrast function of depth is then derived the solution is singularity free in the upper half space over the prism and its singularity in the lower half space containing the prism is resolved by assigning their limit values to the singular terms the numerical stability of the solution is also evaluated through numerical tests hence the solution can be used to compute the gravity anomalies of 3d irregular bodies with variable density contrasts without singularities when computation

robust estimation for ordinary differential equation models summary applied scientists often like to use ordinary differential equations odes to model complex dynamic processes that arise in biology engineering medicine and many other areas it is interesting but challenging to estimate ode parameters from noisy data especially when the data have some outliers we propose a robust method to address this problem the dynamic process is represented with a nonparametric function which is a linear combination of basis functions the nonparametric function is estimated by a robust penalized smoothing method the penalty term is defined with the parametric ode model which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ode model the basis coefficients and ode parameters are estimated in two nested levels of optimization the coefficient estimates are treated as an implicit function of ode parameters which enables one to derive the analytic gradients for optimization using the implicit function theorem simulation studies show that the robust method gives satisfactory estimates for the ode parameters from noisy data with outliers the robust method is demonstrated by estimating a predator prey ode model from real ecological data

optimal control and additive perturbations help in estimating ill posed and uncertain dynamical systems abstractordinary differential equations ode are routinely calibrated on real data for estimating unknown parameters or for reverse engineering nevertheless standard statistical techniques can give disappointing results because of the complex relationship between parameters and states that makes the corresponding estimation problem ill posed moreover ode are mechanistic models that are prone to modeling errors whose influences on inference are often neglected during statistical analysis we propose a regularized estimation framework called tracking that consists in adding a perturbation l2 function to the original ode this perturbation facilitates data fitting and represents also possible model misspecifications so that parameter estimation is done by solving a trade off between data fidelity and model fidelity we show that the underlying optimization problem is an optimal control problem that can be solved by the pontryagin maximum principle for general nonlinear and partially observed ode

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

full block j jacobi method for hermitian matrices abstract the paper considers convergence accuracy and efficiency of a block j jacobi method the method is a proper blas 3 generalization of the known method of veselic for computing the hyperbolic singular value decomposition of rectangular matrices at each step the proposed algorithm diagonalizes the block pivot submatrix the convergence is proved for cyclic strategies which are weakly equivalent to the row cyclic strategy the relative accuracy is proved under the standard conditions numerical tests show improved performance with respect to the block oriented generalization of the original method of veselic combined with the hermitian indefinite factorization the proposed method becomes accurate and efficient eigensolver for hermitian indefinite matrices

2d vector gravity potential and line integrals for the gravity anomaly caused by a 2d mass of depth dependent density contrast using line integrals lis used to calculate the gravity anomaly caused by a 2d mass of complicated geometry and spatially variable density contrast is a computationally efficientalgorithm thatreducesthecalculationfromtwodimensions to one dimension this work has developed a mechanism for defining lis systematically for different types of density functions two dimensional vector gravity potential is defined as a vector the net circulation of which along the closedcontourboundinga2dmass equalsthegravityanomaly caused by the 2d mass two representative types of lis aredefined anliwithanarctangentkernelforanydepth dependent density contrast function which has been studied historically and an li with a simple algebraic kernel for any integrabledensity contrastfunction thepresentworkoffers 1avectorial basedderivationofformulasthatdonotsuffer from the arbitrary sign conventions found in some historical approaches and2asimplealgebraickernelinlineintegrals as an alternative to the historical arctangent kernel with the possibilityofextensiontomoregeneralcases theconceptof 2dvectorgravitypotentialprovidesausefultoolfordefining lis systematically for any mass density function helping us understand how dimensions can be reduced in a calculating gravity anomaly especially when the density contrast varies with space lis have been tested in case studies the maximumdifferencesincalculatedgravityanomaliesbydifferent lis for the case studies were between 5 9310

asymptotic efficiency and finite sample properties of the generalized profiling estimation of parameters in ordinary differential equations ordinary differential equations odes are commonly used to model dynamic behavior of a system because many parameters are unknown and have to be estimated from the observed data there is growing interest in statistics to develop efficient estimation procedures for these parameters among the proposed methods in the literature the generalized profiling estimation method developed by ramsay and colleagues is particularly promising for its computational efficiency and good performance in this approach the ode solution is approximated with a linear combination of basis functions the coefficients of the basis functions are estimated by a penalized smoothing procedure with an ode defined penalty however the statistical properties of this procedure are not known in this paper we first give an upper bound on the uniform norm of the difference between the true solutions and their approximations then we use this bound to prove the consistency and asymptotic normality of this estimation procedure we show that the asymptotic covariance matrix is the same as that of the maximum likelihood estimation therefore this procedure is asymptotically efficient for a fixed sample and fixed basis functions we study the limiting behavior of the approximation when the smoothing parameter tends to infinity we propose an algorithm to choose the smoothing parameters and a method to compute the deviation of the spline approximation from solution without solving the odes

gravity inversion of an interface above which the density contrast varies exponentially with depth mapping of an interface above which the density contrast varies exponentially with depth as is common at the basement surface of sedimentary basins is efficiently achieved by a theoretically precise gravity method which can be applied to either profile data or twodimensional data the contrast in mass above the interface is modeled by an array of vertical rectangular prisms with density contrasts varying exponentially with depth gravity anomalies due to the prisms are calculated in the wavenumber domain and then converted to the space domain the precision of the inverse numerical fourier transform in this procedure is significantly increased by a shift sampling technique based on the discrete fourier deviation equation depth to the interface is determined by iterative adjustment of the vertical extent of the prisms in accordance with observed gravity anomaly data the basement surface of the los angeles basin california calculated by this method closely duplicates the published configuration based

gravity interpretation using fourier transforms and simple geometrical models with exponential density contrast the decrease of density contrast in sedimentary basins can often be approximated by an exponential function theoretical fourier transforms are derived for symmetric trapezoidal vertical fault vertical prism syncline and anticline models this is desirable because there are no equivalent closed form solutions in the space domain for these models combined with an exponential density contrast these transforms exhibit characteristic minima maxima and zero values and hence graphical methods have been developed for interpretation of model parameters after applying end corrections to improve the discrete transforms of observed gravity data the transforms are interpreted for model parameters this method is first tested on two synthetic models then applied to gravity anomalies over the san jacinto graben and los angeles basin

a jacobi eigenreduction algorithm for definite matrix pairs we propose a jacobi eigenreduction algorithm for symmetric definite matrix pairsa j of small to medium size real symmetric matrices withj 2 i j diagonal neitherj nora itself need be definite our jacobi reduction works only on one matrix and usesj orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process for the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization we prove the global convergence of the method the quadratic convergence was observed in all experiments we apply our method in two situations i eigenreducing a single real symmetric matrix and ii eigenreducing an overdamped quadratic matrix pencil in both cases our method is preceded by a symmetric indefinite decomposition and performed in its one sided variant on the thus obtained factors our method outdoes the standard methods like standard jacobi orqr ql in accuracy in spite of the use of hyperbolic transformations which are not orthogonal a theoretical justification of this behaviour is made elsewhere the accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order in addition in working with quadratic pencils our method is shown to either converge or to detect non overdampedness

network reconstruction from high dimensional ordinary differential equations abstractwe consider the task of learning a dynamical system from high dimensional time course data for instance we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points we model the dynamical system nonparametrically as a system of additive ordinary differential equations most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations this is known to be challenging and inefficient we propose a novel approach that does not involve derivative estimation we show that the proposed method can consistently recover the true network structure even in high dimensions and we demonstrate empirical improvement over competing approaches supplementary materials for this article are available online

a hierarchically blocked jacobi svd algorithm for single and multiple graphics processing units we present a hierarchically blocked one sided jacobi algorithm for the singular value decomposition svd targeting both single and multiple graphics processing units gpus the blocking structure reflects the levels of the gpus memory hierarchy the algorithm may outperform magma s textttdgesvd while retaining high relative accuracy to this end we developed a family of parallel pivot strategies on the gpu s shared address space but applicable also to inter gpu communication unlike common hybrid approaches our algorithm in a single gpu setting needs a cpu for the controlling purposes only while utilizing the gpu s resources to the fullest extent permitted by the hardware when required by the problem size the algorithm in principle scales to an arbitrary number of gpu nodes the scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a tesla s2050 system with four gpus versus a single fermi card

the gravity anomaly of two dimensional sources with continuous density distribution and bounded by continuous surfaces the gravity anomaly of a complicated two dimensional 2 d source having arbitrary surfaces and density varying in either horizontal or vertical direction is calculated using a combination of closed form solutions and numerical integration the surfaces and density can be defined by continuous or piecewise continuous two dimensional functions in the integration interval for example the anomalies for intrusions or folded sedimentary units having an arbitrary density in the horizontal direction and a polynomial density distribution in the vertical direction can be calculated using surfaces represented by functions of the horizontal dimension when modeling dipping layered intrusions or sedimentary beds the surfaces are represented by functions of the vertical dimension in which case the density can be an arbitrary function of depth and a polynome function of horizontal coordinate the accuracy of the method is defined by the user the method gives simple and general equations to calculate anomalies of c

gravimetric analysis of uniform polyhedra analytical formulas for the gravity anomaly of a uniform polyhedral body are subject to numerical error that increases with distance from the target while the anomaly decreases this leads to a limited range of target distances in which the formulas are operational beyond which the calculations are dominated by rounding error we analyze the sources of error and propose a combination of numerical and analytical procedures that exhibit advantages over existing methods namely 1 errors that diminish with distance 2 enhanced operating range and 3 algorithmic simplicity the latter is achieved by avoiding the need to transform coordinates and the need to discriminate between projected observation points that lie inside on or outside a target facet boundary our error analysis is verified in computations based on a published code and on a code implementing our methods the former requires a numerical precision of one part in 1016 double precision in problems of geophysical interest whereas our c

a bayesian approach to modeling 2d gravity data using polygons abstractwe apply a bayesian markov chain monte carlo formalism to the gravity inversion of a single localized 2d subsurface object the object is modeled as a polygon described by five parameters the number of vertices a density contrast a shape limiting factor and the width and depth of an encompassing container we first constrain these parameters with an interactive forward model and explicit geologic information then we generate an approximate probability distribution of polygons for a given set of parameter values from these we determine statistical distributions such as the variance between the observed and model fields the area the center of area and the occupancy probability the probability that a spatial point lies within the subsurface object we introduce replica exchange to mitigate trapping in local optima and to compute model probabilities and their uncertainties we apply our techniques to synthetic data sets and a natural data set collected across the rio grande gorge bridge i

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa

on the global convergence of the eberlein method for real matrices the global convergence proof of the column and row cyclic eberlein diagonalization process for real matrices is given the convergence to a fixed matrix in murnaghan form is obtained with the well known exception of complex conjugate pairs of eigenvalues whose real parts are more than double