born inversion with a variable background velocity direct or born inversion has been presented as an approximate solution of the multidimensional seismic acoustic inverse problem e g cohen and bleistein 1979 phinney and frazer 1978 raz 1981 clayton and stolt 1981 the most basic assumption of the born model is that variations or perturbations of some reference model remain small this requirement of smallness is due to the fact that errors are directly related to the value of the perturbations both the local and overall values of the perturbation variable contribute to these errors which are reduced if the known reference model or background velocity closely matches the actual medium velocity

explicit solution of a stochastic irreversible investment problem and its moving threshold we consider a firm producing a single consumption good that makes irreversible investments to expand its production capacity the firm aims to maximize its expected total discounted real profit net of investment on a finite horizont the capacity is modeled as a controlled ita process where the control is the real investment which is not necessarily a rate but more generally a monotone process the result is a singular stochastic control problem we introduce the associated optimal stopping problem that is the optimal cost of not investing its variational formulation turns out to be a parabolic obstacle problem which we explicitly solve in the case of constant relative risk aversion crra production functions the moving free boundary is the threshold at which the shadow value of installed capital exceeds the capital s replacement cost then we use the equation of the free boundary to evaluate the optimal investment policy and its corresponding optimal profits

adaptive multiresolution approach for two dimensional pdes a multiresolution adaptive approach for the solution of two dimensional partial differential equations pdes is presented this methodology is a multidimensional extension of that presented in a previous work comput methods appl mech engrg 191 2002 3909 the method proposed is unconditionally stable by incorporating convection differencing schemes with the tvd property and the grid is dynamically adapted so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed the two desired properties of any pde solver stability and accuracy are therefore retained numerical results for four test problems are presented which serve to demonstrate the robustness and cost effectiveness of the method

approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

a multidimensional stochastic singular control problem via dynkin game and dirichlet form the traditional difficulty with stochastic singular control is characterizing the regularities of the value function and the optimal control policy in this paper a multidimensional singular control problem is considered we found the optimal value function and the optimal control policy of this problem via a dynkin game whose solution is given by the saddle point of the cost function the existence and uniqueness of the solution to this dynkin game are proved through an associated variational inequality problem involving dirichlet form as a consequence the properties of the value function of this dynkin game imply the smoothness of the value function of the stochastic singular control problem in this way we are able to show the existence of a classical solution to this multidimensional singular control problem which was traditionally solved in the sense of viscosity solutions and this enables the application of the verification theorem to prove optimality

instantaneous control of brownian motion a controller continuously monitors a storage system such as an inventory or bank account whose content z zt t 0 fluctuates as a 2 brownian motion in the absence of control holding costs are incurred continuously at rate h zt at any time the controller may instantaneously increase the content of the system incurring a proportional cost of r times the size of the increase or decrease the content at a cost of l times the size of the decrease we consider the case where h is convex on a finite interval and h outside this interval the objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon it is shown that there exists an optimal control limit policy characterized by two parameters a and b a b roughly speaking this policy exerts the minimum amounts of control sufficient to keep zt a b for all t 0 put another way the optimal control limit policy imposes on z a lower reflecting barrier at a an

generalized covariance functions associated with the laplace equation and their use in interpolation and inverse problems this work examines which generalized covariance function when used in the stochastic approach produces the flattest possible estimate of an unknown function that is consistent with the data such an estimate is the plainest possible continuous function thus in a sense eliminating details that are irrelevant or unsupported by data the answer is found from the solution of the following variational problem determine the function that reproduces the data has the smallest gradient in the square norm sense and has a gradient that vanishes at large distances from the observations the generalized covariance functions are shown to be the green s functions for the free space laplace equation the linear distance in one dimension the logarithmic distance in two dimensions and the inverse distance in three dimensions it is demonstrated that they are appropriate covariance functions for intrinsic random fields a modification is proposed to facilitate numerical implementation and a couple of examples are presented to illustrate the applicability of the methodology

quasi linear geostatistical theory for inversing a quasi linear theory is presented for the geostatistical solution to the inverse problem the archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations the unknown is parameterized as a realization of a random field and the estimation problem is solved in two phases structural analysis where the random field is characterized followed by estimation of the log transmissivity conditional on all observations the proposed method generalizes the linear approach of kitanidis and vomvoris 1983 the generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements i e there are enough observations that the variance of estimation error of the log transmissivity is small the methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent the applicability of the methodology is demonstrated through an example that includes structural analysis determination of best estimates and conditional simulations

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

extended local born fourier migration method a migration approach based on a local application of the born approximation within each extrapolation interval contains a singularity that can make direct application unstable previous authors have suggested adding an imaginary part to the vertical wavenumber to eliminate the singularity however their method requires that the reference slowness must be the maximum slowness of a given layer consequently the slowness perturbations are larger than those when the average slowness is selected as a reference slowness therefore its applicability is limited we develop an extended local born fourier migration method that circumvents the singularity problem of the local born solution and makes it possible to choose the average slowness as a reference slowness it is computationally efficient because of the use of a fast fourier transform algorithm it can handle wider angles or steeper interfaces and scattering effects of heterogeneities more accurately than the split step fourier ssf method which acco

connections between optimal stopping and singular stochastic control ii reflected follower problems the stochastic control problem of following a brownian path by a process of bounded variation and subject to a reflecting barrier at the origin reflected follower is reduced to a question of optimal stopping with absorption direct probabilistic arguments are used to establish the equivalence of the two problems under suitable conditions on the cost functions

parabolic variational inequality with parameter and gradient constraints abstract this paper concerns a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints the method is to transform a two dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies moreover we analyze the behaviors of the free boundary surfaces the main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter

multiresolution schemes on triangles for scalar conservation laws this paper proposes a multiresolution procedure adapted to triangular cell averages to improve the performance of finite volume schemes by reducing flux evaluation cost using the approach introduced by a harten a specific coarse to fine prediction scheme is proposed that ensures the stability of the computations even when a large number of scales are involved numerical tests are presented that illustrate the computational gain as well as the order of accuracy of the scheme

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

the sensitivity of born inversion to the choice of reference velocity a simple example we examine the sensitivity of the born model to the input background velocity we use a one dimensional analytic example to point out the difference between a corrective procedure and merely a perturbative one we examine various aspects of the sensitivity issue including the trade off between velocity determination and mapping of reflector location although this problem is discussed within the context of the born model it is an issue common to all perturbative methods e g migration methods which transform surface reflection data into a map of subsurface reflectors

least squares wave equation migration for avp ava inversion we present an acoustic migration inversion algorithm that uses extended double square root wave equation migration and modeling operators to minimize a constrained least squares data misfit function least squares migration we employ an imaging principle that allows for the extraction of ray parameter domain common image gathers cigs from the propagated seismic wavefield the cigs exhibit amplitude variations as a function of half offset ray parameter avp closely related to the amplitude variation with reflection angle ava our least squares wave equation migration inversion is constrained by a smoothing regularization along the ray parameter this approach is based on the idea that rapid amplitude changes or discontinuities along the ray parameter axis result from noise incomplete wavefield sampling and numerical operator artifacts these discontinuities should therefore be penalized in the inversion the performance of the proposed algorithm is examined with two synthetic examples in the first

optimal switching in an economic activity under uncertainty this paper considers the problem of finding the optimal sequence of opening starting and closing stopping times of a multi activity production process given the costs of opening running and closing the activities and assuming that the state of the economic system is a stochastic process the problem is formulated as an extended impulse control problem and solved using stochastic calculus as an application the optimal starting and stopping strategy are explicitly found for a resource extraction when the price of the resource is following a geometric brownian motion

long term optimal investment strategies in the presence of adjustment costs we consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment in particular we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project s output commodity we model this economic indicator by means of a general one dimensional ergodic diffusion at any time the project s capacity level can be increased or decreased at given proportional costs the aim is to maximize an ergodic performance criterion that reflects the long term average payoff resulting from the project s management we solve this genuinely two dimensional stochastic control problem by constructing an explicit solution to an appropriate hamilton jacobi bellman equation and by fully characterizing an optimal investment strategy

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

a model for reversible investment capacity expansion we consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and or demand fluctuations we model market uncertainty by means of a geometric brownian motion and we consider general running payoff functions our model allows for capacity expansion as well as for capacity reduction with each of these actions being associated with proportional costs the resulting optimization problem takes the form of a singular stochastic control problem that we solve explicitly we illustrate our results by means of the so called cobb douglas production function the problem that we study presents a model in which the associated hamilton jacobi bellman equation admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function which may be identically equal to infty thus our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimization applications in finance and economics as well as in other fields

an optimal true amplitude least squares prestack depth migration operator in order to define an optimal true amplitude prestack depth migration for multishot and multitrace data we develop a general methodology based on the least squares data misfit function associated with a forward model the amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data the migrated section is then the gradient of the cost function multiplied by a weight matrix a study of the hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this hessian thanks to this matrix the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least squares migrations applied to a forward model based on born ray tracing and diffracting points approximation this optimal migration outperforms or at least equals the classic kirchhoff formula since the latter belongs to the class of least squares migrations and is only optimal for one shot and an infinite aperture numerical results illustrate this construction and confirm the above expectations

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

optimal control under a dynamic fuel constraint we present a new approach to solve optimal control problems of the monotone follower type the key feature of our approach is that it allows us to include an arbitrary dynamic fuel constraint instead of dynamic programming we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the snell envelope of the objective functional s gradient at the optimum the optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in bank and el karoui 2004 ann probab 32 pp 1030 1067 as an illustration we show how our methodology allows us to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics

generalized linear inversion and the first born theory for acoustic media a procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional born inversion method the method we present is a more general born theory which can accommodate insufficient and inaccurate data this general method reduces to the ordinary born procedure when the data requirements of the latter technique are satisfied

a new sensitivity controlled focusing regularization scheme for the inversion of induced polarization data based on the minimum gradient support we present a new application of a focusing regularization scheme for the inversion of resistivity and induced polarization ip data that supports large resistivity magnitude and phase contrasts similar approaches so far have only been used for the interpretation of gravity magnetic or seismic data sets unlike methods based on smoothness constraints the approach is able to resolve sharp boundaries of bodies and layers and it allows slight parameter variations within them therefore it can be used in hydrogeologic applications where we need focused images to resolve high contrast aquifer boundaries our approach is based on the minimum gradient support which seeks to minimize the occurrence of parameter contrasts independent of their magnitude we study the effects of a variable control parameter on the reweighting optimization allowing a continuous transition from smooth to sharp images we also take the spatially varying sensitivity into account to allow focusing even where sensitivities are small the implemented weighting leads to increased smoothing in well resolved areas and a decrease in regions of lower sensitivity the opposite approach is examined as well this gradient dependent sensitivity weighting is basically an extension of depth dependent sensitivity weighting we demonstrate the effectiveness and limitations of the approach and the influence of the control parameter using different synthetic models and field data from a hydrogeophysical test site the technique has proven particularly suitable for revealing sharp parameter contrasts

the effects of noise on occam s inversion of resistivity tomography data an occam s inversion algorithm for crosshole resistivity data that uses a finite element method forward solution is discussed for the inverse algorithm the earth is discretized into a series of parameter blocks each containing one or more elements the occam s inversion finds the smoothest 2 d model for which the chi squared statistic equals an a priori value synthetic model data are used to show the effects of noise and noise estimates on the resulting 2 d resistivity images resolution of the images decreases with increasing noise the reconstructions are underdetermined so that at low noise levels the images converge to an asymptotic image not the true geoelectrical section if the estimated standard deviation is too low the algorithm cannot achieve an adequate data fit the resulting image becomes rough and irregular artifacts start to appear when the estimated standard deviation is larger than the correct value the resolution decreases substantially the image is too smooth the same effects are demonstrated for field data from a site near livermore california however when the correct noise values are known the occam s results are independent of the discretization used a case history of monitoring at an enhanced oil recovery site is used to illustrate problems in comparing successive images over time from a site where the noise level changes in this case changes in image resolution can be misinterpreted as actual geoelectrical changes one solution to this problem is to perform smoothest but non occam s inversion on later data sets using parameters found from the background data set

explicit solution to an optimal switching problem in the two regime case this paper considers the problem of determining the optimal sequence of stopping times for a diffusion process subject to regime switching decisions this is motivated in the economics literature by the investment problem under uncertainty for a multi activity firm involving opening and closing decisions we use a viscosity solutions approach combined with the smooth fit property and explicitly solve the problem in the two regime case when the state process is of geometric brownian nature the results of our analysis take several qualitatively different forms depending on model parameter values

on the starting and stopping problem application in reversible investments in this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process we use backward stochastic differential equations bsdes and snell envelopes finally we give some numerical results

prestack migration by split step dsr the double square root dsr prestack migration equation though defined for depth variable velocity can be used to image media with strong velocity variations using a phase shift plus interpolation pspi or split step correction the split step method is based on applying a phase shift correction to the extrapolated wavefield a correction that attempts to compensate for the lateral velocity variations i show how to extend dsr prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the marmousi data set the split step dsr migration is very fast and offers excellent imaging results

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme

finite horizon optimal multiple switching with signed switching costs this paper is concerned with optimal switching over multiple modes in continuous time and on a finite horizon the performance index includes a running reward terminal reward and switching costs that can belong to a large class of stochastic processes particularly the switching costs are modelled by right continuous with left limits processes that are quasi left continuous and can take positive and negative values we provide sufficient conditions leading to a well known probabilistic representation of the value function for the switching problem in terms of interconnected snell envelopes we also prove the existence of an optimal strategy in a suitable class of admissible controls defined iteratively in terms of the snell envelope processes

connections between singular control and optimal switching this paper builds a new theoretical connection between singular control of finite variation and optimal switching problems this correspondence provides a novel method for solving high dimensional singular control problems and enables us to extend the theory of reversible investment sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions consequently our regularity result links singular controls and dynkin games through sequential optimal stopping problems

multiresolution schemes for the numerical solution of 2 d conservation laws i a generalization of harten s multiresolution algorithms to two dimensional 2 d hyperbolic conservation laws is presented given a cartesian grid and a discretized function on it the method computes the local scale components of the function by recursive diadic coarsening of the grid since the function s regularity can be described in terms of its scale or multiresolution analysis the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth instead in those locations the divergence of the solution is interpolated from the next coarser grid level first the basic 2 d essentially nonoscillatory eno scheme is presented then the 2 d multiresolution analysis is developed and finally the subsequent scheme is tested numerically the computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well

a born wkbj inversion method for acoustic reflection data density and bulk modulus variations in an acoustic earth are separately recoverable from standard reflection surveys by utilizing the amplitude versus offset information present in the observed wave fields both earth structure and a variable background velocity can be accounted for by combining the born and wkbj approximations in a before stack migration with two output sections one for density variations and the other for bulk modulus variations for the inversion the medium is considered to be composed of a known low spatial frequency variation the background plus an unknown high spatial frequency variation in bulk modulus and density the reflectivity the division between the background and the reflectivity depends upon the frequency content of the source for constant background parameters computations are done in the fourier domain where the first part of the algorithm includes a frequency shift identical to that in an f k migration the modulus and density variations are then determined by observing in a least squares sense amplitude versus offset wavenumber for a spatially variable background wkbj green s operators that model the direct wave in a medium with a smoothly varying background are used a downward continuation with these operators removes the effects of variable velocity from the problem and consequently the remainder of the inversion essentially proceeds as if the background were constant if the background is strictly depth dependent the inversion can be expressed in closed form the method neglects multiples and surface waves and it is restricted to precritical reflections density is distinguishable from bulk modulus only if a sufficient range of precritical incident angles is present in the data

two ways to quantify uncertainty in geophysical inverse problems we present two approaches to invert geophysical measurements and estimate subsurface properties and their uncertainties when little is known a priori about the size of the errors associated with the data we illustrate these approaches by inverting first arrival traveltimes of seismic waves measured in a vertical well to infer the variation of compressional slowness in depth first we describe a bayesian formulation based on probability distributions that define prior knowledge about the slowness and the data errors we use an empirical bayes approach where hyperparameters are not well known ahead of time e g the variance of the data errors and are estimated from their most probable value given the data the second approach is a non bayesian formulation that we call spectral in the sense that it uses the power spectral density of the traveltime data to constrain the inversion e g to estimate the variance of the data errors in the spectral approach we vary assumptions made about the characteristics of the slowness signal and evaluate the resulting slowness estimates and their uncertainties this approach is computationally simple and starts from a few assumptions these assumptions can be checked during the analysis on the other hand it requires evenly spaced traveltime measurements and it cannot be extended easily e g to data that have gaps in contrast the bayesian framework is based on a general theory that can be generalized immediately but it is more involved computationally despite the conceptual and practical differences we find that the two approaches give the same results when they start from the same assumptions the allegiance to a bayesian or non bayesian formulation matters less than what one is willing to assume when solving the inverse problem

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

optimal dynamic risk taking we analyze a continuous time stochastic control problem that arises in the study of several important issues in financial economics an agent controls the drift and volatility of a diffusion output process by dynamically selecting one of an arbitrary but finite number of projects and the termination time the optimal policy depends on the projects risk adjusted drifts that are determined by their drifts volatilities and the curvature or relative risk aversion of the agent s payoff function we prove that the optimal policy only selects projects in the spanning subset furthermore if the projects risk adjusted drifts are consistently ordered for all output values then the optimal policy is characterized by at most k 1 switching triggers where k is the number of projects in the spanning subset we also characterize the optimal policy when the consistent ordering condition does not hold and we outline a general and tractable computational algorithm to derive the optimal policies

2 d and 3 d interpretation of electrical tomography measurements part 2 the inverse problem the interpretation of borehole to borehole electrical measurements requires solving an inverse problem for a given class of model geometries the usual approach to an inverse problem includes a model dependent task i e forward modeling and a generic task i e an optimization scheme we have developed an optimization algorithm using a nonlinear inversion technique this algorithm allows recovery of a possible resistivity distribution in an investigated zone between two boreholes or in the vicinity of them this resistivity distribution is defined as a set of 2 d or 3 d volumes of constant resistivity the inversion procedure minimizes a least squares term plus a damping term this latter term seeks to minimize the roughness of the solution an improved form of this smoothness term may enhance the spatial resolution of the resistivity image assuming that the resistivity contrast is known a priori this reconstruction algorithm has been tested for both 2 d and 3 d geometries these inversion tests we

a class of singular control problems and the smooth fit principle this paper analyzes a class of singular control problems for which value functions are not necessarily smooth necessary and sufficient conditions for the well known smooth fit principle along with the regularity of the value functions are given explicit solutions for the optimal policy and for the value functions are provided in particular when payoff functions satisfy the usual inada conditions the boundaries between action and continuation regions are smooth and strictly monotonic as postulated and exploited in the existing literature see a k dixit and r s pindyck investment under uncertainty princeton university press princeton nj 1994 m h a davis et al adv in appl probab 19 1987 pp 156 176 t o kobila stochastics stochastics rep 43 1993 pp 29 63 a b abel and j c eberly j econom dynam control 21 1997 pp 831 852 a oksendal finance stoch 4 2000 pp 223 250 j a scheinkman and t zariphopoulou j econom theory 96 2001 pp 180 207 a merhi and m zervos siam j control optim 46 2007 pp 839 876 and l h alvarez a general theory of optimal capacity accumulation under price uncertainty and costly reversibility working paper helsinki center of economic research helsinski finland 2006 illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions

variational bayesian inversion of synthetic 3d controlled source electromagnetic geophysical data abstractinversion of controlled source electromagnetic data is dealt with for a geophysical application the goal is to retrieve a map of conductivity of an unknown body embedded in a layered underground from measurements of the scattered electric field that results from its interaction with a known interrogating wave this constitutes an inverse scattering problem whose associated forward problem is described by means of electric field domain integral equations the inverse problem is solved in a bayesian framework in which prior information is introduced via a gauss markov potts model this model describes the body as being composed of a finite number of different materials distributed into compact homogeneous regions the posterior distribution of the unknowns is approached by means of the variational bayesian approximation as a separable distribution that minimizes the kullback leibler divergence with respect to the posterior law thus we get a parametric model for the distributions of the induced cu

target oriented wave equation inversion a target oriented strategy can be applied to estimate a wave equation least squares inverse lsi image by explicitly computing the wave equation hessian the lsi image is obtained as the solution of a nonstationary least squares inverse filtering problem the rows of the hessian are the nonstationary filters containing information about the acquisition geometry the velocity model and the band limited characteristics of the seismic data by exploiting the sparsity and the structure of the hessian matrix a large number of iterations necessary to achieve convergence can be computed cheaply the results on a structurally complex model show the improvements of the lsi image versus the migrated image

robust and accelerated bayesian inversion of marine controlled source electromagnetic data using parallel tempering bayesian methods can quantify the model uncertainty that is inherent in inversion of highly nonlinear geophysical problems in this approach a model likelihood function based on knowledge of the data noise statistics is used to sample the posterior model distribution which conveys information on the resolvability of the model parameters because these distributions are multidimensional and nonlinear we used markov chain monte carlo methods for highly efficient sampling because a single markov chain can become stuck in a local probability mode we run various randomized markov chains independently to some extent this problem can be mitigated by running independent markov chains but unless a very large number of chains are run biased results may be obtained we got around these limitations by running parallel interacting markov chains with annealed or tempered likelihoods which enable the whole system of chains to effectively escape local probability maxima we tested this approach using a transdimensional algorithm where the number of model parameters as well as the parameters themselves were treated as unknowns during the inversion this gave us a measure of uncertainty that was independent of any particular parameterization we then subset the ensemble of inversion models to either reduce uncertainty based on a priori constraints or to examine the probability of various geologic scenarios we demonstrated our algorithms fast convergence to the posterior model distribution with a synthetic 1d marine controlled source electromagnetic data example the speed up gained from this new approach will facilitate the practical implementation of future 2d and 3d bayesian inversions where the cost of each forward evaluation is significantly more expensive than for the 1d case

bounded variation singular stochastic control and dynkin game we consider a bounded variation singular stochastic control problem with value v in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation bsde associated with this is a dynkin game with value u we establish the well known relation frac partial partial x v u for this general situation a saddle point for the dynkin game is given by the pair of first action times of an optimal control the methods are from stochastic analysis and include a priori estimates pathwise construction and comparison theorems for forward stochastic differential equations fsde and bsde

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

optimal switching over multiple regimes this paper studies the optimal switching problem for a general one dimensional diffusion with multiple more than two regimes this is motivated in the real options literature by the investment problem of a firm managing several production modes while facing uncertainties a viscosity solutions approach is employed to carry out a fine analysis on the associated system of variational inequalities leading to sharp qualitative characterizations of the switching regions these characterizations in turn reduce the switching problem into one of finding a finite number of threshold values in a state that would trigger switchings the results of our analysis take several qualitatively different forms depending on model parameters and the issue of when and where it is optimal to switch is addressed the general results are then demonstrated by the three regime case where a quasi explicit solution is obtained and a numerical procedure to find these critical values is devised in terms of the expectation functionals of hitting times for one dimensional diffusions

impulse control of multidimensional jump diffusions this paper studies regularity properties of the value function for an infinite horizon discounted cost impulse control problem where the underlying controlled process is a multidimensional jump diffusion with possibly infinite activity jumps surprisingly despite these jumps we obtain the same degree of regularity as for the diffusion case at least when the jump satisfies certain integrability conditions

resolution and uncertainty in 1d csem inversion a bayesian approach and open source implementation we show that resolution and uncertainty in csem inversion is most naturally approached using a bayesian framework resolution can be inferred by either hierarchical models with free parameters for effective correlation lengths bayesian smoothing or model choice frameworks applied to variable resolution spatial models bayesian splitting merging we find that typical 1d csem data can be modelled with quite parsimonious models typically o 10 parameters per common midpoint efficient opt imizations for the csem problem must address the challenges of poor scaling strong nonlinearity multimodality and the necessity of bound constraints the posterior parameter uncertainties are frequently controlled by the nonlinearity and linearised approaches to uncertainty are usually very poor in markov chain monte carlo mcmc approaches the nonlinearity and poor scaling make good mixing hard to achieve a novel approximate frequentist method we call the bayesianized parametric bootstrap sometimes called randomized maximum likelihood is much more efficient than mcmc in this problem is considerably better than linearized analysis but tends to modestly overstate uncertainties the software that implements these ideas for the 1d csem problem is made available under an open source license agreement

a multiresolution finite volume scheme for two dimensional hyperbolic conservation laws in this paper high resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two dimensional hyperbolic conservation laws the method is conservative moreover it is stable which is proven numerically in this paper the computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed using this proposed scheme we compute several two dimensional model problems and a compressive rate ranging from about 5 10 is observed in all simulations

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

adaptive multiresolution schemes for shock computations in this paper we present adaptive multiresolution schemes for the computation of discontinuous solutions of hyperbolic conservation laws starting with the given grid we consider the grid averages of the numerical solution for a hierarchy of nested grids which is obtained by diadic coarsening and we compute its equivalent multiresolution representation this representation of the numerical solution consists of the grid averages of the coarsest grid and the set of errors in predicting the grid averages of each level of resolution from the next coarser one these errors depend on the size of the grid and the local regularity of the solution at a jump discontinuity they remain the size of the jump independent of the level of resolution this observation enables us to identify the location of discontinuities in the numerical solution in a region of smoothness once the numerical solution is resolved to our satisfaction at a certain locality of some grid then the prediction errors there for this grid and all finer ones are small this enables us to obtain data compression by setting to zero the terms of the multiresolution representation that fall below a specified tolerance the numerical flux of the adaptive scheme is taken to be that of a standard centered scheme unless it corresponds to an identified discontinuity in which case it is taken to be the flux of an eno scheme we use the data compression of the numerical solution in order to reduce the number of numerical flux evaluations as follows we start with the computation of the exact numerical fluxes at the few grid points of the coarsest grid and then we proceed through diadic refinement to the given grid at each step of refinement we add values for the numerical flux at the new grid points which are the centers of the coarser cells wherever the solution is locally well resolved i e the corresponding prediction error is below the specified tolerance the costly exact value of the numerical flux function is replaced by an accurate enough approximate value which is obtained by an inexpensive interpolation from the values of the coarser grid

bayesian model choice asymptotics and exact calculations abstract model determination is a fundamental data analytic task here we consider the problem of choosing amongst a finite with loss of generality we assume two set of models after briefly reviewing classical and bayesian model choice strategies we present a general predictive density which includes all proposed bayesian approaches we are aware of using laplace approximations we can conveniently assess and compare asymptotic behavior of these approaches concern regarding the accuracy of these approximation for small to moderate sample sizes encourages the use of monte carlo techniques to carry out exact calculations a data set fit with nested non linear models enables comparison between proposals and between exact and asymptotic values

asymptotic properties of criteria for selection of variables in multiple regression in normal linear regression analysis many model selection rules proposed from various viewpoints are available for the information criteria aic fpe c p pss and bic the asymptotic distribution of the selected model and the asymptotic quadratic risk based on each criterion are explicitly obtained

on the one dimensional optimal switching problem we explicitly solve the optimal switching problem for one dimensional diffusions by directly using the dynamic programming principle and the excessive characterization of the value function the shape of the value function and the smooth fit principle then can be proved using the properties of concave functions