delay estimation by expected value the estimation of time delay between narrowband signals is discussed a minimum mean square error mmse estimator is used that utilizes the maximum likelihood ml window derived by knapp and carter in obtaining the required conditional probability density function from which the delay estimate is calculated the results of simulations using a narrowband signal are presented which evaluate and compare the mean square error of the mmse and ml processors the mmse estimator shows improvement over the ml estimator particularly in regions below the threshold signal to noise ratio conditions for application of the mmse processor are discussed as well as some important considerations for its implementation

quickest detection of a random pulse in white gaussian noise a class of stochastic processes characterized by rapid transitions in their structure is considered and the quickest detection of such transitions is studied in a bayesian framework the emphasis is on stochastic processes consisting of a randomly arrived causal pulse possibly with a set of random parameters such as amplitude and duration and an additive white gaussian noise in this model the pulse shape and the prior joint density of the arrival time and other random parameters are assumed known the task of quickest detection in this paper is described mathematically by minimizing the expected detection error the detection error is represented by a nonlinear function of the distance between the actual transition time and its associated detection time the assumptions on this function are fairly mild and allow to flexibly design its shape for a desired trade off between the detection delay and the false alarm rate two special cases of such design are well known error measures mean squared and mean absolute error the quickest detection problem a subclass of optimal stopping time problems is formulated as a stochastic optimal control problem and is resolved using dynamic programming the optimal detection rule is determined in terms of the solution of an integral equation that cannot be directly solved due to its complexity this equation is later used to develop a class of suboptimal detection rules and a lower bound on the minimum error using this lower bound it is shown for a numerical example that the suboptimal detector is nearly optimal

discrete time techniques for time delay estimation basic aspects of time delay estimation tde based on sampled signals are investigated the direct cross correlation method is analyzed and compared to the average square difference function asdf and the addition only based average magnitude difference function amdf estimators their relative accuracy is theoretically evaluated and previous empirical results are explained it is shown that both the asdf and the amdf based estimators outperform the direct cross correlation based estimator for medium high signal to noise ratios moreover the amdf based estimator which avoids any multiplications significantly reduces the computational complexity of the estimation procedure while offering only a moderate performance loss with respect to the asdf based estimator

estimation of input pulse locations from the response of an all pole transfer system using tapered rank reduction when ordinary rank reduction is used to remove the characteristics of the transfer system and to estimate the multipulse input time series the nonsignificant singular values are sharply cut off by low order truncation if the system q quality factor is high and then a large ripple occurs around each pulse location in order to avoid these difficulties a method is proposed in which the multipulse time series is estimated by a rank reduction using a tapering window in order to suppress the ripple due to the low order sharp truncation and then by applying the pole estimation method to the inverse fourier transform of the resultant time series thus the pulse locations are accurately estimated by using the pulse locations as the initial estimates the maximum likelihood estimates of the pulse locations are obtained from simulation experiments these principles are confirmed

a new integral equation method to solve highly nonlinear inverse scattering problems a family of new integral equations nie is proposed in this paper which are transformed from the original lippmann schwinger integral equation it can be shown that the nie can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects introduced by the multiple scattering behaviors in estimating the contrast equipping the previously proposed twofold subspace based optimization method with such nie the new inversion method is able to solve inverse scattering problems with strong scatterers like with high contrast and or large dimensions in terms of wavelength ones furthermore such a family of nie could provide a convenient tool to appraise reconstructed results several representative numerical tests are carried out using both synthetic and experimental data to verify the efficacy of the new inversion method

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

time domain analysis of cross correlation for time delay estimation with an autocorrelated signal the estimation accuracy of the time difference of arrival tdoa of an exponentially autocorrelated signal at two sensors in white noise is analyzed the estimate is obtained by cross correlating samples taken as short term integrals of the noisy signals from the two sensors this technique avoids ambiguities in the cross correlation and it is shown that the best sampling rate is double the nyquist rate for which the cramer rao lower bound crlb is met in practice

an experimental comparison of optimum and sub optimum filters effectiveness in the generalized correlator abstract various optimum and sub optimum filters have been added to the basic cross correlator in order to enhance its detection and estimation capabilities in the presence of noise these pre detection filters include the hannan thomson the eckart the smoothed coherence transform some least squares and two optimum filters recently derived by the authors here a quantitative comparison of all six filters has been conducted by extensive simulation for various input signal to noise ratios and signal and noise bandwidths the results define the actual enhancement in time delay detection and estimation when using the optimum filters and the amount of performance degradation from the optimum for the other filters the rank ordering of the filters is carried out histograms of the time delay estimates their mean variance and probability of detection are presented wherever available comparison of analytical predictions with the experimental results is noted

lower bounds on worst case probability of large error for two channel time delay estimation a lower bound on the worst case probability of large error for two channel time delay estimation using random signals is developed the bound is based on the minimum probability of error m ary hypothesis test using a maximum likelihood estimator mle the bound can only be evaluated approximately however it can be determined arbitrarily well via simulation if the mle can be instrumented the bound is evaluated in detail for uniform lowpass signal and noise spectra and large bandwidth time product it is then compared with another lower bound based on a binary hypothesis test with computer simulation results for an approximate mle instrumentation and with an upper bound established via computer simulation for a correlator for a uniform lowpass spectrum the m ary bound is shown to be very nearly a greatest lower bound over the full range of input signal to noise ratio further the correlator is shown to be very nearly an optimal instrumentation in the sense of reaching the minimum probability of large error

robust adapative kalman filtering for systems with unknown step inputs and non gaussian measurement errors target tracking with kalman filters is hampered by target maneuvering and unknown process and measurement noises we show that moving data windows may be used to analyze state and measurement error sequences determining robust estimates of bias and covariance for steps in the system forcing functions and non gaussian measurement errors the robust estimators yield improvements over linear bias and covariance estimators extensive simulations compare conventional linear adaptive and robust adaptive average step responses of a first order system filter quantities examined are state estimate state error process and measurement covariance estimates kalman gain and input step estimate

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

an inexact newton based approach to microwave imaging within the contrast source formulation in this paper an approach to microwave imaging based on an inexact newton method within the contrast source formulation of the inverse scattering problem is considered in this framework the iteratively linearized equations of the inverse problem are solved by means of the regularizing truncated landweber algorithm this approach allows an effective and robust tool to inspect dielectric objects as it is shown by the reported numerical results moreover thanks to the structure of the involved integral equations within the contrast source formulation computer memory saving can be obtained with respect to the electric field integral equation framework finally the proposed method is validated against experimental data kindly provided by the institute fresnel marseille france

delay estimation using narrow band processes array processing of narrow band gaussian signals is studied with emphasis on delay estimation the barankin bound is used to examine the effect of ambiguity on mean square measurement error when the bound is plotted as a function of signal to noise ratio one observes a distinct threshold above the critical signal to noise ratio the lower bound on mean square error is given by the cramer rao inequality which is approached by the barankin inequality under these conditions below the threshold the barankin bound can exceed the cramer rao bound by large factors the relative magnitude of the bounds in that region depends critically on the ratio of signal center frequency to signal bandwidth

a nested multi scaling inexact newton iterative approach for microwave imaging a microwave imaging technique based on the integration of the inexact newton method within a multi scaling strategy is proposed in the framework of the contrast field formulation of the electromagnetic inverse scattering the inversion problem is solved by means of a nested procedure that considers three different logical levels a an outer multi focusing loop aimed at implementing a synthetic zoom for focusing the scatterer support within the investigation domain b a local linearization of the original full nonlinear inverse scattering function and c a truncated landweber inner loop devoted to regularize the arising ill posed linear problem thanks to the features of the integrated approach a reliable inversion technique able to suitably face the non linearity and the ill posedness ill conditioning issues of the imaging problem is designed a numerical validation dealing with different objects measurement setups and noise conditions is carried out to assess the features and the potentialities as well as the limitations of the proposed strategy comparisons with bare approaches and other multi resolution formulations are presented as well

a schauder approach to degenerate parabolic partial differential equations with unbounded coefficients abstract motivated by applications to probability and mathematical finance we consider a parabolic partial differential equation on a half space whose coefficients are suitably holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half space we establish existence and uniqueness of solutions in weighted holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients in our companion article feehan and pop 12 we apply the main result of this article to show that the martingale problem associated with a degenerate elliptic partial differential operator is well posed in the sense of stroock and varadhan

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

nonlinear filters for bayesian estimation of pulse arrival time in additive white gaussian noise bayesian estimation of the arrival time of a single pulse is considered in the presence of a white noise the employed model is a stochastic process consisted of a randomly delayed causal pulse and an additive white gaussian noise where the prior density of the delay is known the history of this stochastic process is given at every point in time and the problem is to obtain the conditional expectation of an arbitrary function of the arrival time the paper adopts a stochastic differential equation approach to develop nonlinear filters to efficiently compute this estimation the filtering problem is resolved for a number of pulse shapes which allow for a finite dimensional solution these pulse shapes include step exponential and rectangular functions as well as a piecewise constant function which well approximates a broad class of waveforms the application of this filtering problem in real time pulse arrival detection is discussed the task of this operation is to report the event of pulse arrival as soon after occurrence as possible the performance of nonlinear filtering is numerically verified for this application

analysis of wide band cross correlation for time delay estimation this correspondence deals with the estimation of the time delay of a wide band signal based on cross correlation of samples of the received noisy signals at two locations the accuracy of the time delay estimate is expressed explicitly in terms of the sampling period resolution cell the signal to noise ratio snr and the number of samples in the data window simulations that confirm the validity of the estimation accuracy expressions for snr as low as 6 db for a 128 sample window are shown

c1 1 regularity for degenerate elliptic obstacle problems abstract the heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half plane the generator of this process with killing called the elliptic heston operator is a second order degenerate elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half plane in mathematical finance solutions to the obstacle problem for the elliptic heston operator correspond to value functions for perpetual american style options on the underlying asset with the aid of weighted sobolev spaces and weighted holder spaces we establish the optimal c 1 1 regularity up to the boundary of the half plane for solutions to obstacle problems for the elliptic heston operator when the obstacle functions are sufficiently smooth

a new generalized cross correlator a new generalized cross correlator gcc for the passive time delay estimation problem is presented the interpretation of this gcc is that of estimating the cross correlation function by cross correlating the least mean square estimates of the signal component in each of the observed waveforms the implementation is simply a gcc with the weighting filter equal to the magnitude coherency squared numerical evaluation of the performance of this processor and a robust version indicate that they compare favorably to some of the well known gcc procedures

error analysis and bounds in time delay estimation we have analyzed the error propagation in the classical delay estimation based on the identification of the maximum of the cross correlation between the input signals the resulting error propagation model has the form of a linear combination of the errors additional to the signal samples these errors are weighted by the derivative of the input signals at the sampling instants based on the error propagation model we have further found an upper bound to the square of the estimation error the use of the model is demonstrated by two simple examples

stochastic optimization methods applied to microwave imaging a review stochastic methods are now very common in electromagnetics among the various applications they have been recently proposed for solving inverse problems arising in radio frequency and microwave imaging some of the recently proposed stochastic inversion procedures are critically discussed e g genetic algorithms differential evolution methods memetic algorithms particle swarm optimizations hybrid techniques etc and the way they have been applied in this area the use of the ant colony optimization method which is a relatively new method in electromagnetics is also proposed various imaging modalities are considered tomography buried object detection and borehole sensing finally the main features making these approaches useful for imaging purposes are discussed and the currently considered strategies to reduce the computational load associated with stochastic optimization procedures delineated

delay estimation simulations and a normalized comparison of published results three recent publications have reported delay estimate variances from simulations using the generalized cross correlator method on both stationary random signals with noise and deterministic signals with noise further simulations with finite energy signals are reported here a method of comparing the three reported data sets is developed and a remarkable consistency is observed even though the simulation parameters have varied widely

application of the multiplicative regularized gauss newton algorithm for three dimensional microwave imaging we apply the so called multiplicative regularized gauss newton inversion algorithm for solving three dimensional electromagnetic microwave inverse problems this inversion algorithm automatically adjusts the regularization parameter and when combined with the total variation type regularization function it can provide inversion results with excellent edge preserving characteristics in addition in order to deal with an extensive memory requirement for the gauss newton method we employ an implicit jacobian calculation scheme by using this scheme we do not have to explicitly store the jacobian matrix hence we are able to significantly reduce the memory requirement of the gauss newton method albeit at an additional computational overhead furthermore in order to be able to handle a large scale computational problem both the forward and the inversion algorithms are parallelized using the mpi library where we obtain a nearly linear speedup factor we demonstrate efficiency and robustness of this algorithm by inverting synthetic data fresnel experimental data and biomedical experimental data

microwave data inversions using the source receiver compression scheme we apply a source receiver compression approach to reduce the computational time and memory usage of the nonlinear inversion approaches for interpreting three dimensional microwave data by detecting and quantifying the extent of redundancy in the data we assemble a reduced set of simultaneous sources and receivers that are weighted sums of the physical sources and receivers employed in the measurement setup because the number of these simultaneous sources and receivers can be significantly less than those of the physical sources and receivers the computational time and memory usage of any inversion method such as steepest descent nonlinear conjugate gradient contrast source inversion and quasi newton can be tremendously reduced the scheme is based on decomposing the data into their principal components using a singular value decomposition approach and the data compression is done through the elimination of eigenvectors corresponding to small eigenvalues consequently this will also suppress the effect of noise in the data as a concept demonstration we show that this approach has the potential of significantly reducing both computational time and memory usage of the gauss newton inversion method by few orders of magnitude

time delay estimation via cross correlation in the presence of large estimation errors the estimate of the difference in time of arrival of a common random signal received at two sensors each of which also receives uncorrelated noise is examined for both small and large estimation errors it is shown that as the post integration signal to noise ratio decreases the correlator exhibits a thresholding effect that is the probability of a large error an anomalous estimate increases rapidly approximate theoretical results for the probability of an anomaly are presented and are verified experimentally the variance of the time delay estimate is examined for both a gated mode in which the time delay corresponding to the correlation peak closest to the true time delay is used as the estimate of time delay and an ungated mode in which the time delay corresponding to the largest peak over the full range of the correlator delay times is used as the estimate the observed variance for both modes is compared with the theoretical variance based on a small error analysis for the gated modes the signal to noise ratio below which the observed variance begins to differ significantly from the small error theory can be reliably predicted from a linearity criterion it is shown however that the expected variance for the ungated mode can depart from the small error theory at a higher signal to noise ratio than for the gated modes thus the variance due to anomalies can be the most important factor in determining the region of applicability of the small error analysis

nonparametric narrowband wavefront estimation from wavefront gradients the wavefield sampled by a two dimensional 2 d planar array can he characterized by a delayed signal this correspondence investigates the problem of the nonparametric estimation of the delay function or wavefront when the waveform is narrowband the wavefront is estimated by integrating the local estimates of the differential time of delay tod of the waveform between neighboring sensors the ambiguity of the tod estimator for the narrowband waveform can be conveniently limited by the a priori search interval for the cross correlation method the reconstruction of the wavefronts is formulated as a least squares problem that is solved iteratively errors of the solution are quantified and the contribution caused by noise is shown to be negligible

alignment blur in coherently averaged images blurring of coherently averaged images due to imperfect alignment is studied and two restoration methods are proposed and evaluated it is shown that iterative realignment is more powerful than post filtering in reducing blur the value of averaging and restoration is illustrated on human subjects in noisy video sequences

on the determination of the position of extrema of sampled correlators the problem of determination of the position of the extremum of a continuous correlator function from its samples if the spectrum of the correlator is not bandlimited is discussed searching for the position of the extremum by fitting a parabola through three samples around that extremum leads to clustering of the estimates around values corresponding with the sampling moments

poisson models and mean squared error for correlator estimator of time delay a method for modeling large errors in correlation based time delay estimation is developed in terms of level crossing probabilities the level crossing interpretation for peak ambiguity leads directly to an exact expression for the probability of large error involving the hazard function associated with the level crossing process two models for the distribution of the error over the level crossing time yield approximations to the mean square error mse that involve the low order

shrinkage thresholding enhanced born iterative method for solving 2d inverse electromagnetic scattering problem a numerical framework that incorporates recently developed iterative shrinkage thresholding ist algorithms within the born iterative method bim is proposed for solving the two dimensional inverse electromagnetic scattering problem ist algorithms minimize a cost function weighted between measurement data misfit and a zeroth first norm penalty term and therefore promote sharpness in the solution consequently when applied to domains with sharp variations discontinuities or sparse content the proposed framework is more efficient and accurate than the classical bim that minimizes a cost function with a second norm penalty term indeed numerical results demonstrate the superiority of the ist bim over the classical bim when they are applied to sparse domains permittivity and conductivity profiles recovered using the ist bim are sharper and more accurate and converge faster

on the simulation of a class of time delay estimation algorithms the time delay between signals received at two or more sensors has proven to be a useful parameter in passive sonar for estimating the location of an acoustic source this paper presents the results of a simulation comparing the smoothed coherence transform and maximum likelihood estimation methods to the basic cross correlation technique for time delay estimation band limited random signals which are corrupted by white noise and received at two sensors are considered at various signal to noise ratios the variance of the time delay estimates are compared to the minimum variance obtainable in theory as given by the cramer rao lower bound

the structure and performance of estimators for real time estimation of randomly varying time delay structures for time delay estimation are described which are deduced using the maximum likelihood technique the performance of estimators designed for time invariant delays is discussed for the case of signals with a time variable delay a two step algorithm for the estimation of a randomly varying time delay between two stochastic signals is developed in the first step the maximum likelihood estimate is computed over an observation interval small enough to consider the delay to be approximately constant due to the short averaging interval the probability of ambiguous peaks in the estimation function and the variance of the estimate are greatly increased therefore in the second step a decision directed adaptive postfiltering algorithm is presented that effectively suppresses outliers and reduces the variance

analysis of discrete implementation of generalized cross correlator a common discrete implementation of the cross correlator uses a parabolic fit to the peak when the delay is not an integral multiple of the sampling period this correspondence analyzes and assesses the pitfalls of this approach it is shown that this yields a biased estimate of the time delay with both the bias and variance of the estimate dependent on the location of the delay between samples snr signal and noise bandwidths and the prefilter or window used in the generalized correlator

schauder a priori estimates and regularity of solutions to boundary degenerate elliptic linear second order partial differential equations abstract we establish schauder a priori estimates and regularity for solutions to a class of boundary degenerate elliptic linear second order partial differential equations furthermore given a c smooth source function we prove c regularity of solutions up to the portion of the boundary where the operator is degenerate boundary degenerate elliptic operators of the kind described in our article appear in a diverse range of applications including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance 10 25 generators of diffusion processes arising in mathematical biology 3 11 and the study of porous media 7 8

robust eckart filters for time delay estimation minimax or robust filters which optimize worst case performance have been considered previously for matched and wiener filtering in this paper we show how similar ideas can be applied to obtain robust versions of one particular filter the eckart filter for time delay estimation the use of a robust filter can eliminate the requirement of estimation of the signal and noise power spectral densities psd s and cross psd s robust filters being less sensitive in their performance to variations of the input psd s within specified classes of psd s explicit solutions for specific classes of psd s are derived and several numerical examples and simulation results are given similar formulations for robust filters should also be possible for other types of filters used in time delay estimation

passive bearing estimation of a broad band source the variance on the bearing estimates of a wide band source has been given wide attention in the literature under the assumption of unbiased estimation process this paper relaxes such a condition and derives the resulting statistical measures those lower bound measures depend on the tangent of the true bearing pt to the source and the bearing variance 2 u as derived with an unbiased condition in the far field the resulting bias isgiven simply by a tan t 2 u where a is a constant the bias size can be in the order of the unbiased bearing variance 2 u the variance is the sum of the unbiased variance 2 u and a combination of the bias and 2 u the variance is contrasted against the cramer rao bound developed for the biased estimate such statistical results enable the sizing of the bias and its deteriorating effect on the estimation process for the specific case of interest when the bias is unacceptable averaging should be applied to the time delays rather than the bearings the bias impact is also of interest when noisy bearings are applied to estimate other functions such as passive source motion at long ranges

new perspectives for maximum likelihood time delay estimation this paper introduces a new realization for maximum likelihood time delay estimation tde that illuminates the relationships between maximum likelihood tde and other methods we obtain the result by deriving the likelihood function using a fundamental method that appears to be new to the field of array processing this method is a natural complement to the generalized karhunen loeve expansion

exact maximum likelihood time delay estimation for short observation intervals an exact solution is presented to the problem of maximum likelihood time delay estimation for a gaussian source signal observed at two different locations in the presence of additive spatially uncorrelated gaussian white noise the solution is valid for arbitrarily small observation intervals that is the assumption t tau sub c mod d mod made in the derivation of the conventional asymptotic maximum likelihood aml time delay estimator where tau sub c is the correlation time of the various random processes involved and d is the differential time delay is relaxed the resulting exact maximum likelihood eml instrumentation is shown to consist of a finite time delay and sum beamformer followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one dimensional integral equation with nonconstant weight the solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities finally the performance of the eml and aml estimators is compared by means of computer simulations

blind identification of simo systems and simultaneous estimation of multiple time delays from hos based inverse filter criteria higher order statistics based inverse filter criteria ifc have been effectively used for blind equalization of single input multiple output simo systems recently chi and chen reported a relationship between the unknown simo system and the optimum equalizer designed by the ifc for finite signal to noise ratio snr in this paper based on this relationship an iterative fast fourier transform fft based nonparametric blind system identification bsi algorithm and an fft based multiple time delay estimation mtde algorithm are proposed with a given set of non gaussian measurements the proposed bsi algorithm allows the unknown simo system to have common subchannel zeros and its performance estimation accuracy is superior to that of the conventional ifc based methods the proposed mtde algorithm can simultaneously estimate all the p 1 time delays with respect to a reference sensor with space diversity of sensors exploited therefore its performance estimation accuracy is robust to the nonuniform distribution of snrs of p spl ges 2 sensors due to channel fading some simulation results are presented to support the efficacy of the proposed bsi algorithm and mtde algorithm

a classical perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate elliptic operators via holomorphic maps abstract we prove existence of solutions to boundary value problems and obstacle problems for degenerate elliptic linear second order partial differential operators with partial dirichlet boundary conditions using a new version of the perron method the elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by daskalopoulos and hamilton 9 in their study of the porous medium equation or the degeneracy of the heston operator 21 in mathematical finance existence of a solution to the partial dirichlet problem on a half ball where the operator becomes degenerate on the flat boundary and a dirichlet condition is only imposed on the spherical boundary provides the key additional ingredient required for our perron method surprisingly proving existence of a solution to this partial dirichlet problem with mixed boundary conditions on a half ball is more challenging than one might expect due to the difficulty in developing a global schauder estimate and due to compatibility conditions arising where the degenerate and non degenerate boundaries touch one cannot directly apply the continuity or approximate solution methods however in dimension two there is a holomorphic map from the half disk onto the infinite strip in the complex plane and one can extend this definition to higher dimensions to give a diffeomorphism from the half ball onto the infinite slab the solution to the partial dirichlet problem on the half ball can thus be converted to a partial dirichlet problem on the slab albeit for an operator which now has exponentially growing coefficients the required schauder regularity theory and existence of a solution to the partial dirichlet problem on the slab can nevertheless be obtained using previous work of the author and c pop 16 our perron method relies on weak and strong maximum principles for degenerate elliptic operators concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate elliptic operators and maximum and comparison principle estimates previously developed by the author 13