Gaussian Realizations Research Articles

Forecast of cross-correlation of Chinese Survey Space Telescope cosmic shear tomography with Ali CMB Polarization Telescope cosmic microwave background lensing

ABSTRACT We present a forecast study on the cross-correlation between cosmic shear tomography from the Chinese Survey Space Telescope (CSST) and cosmic microwave background (CMB) lensing from Ali CMB Polarization Telescope (AliCPT-1) in Tibet. The correlated galaxy and CMB lensing signals were generated from Gaussian realizations based on inputted auto and cross-spectra. To account for the error budget, we considered the CMB lensing reconstruction noise based on the AliCPT-1 lensing reconstruction pipeline; shape noise of the galaxy lensing measurement; CSST photo-z error; photo-z bias; intrinsic alignment (IA) effect; and multiplicative bias. The AliCPT-1 CMB lensing mock data were generated according to two experimental stages, namely the ‘4 modules*yr’ and ‘48 modules*yr’ cases. We estimate the cross-spectra in four tomographic bins according to the CSST photo-z distribution in the range of z ∈ [0, 4). After reconstructing the pseudo-cross-spectra from the realizations, we calculate the signal-to-noise ratio (SNR). By combining the four photo-z bins, the total cross-correlation SNR ≈ 15 (AliCPT-1 ‘4 modules*yr’) and SNR ≈ 22 (AliCPT-1 ‘48 modules*yr’). Finally, we study the cosmological application of this cross-correlation signal. Excluding IA in the template fitting would lead to roughly a 0.6σ increment in σ8 due to the negative IA contribution to the galaxy lensing data. For AliCPT-1 first and second stages, the cross-correlation of CSST cosmic shear with CMB lensing gives errors on the clustering amplitude $\sigma _{\sigma _8}=^{+0.043}_{-0.038}$ or $\sigma _{S_8}=\pm 0.031$ and $\sigma _{\sigma _8}=^{+0.030}_{-0.027}$ or $\sigma _{S_8}=\pm 0.018$, respectively.

The miniJPAS survey quasar selection – I. Mock catalogues for classification

ABSTRACT In this series of papers, we employ several machine learning (ML) methods to classify the point-like sources from the miniJPAS catalogue, and identify quasar candidates. Since no representative sample of spectroscopically confirmed sources exists at present to train these ML algorithms, we rely on mock catalogues. In this first paper, we develop a pipeline to compute synthetic photometry of quasars, galaxies, and stars using spectra of objects targeted as quasars in the Sloan Digital Sky Survey. To match the same depths and signal-to-noise ratio distributions in all bands expected for miniJPAS point sources in the range 17.5 ≤ r < 24, we augment our sample of available spectra by shifting the original r-band magnitude distributions towards the faint end, ensure that the relative incidence rates of the different objects are distributed according to their respective luminosity functions, and perform a thorough modelling of the noise distribution in each filter, by sampling the flux variance either from Gaussian realizations with given widths, or from combinations of Gaussian functions. Finally, we also add in the mocks the patterns of non-detections which are present in all real observations. Although the mock catalogues presented in this work are a first step towards simulated data sets that match the properties of the miniJPAS observations, these mocks can be adapted to serve the purposes of other photometric surveys.

Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every ε-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.

Not all peaks are created equal: the early growth of supermassive black holes

ABSTRACT In this work, we use the constrained Gaussian realization technique to study the early growth of supermassive black holes (SMBHs) in cosmological hydrodynamic simulations, exploring its relationship with features of the initial density peaks on large scales, ∼ 1 h−1Mpc. Our constrained simulations of volume (20 h−1Mpc)3 successfully reconstruct the large-scale structure as well as the black hole growth for the hosts of the rare 109 M⊙ SMBHs found in the bluetides simulation at z ∼ 7. We run a set of simulations with constrained initial conditions by imposing a $5 \sigma _0(R_\mathrm{G})$ peak on the scale of $\\ R_\mathrm{ G} = 1$ h−1Mpc varying different peak features, such as the shape and compactness as well as the tidal field surrounding the peak. We find that initial density peaks with high compactness and low tidal field induce the most rapid BH growth at early epochs. This is because compact density peaks with a more spherical large-scale matter distribution lead to the formation of the highest gas inflows (mostly radial) in the centres of haloes which boost the early BH accretion. Moreover, such initially compact density peaks in low tidal field regions also lead to a more compact BH host galaxy morphology. Our findings can help explain the tight correlation between BH growth and host galaxy compactness seen in observations.

Decomposition of multivariate spatial data into latent factors

Simulating spatial Gaussian realizations is one of the core components of geostatistics and numerous other fields involving uncertainty, risk, and reliability. The use of decorrelation methods and truncated Gaussian algorithms further promotes the use of Gaussian realizations. In multivariate cases, geological variables may represent different scales and exhibit different spatial structures and anisotropy that complicates decorrelation methods. For cases where spatial dependencies cannot be removed using techniques such as the projection pursuit multivariate transform coupled with maximum autocorrelation factors, cokriging is advocated and requires a model of coregionalization. However, blind source separation represents the original multivariate problem as a linear combination of latent source variables, each one having a spatial structure from the linear model of coregionalization (LMC). The latent source variables or factors are independent and follow a standard normal distribution facilitating the use of Gaussian simulation algorithms. Moreover, different algorithms may be utilized for each variable given that some are more efficient at generating realizations for different spatial covariance functions. Recovering the original variables afterwards is straightforward. The theory for this decomposition is presented. A small numerical example is used to explain the theory and the proposed approach is illustrated through a case study with geochemical sample data.

The early growth of supermassive black holes in cosmological hydrodynamic simulations with constrained Gaussian realizations

ABSTRACT The paper examines the early growth of supermassive black holes (SMBHs) in cosmological hydrodynamic simulations with different BH seeding scenarios. Employing the constrained Gaussian realization, we reconstruct the initial conditions in the large-volume bluetides simulation and run them to z = 6 to cross-validate that the method reproduces the first quasars and their environments. Our constrained simulations in a volume of $(15 \, h^{-1} {\rm Mpc})^3$ successfully recover the evolution of large-scale structure and the stellar and BH masses in the vicinity of a ${\sim}10^{12} \, M_{\odot }$ halo which we identified in bluetides at z ∼ 7 hosting a ${\sim}10^9 \, M_{\odot }$ SMBH. Among our constrained simulations, only the ones with a low-tidal field and high-density peak in the initial conditions induce the fastest BH growth required to explain the z > 6 quasars. We run two sets of simulations with different BH seed masses of 5 × 103, 5 × 104, and $5 \times 10^5 \, h^{-1} M_{\odot }$, (i) with the same ratio of halo to BH seed mass and (ii) with the same halo threshold mass. At z = 6, all the SMBHs converge in mass to ${\sim}10^9 \, M_{\odot }$ except for the one with the smallest seed in (ii) undergoing critical BH growth and reaching 108 – $10^9 \, M_{\odot }$, albeit with most of the growth in (ii) delayed compared to set (i). The finding of eight BH mergers in the small-seed scenario (four with masses 104 – $10^6 \, M_{\odot }$ at z > 12), six in the intermediate-seed scenario, and zero in the large-seed scenario suggests that the vast BHs in the small-seed scenario merge frequently during the early phases of the growth of SMBHs. The increased BH merger rate for the low-mass BH seed and halo threshold scenario provides an exciting prospect for discriminating BH formation mechanisms with the advent of multimessenger astrophysics and next-generation gravitational wave facilities.

Suppressing cosmic variance with paired-and-fixed cosmological simulations: average properties and covariances of dark matter clustering statistics

ABSTRACT Making cosmological inferences from the observed galaxy clustering requires accurate predictions for the mean clustering statistics and their covariances. Those are affected by cosmic variance – the statistical noise due to the finite number of harmonics. The cosmic variance can be suppressed by fixing the amplitudes of the harmonics instead of drawing them from a Gaussian distribution predicted by the inflation models. Initial realizations also can be generated in pairs with 180○ flipped phases to further reduce the variance. Here, we compare the consequences of using paired-and-fixed versus Gaussian initial conditions on the average dark matter clustering and covariance matrices predicted from N-body simulations. As in previous studies, we find no measurable differences between paired-and-fixed and Gaussian simulations for the average density distribution function, power spectrum, and bispectrum. Yet, the covariances from paired-and-fixed simulations are suppressed in a complicated scale- and redshift-dependent way. The situation is particularly problematic on the scales of Baryon acoustic oscillations where the covariance matrix of the power spectrum is lower by only $\sim 20{{\ \rm per\ cent}}$ compared to the Gaussian realizations, implying that there is not much of a reduction of the cosmic variance. The non-trivial suppression, combined with the fact that paired-and-fixed covariances are noisier than from Gaussian simulations, suggests that there is no path towards obtaining accurate covariance matrices from paired-and-fixed simulations – result, that is theoretically expected and accepted in the field. Because the covariances are crucial for the observational estimates of galaxy clustering statistics and cosmological parameters, paired-and-fixed simulations, though useful for some applications, cannot be used for the production of mock galaxy catalogues.

Hammurabi X: Simulating Galactic Synchrotron Emission with Random Magnetic Fields

Abstract We present version X of the hammurabi package, the HEALPix-based numeric simulator for Galactic polarized emission. Improving on its earlier design, we have fully renewed the framework with modern C++ standards and features. Multithreading support has been built in to meet the growing computational workload in future research. For the first time, we present precision profiles of the hammurabi line-of-sight integral kernel with multilayer HEALPix shells. In addition to fundamental improvements, this report focuses on simulating polarized synchrotron emission with Gaussian random magnetic fields. Two fast methods are proposed for realizing divergence-free random magnetic fields either on the Galactic scale where field alignment and strength modulation are imposed, or on a local scale where more physically motivated models like a parameterized magnetohydrodynamic (MHD) turbulence can be applied. As an example application, we discuss the phenomenological implications of Gaussian random magnetic fields for high Galactic latitude synchrotron foregrounds. In this, we numerically find B/E polarization-mode ratios lower than unity based on Gaussian realizations of either MHD turbulent spectra or in spatially aligned magnetic fields.

An extension of Phase Linearity Measurement for revealing cross frequency coupling among brain areas

BackgroundBrain areas need to coordinate their activity in order to enable complex behavioral responses. Synchronization is one of the mechanisms neural ensembles use to communicate. While synchronization between signals operating at similar frequencies is fairly straightforward, the estimation of synchronization occurring between different frequencies of oscillations has proven harder to capture. One specifically hard challenge is to estimate cross-frequency synchronization between broadband signals when no a priori hypothesis is available about the frequencies involved in the synchronization.MethodsIn the present manuscript, we expand upon the phase linearity measurement, an iso-frequency synchronization metrics previously developed by our group, in order to provide a conceptually similar approach able to detect the presence of cross-frequency synchronization between any components of the analyzed broadband signals.ResultsThe methodology has been tested on both synthetic and real data. We first exploited Gaussian process realizations in order to explore the properties of our new metrics in a synthetic case study. Subsequently, we analyze real source-reconstructed data acquired by a magnetoencephalographic system from healthy controls in a clinical setting to study the performance of our metrics in a realistic environment.ConclusionsIn the present paper we provide an evolution of the PLM methodology able to reveal the presence of cross-frequency synchronization between broadband data.

An inpainting approach to tackle the kinematic and thermal SZ induced biases in CMB-cluster lensing estimators

A galaxy cluster's own Sunyaev-Zel'dovich (SZ) signal is known to be a major contaminant when reconstructing the cluster's underlying lensing potential using cosmic microwave background (CMB) temperature maps. In this work, we develop a modified quadratic estimator (QE) that is designed to mitigate the lensing biases due to the kinematic and thermal SZ effects. The idea behind the approach is to use inpainting technique to eliminate the cluster's own emission from the large-scale CMB gradient map. In this inpainted gradient map, we fill the pixel values at the cluster location based on the information from surrounding regions using a constrained Gaussian realization. We show that the noise induced due to inpainting process is small compared to other noise sources for upcoming surveys and has negligible impact on the final lensing signal-to-noise. Without any foreground cleaning, we find a stacked mass uncertainty of 6.5% for the CMB-S4 experiment on a cluster sample containing 5000 clusters with M200c = 2 × 1014 M⊙ at z = 0.7. In addition to the SZ-induced lensing biases, we also quantify the low mass bias arising due to the contamination of the CMB gradient by the cluster convergence. For the fiducial cluster sample considered in this work, we find that this bias is negligible compared to the statistical uncertainties for both the standard and the modified QE even when modes up to ∼ 2700 are used for the gradient estimation. With more gradient modes, we demonstrate that the sensitivity can be increased by 14% compared to the fiducial result quoted above using gradient modes up to 2000.

Simulation of fine-scale electrical conductivity fields using resolution-limited tomograms and area-to-point kriging

SUMMARY Deterministic geophysical inversion approaches yield tomographic images with strong imprints of the regularization terms required to solve otherwise ill-posed inverse problems. While such tomograms enable an adequate assessment of the larger-scale features of the probed subsurface, the finer-scale details tend to be unresolved. Yet, representing these fine-scale structural details is generally desirable and for some applications even mandatory. To address this problem, we have developed a two-step methodology based on area-to-point kriging to generate fine-scale multi-Gaussian realizations from smooth tomographic images. Specifically, we use a co-kriging system in which the smooth, low-resolution tomogram is related to the fine-scale heterogeneity through a linear mapping operation. This mapping is based on the model resolution and the posterior covariance matrices computed using a linearization around the final tomographic model. This, in turn, allows us for analytical computations of covariance and cross-covariance models. The methodology is tested on a heterogeneous synthetic 2-D distribution of electrical conductivity that is probed with a surface-based electrical resistivity tomography (ERT) survey. The results demonstrate the ability of this technique to reproduce a known geostatistical model characterizing the fine-scale structure, while simultaneously preserving the large-scale structures identified by the smoothness-constrained tomographic inversion. Small discrepancies between the geophysical forward responses of the realizations and the reference synthetic data are attributed to the underlying linearization. Overall, the method provides an effective and fast alternative to more comprehensive, but computationally more expensive approaches, such as, for example, Markov chain Monte Carlo techniques. Moreover, the proposed method can be used to generate fine-scale multivariate Gaussian realizations from virtually any smoothness-constrained inversion results given the corresponding resolution and posterior covariance matrices.

Planckintermediate results

We study the statistical properties of interstellar dust polarization at high Galactic latitude, using the Stokes parameter Planck maps at 353 GHz. Our aim is to advance the understanding of the magnetized interstellar medium (ISM), and to provide a model of the polarized dust foreground for cosmic microwave background component-separation procedures. Focusing on the southern Galactic cap, we examine the statistical distributions of the polarization fraction ($p$) and angle ($\psi$) to characterize the ordered and turbulent components of the Galactic magnetic field (GMF) in the solar neighbourhood. We relate patterns at large angular scales in polarization to the orientation of the mean (ordered) GMF towards Galactic coordinates $(l_0,b_0)=(70^\circ \pm 5^\circ,24^\circ \pm 5^\circ)$. The histogram of $p$ shows a wide dispersion up to 25 %. The histogram of $\psi$ has a standard deviation of $12^\circ$ about the regular pattern expected from the ordered GMF. We use these histograms to build a phenomenological model of the turbulent component of the GMF, assuming a uniform effective polarization fraction ($p_0$) of dust emission. To model the Stokes parameters, we approximate the integration along the line of sight (LOS) as a sum over a set of $N$ independent polarization layers, in each of which the turbulent component of the GMF is obtained from Gaussian realizations of a power-law power spectrum. We are able to reproduce the observed $p$ and $\psi$ distributions using: a $p_0$ value of (26 $\pm$ 3)%; a ratio of 0.9 $\pm$ 0.1 between the strengths of the turbulent and mean components of the GMF; and a small value of $N$. We relate the polarization layers to the density structure and to the correlation length of the GMF along the LOS.

Optimization of variograms used for truncated plurigaussian simulation

Abstract Stochastic simulation of facies is a continuing and important area of research. To increase model performance and reproduce realistic relationships between categories (facies), an understanding of the geometries and internal architecture of the domain is required. Modelling of spatial categorical data with truncated bi-Gaussian simulation and generating the required mask that reproduces the desired category’s spatial relationships is an important, initial step in categorical variable modelling. Truncated bi-Gaussian simulation is typically used when there are known, complex spatial relationships between the categories. Truncation rules based on thresholds applied to the Gaussian realizations (i.e. the mask) control the proportions and ordering of categories in the simulation. The choice of these thresholds has a large effect on the final models. This work describes a program that is developed for truncated Gaussian simulation where the truncation rules are linear, but are locally varying to account for locally varying proportions. The appropriate truncation rules are calculated based on the user supplied locally varying proportion maps. Moreover, an optimization framework to determine the input variograms used to generate the initial Gaussian realizations is presented. Initially, the optimization is brute force with the best set of variograms carried forward; the second local refinement step is important in obtaining reasonable bi-Gaussian models. A case study simulating rock types at a mineral deposit is presented to illustrate the implementation of the proposed methodology.

Adiabatic Landau–Zener transitions at avoided level crossings with fast noise

Effects of a fast classical noise on adiabatic Landau–Zener (LZ) transitions between the (2S+1) Zeeman multiplets (diabatic states) of an arbitrary spin S at an avoided level crossing are investigated. The spin system is simultaneously coupled to a slow regular magnetic field and a fast random field with Gaussian realizations. In the longitudinal direction, the magnetic field changes its sign at the degeneracy point (and is unbounded at large positive and negative times t=±∞ far from the degeneracy point) while in its single transverse direction, it remains of constant amplitude. The noise is considered in the limit where its characteristic correlation time (decay time) is small enough compared to the characteristic time of adiabatic LZ transitions. With these considerations, the condition for adiabatic evolution allows us to analytically evaluate the populations of diabatic levels and coherence factors. The study is first implemented for two- (S=1/2) and three- (S=1) state systems and finally extended to arbitrary S. A numerical study is implemented allowing us to check/confirm the range of validity of our analytical solutions. We found a satisfactory quantitative agreement between numerical and analytical data.

Extreme value statistics of cosmic microwave background lensing deflection angles

The smaller the angular scales on which the anisotropies of the cosmic microwave background (CMB) are probed the more important their distortion due to gravitational lensing becomes. Here we investigate the maxima and minima of the CMB lensing deflection field using general extreme value statistics. Since general extreme value statistics applies to uncorrelated data in first place we consider appropriately low-pass filtered deflection maps. Besides the suppression of correlations filtering is required for another reason: The lensing field itself is not directly observable but needs to be (statistically) reconstructed from the lensed CMB by means of a quadratic estimator. This reconstruction, though, is noise dominated and therefore requires smoothing, too. In idealized Gaussian realizations as well as in realistically reconstructed data we find that both maxima and minima of the deflection angle components follow consistently a general extreme value distribution of Weibull-type. However, its shape, location and scale parameters vary significantly between different realizations. The statistics' potential power to constrain cosmological models appears therefore rather limited.

Statistically anisotropic Gaussian simulations of the CMB temperature field

Although theoretically expected to be Statistically Isotropic (SI), the observed Cosmic Microwave Background (CMB) temperature \& polarization field would exhibit SI violation due to various inevitable effects like weak lensing, Doppler boost and practical limitations of observations like non-circular beam, masking etc. However, presence of any SI violation beyond these effects may lead to a discovery of inherent cosmic SI violation in the CMB temperature \& polarization field. Recently, Planck presented strong evidence of SI violation as a dipolar power asymmetry of the CMB temperature field in two hemispheres. Statistical studies of SI violation effect require non-SI (nSI) Gaussian realizations of CMB temperature field. The nSI Gaussian temperature field leads to non-zero off-diagonal terms in the Spherical Harmonics (SH) space covariance matrix encoded in the coefficients of the Bipolar Spherical Harmonics (BipoSH) representation. We discuss an effective numerical algorithm, Code for Non-Isotropic Gaussian Sky (CoNIGS) to generate nSI realizations of Gaussian CMB temperature field of Planck like resolution with specific cases of SI violation. Realizations of nSI CMB temperature field are obtained for non-zero quadrupolar $(L=2)$ BipoSH measurements by WMAP, dipolar asymmetry (resembles $L=1$ BipoSH coefficients) with a \emph{scale dependent} modulation field as measured by Planck and for Doppler boosted CMB temperature field which also leads to $L=1$ BipoSH spectra. Our method, CoNIGS can incorporate any kind of SI violation and can produce nSI realizations efficiently.

A MULTI-LEVEL SOLVER FOR GAUSSIAN CONSTRAINED COSMIC MICROWAVE BACKGROUND REALIZATIONS

We present a multi-level solver for drawing constrained Gaussian realizations or finding the maximum likelihood estimate of the CMB sky, given noisy sky maps with partial sky coverage. The method converges substantially faster than existing Conjugate Gradient (CG) methods for the same problem. For instance, for the 143 GHz Planck frequency channel, only 3 multi-level W-cycles result in an absolute error smaller than 1 microKelvin in any pixel. Using 16 CPU cores, this translates to a computational expense of 6 minutes wall time per realization, plus 8 minutes wall time for a power spectrum-dependent precomputation. Each additional W-cycle reduces the error by more than an order of magnitude, at an additional computational cost of 2 minutes. For comparison, we have never been able to achieve similar absolute convergence with conventional CG methods for this high signal-to-noise data set, even after thousands of CG iterations and employing expensive preconditioners. The solver is part of the Commander 2 code, which is available with an open source license at http://commander.bitbucket.org/.

ON THE BISPECTRUM OF COSMIC STRING SEEDED CMB FLUCTUATIONS

I compute the bispectrum of maps of cosmic microwave background (CMB) fluctuations seeded by cosmic strings from large to 30 arcminute scales. Examining the distribution of triangle configurations and comparing with Gaussian realizations with the same power spectrum, I conclude that the CMB bispectrum cannot pick up the mild non-Gaussianity present in the maps and thus that it cannot characterize cosmic string induced non-Gaussianity produced in the regimes probed by these maps.

On the Likelihood and Fluctuations of Gaussian Realizations

The likelihood of Gaussian realizations, as generated by the Cholesky simulation method, is analyzed in terms of Mahalanobis distances and fluctuations in the variogram reproduction. For random sampling, the probability to observe a Gaussian realization vector can be expressed as a function of its Mahalanobis distance, and the maximum likelihood depends only on the vector size. The Mahalanobis distances are themselves distributed as a Chi-square distribution and they can be used to describe the likelihood of Gaussian realizations. Their expected value and variance are only determined by the size of the vector of independent random normal scores used to generate the realizations. When the vector size is small, the distribution of Mahalanobis distances is highly skewed and most realizations are close to the vector mean in agreement with the multi-Gaussian density model. As the vector size increases, the realizations sample a region increasingly far out on the tail of the multi-Gaussian distribution, due to the large increase in the size of the uncertainty space largely compensating for the low probability density. For a large vector size, realizations close to the vector mean are not observed anymore. Instead, Gaussian vectors with Mahalanobis distance in the neighborhood of the expected Mahalanobis distance have the maximum probability to be observed. The distribution of Mahalanobis distances becomes Gaussian shaped and the bulk of realizations appear more equiprobable. However, the ratio of their probabilities indicates that they still remain far from being equiprobable. On the other hand, it is observed that equiprobable realizations still display important fluctuations in their variogram reproduction. The variance level that is expected in the variogram reproduction, as well as the variance of the variogram fluctuations, is dependent on the Mahalanobis distance. Realizations with smaller Mahalanobis distances are, on average, smoother than realizations with larger Mahalanobis distances. Poor ergodic conditions tend to generate higher proportions of flatter variograms relative to the variogram model. Only equiprobable realizations with a Mahalanobis distance equal to the expected Mahalanobis distance have an expected variogram matching the variogram model. For large vector sizes, Cholesky simulated Gaussian vectors cannot be used to explore uncertainty in the neighborhood of the vector mean. Instead uncertainty is explored around the n-dimensional elliptical envelop corresponding to the expected Mahalanobis distance.

Filling in cosmic microwave background map missing data using constrained Gaussian realizations

For analyzing maps of the cosmic microwave background sky, it is necessary to mask out the region around the galactic equator where the parasitic foreground emission is strongest as well as the brightest compact sources. Since many of the analyses of the data, particularly those searching for non-Gaussianity of a primordial origin, are most straightforwardly carried out on full-sky maps, it is of great interest to develop efficient algorithms for filling in the missing information in a plausible way. We explore practical algorithms for filling in based on constrained Gaussian realizations. Although carrying out such realizations is in principle straightforward, for finely pixelized maps as will be required for the Planck analysis a direct brute force method is not numerically tractable. We present some concrete solutions to this problem, both on a spatially flat sky with periodic boundary conditions and on the pixelized sphere. One approach is to solve the linear system with an appropriately preconditioned conjugate gradient method. While this approach was successfully implemented on a rectangular domain with periodic boundary conditions and worked even for very wide masked regions, we found that the method failed on the pixelized sphere for reasons that we explain here. We present an approach that works for full-sky pixelized maps on the sphere involving a kernel-based multi-resolution Laplace solver followed by a series of conjugate gradient corrections near the boundary of the mask.