17 research outputs found
Gradient accelerated cosmology
Cosmology is currently sitting on a ticking time bomb that will result
in an unprecedented explosion in the quantity and quality of data. In
preparation, physicists are starting to incorporate into their theoretical
predictions more of the physical, observational and instrumental effects
which, until now, could be overlooked. In practice, this translates into
a dramatic increase in the number of parameters that future analyses
will have to consider. This combination of large data sets with complex
models will (and in many cases already does) overwhelm the inference
methods we currently use to constrain the values of these parameters.
In this thesis, we propose two solutions to this problem. First, we show
how gradient-based inference algorithms can dramatically speed up the
numerical marginalisation of high dimensional parameter spaces. Second,
we show how analytical marginalisation schemes, such as the Laplace ap-
proximation, can achieve similar speed increases. Crucially, both these
methods rely on having access to computationally affordable gradients of
the cosmological models, stressing the importance of developing differen-
tiable analyses pipelines for future cosmological surveys
Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions
In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results
Capse.jl: efficient and auto-differentiable CMB power spectra emulation
We present Capse.jl, a novel emulator that utilizes neural networks to
predict Cosmic Microwave Background (CMB) temperature, polarization and lensing
angular power spectra. The emulator computes predictions in just a few
microseconds with emulation errors below 0.1 for all the scales
relevant for the planned CMB-S4 survey. Capse.jl can also be trained in an
hour's time on a CPU. As a test case, we use Capse.jl to analyze Planck 2018
data and ACT DR4 data. We obtain the same result as standard analysis methods
with a computational efficiency 3 to 6 order of magnitude higher. We take
advantage of the differentiability of our emulators to use gradients-based
methods, such as Pathfinder and Hamiltonian Monte Carlo (HMC), which speed up
the convergence and increase sampling efficiency. Together, these features make
Capse.jl a powerful tool for studying the CMB and its implications for
cosmology. When using the fastest combination of our likelihoods, emulators,
and analysis algorithm, we are able to perform a Planck TT + TE + EE analysis
in less than a second. To ensure full reproducibility, we provide open access
to the codes and data required to reproduce all the results of this work.Comment: 16 pages, 4 figure
Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds
The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker
optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the
addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical
examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient
conditions for weakly efficient Pareto points to the constrained vector optimization problem are
presented. The results described in this article generalize results obtained by Gong (2008) andWei
and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed
spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of
Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds
Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds
This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivatives, second-order pseudoinvexity functions, and the second-order Karush-Kuhn-Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization, respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of "Higgs Boson like" potentials, among others
Analytical marginalisation over photometric redshift uncertainties in cosmic shear analyses
As the statistical power of imaging surveys grows, it is crucial to account
for all systematic uncertainties. This is normally done by constructing a model
of these uncertainties and then marginalizing over the additional model
parameters. The resulting high dimensionality of the total parameter spaces
makes inferring the cosmological parameters significantly more costly using
traditional Monte-Carlo sampling methods. A particularly relevant example is
the redshift distribution, , of the source samples, which may require
tens of parameters to describe fully. However, relatively tight priors can be
usually placed on these parameters through calibration of the associated
systematics. In this paper we show, quantitatively, that a linearisation of the
theoretical prediction with respect to these calibratable systematic parameters
allows us to analytically marginalise over these extra parameters, leading to a
factor reduction in the time needed for parameter inference, while
accurately recovering the same posterior distributions for the cosmological
parameters that would be obtained through a full numerical marginalisation over
160 parameters. We demonstrate that this is feasible not only with
current data and current achievable calibration priors but also for future
Stage-IV datasets.Comment: 11 pages, 8 figures, prepared for submission to MNRAS, comments
welcom
Analytical marginalization over photometric redshift uncertainties in cosmic shear analyses
As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties. This is normally done by constructing a model of these uncertainties and then marginalizing over the additional model parameters. The resulting high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using traditional Monte Carlo sampling methods. A particularly relevant example is the redshift distribution, p(z ), of the source samples, which may require tens of parameters to describe fully. However, relatively tight priors can be usually placed on these parameters through calibration of the associated systematics. In this paper, we show, quantitatively, that a linearization of the theoretical prediction with respect to these calibrated systematic parameters allows us to analytically marginalize over these extra parameters, leading to a factor of ∼30 reduction in the time needed for parameter inference, while accurately recovering the same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalization over 160 p(z ) parameters. We demonstrate that this is feasible not only with current data and current achievable calibration priors but also for future Stage-IV data sets
Cosmology with 6 parameters in the Stage-IV era: efficient marginalisation over nuisance parameters
The analysis of photometric large-scale structure data is often complicated
by the need to account for many observational and astrophysical systematics.
The elaborate models needed to describe them often introduce many ``nuisance
parameters'', which can be a major inhibitor of an efficient parameter
inference. In this paper we introduce an approximate method to analytically
marginalise over a large number of nuisance parameters based on the Laplace
approximation. We discuss the mathematics of the method, its relation to
concepts such as volume effects and profile likelihood, and show that it can be
further simplified for calibratable systematics by linearising the dependence
of the theory on the associated parameters. We quantify the accuracy of this
approach by comparing it with traditional sampling methods in the context of
existing data from the Dark Energy Survey, as well as futuristic Stage-IV
photometric data. The linearised version of the method is able to obtain
parameter constraints that are virtually equivalent to those found by exploring
the full parameter space for a large number of calibratable nuisance
parameters, while reducing the computation time by a factor 3-10. Furthermore,
the non-linearised approach is able to analytically marginalise over a large
number of parameters, returning constraints that are virtually
indistinguishable from the brute-force method in most cases, accurately
reproducing both the marginalised uncertainty on cosmological parameters, and
the impact of volume effects associated with this marginalisation. We provide
simple recipes to diagnose when the approximations made by the method fail and
one should thus resort to traditional methods. The gains in sampling efficiency
associated with this method enable the joint analysis of multiple surveys,
typically hindered by the large number of nuisance parameters needed to
describe them.Comment: 17 pages, 6 figures, 3 table
LimberJack.jl: auto-differentiable methods for angular power spectra analyses
We present LimberJack.jl, a fully auto-differentiable code for cosmological analyses of 2 point auto- and cross-correlation measurements from galaxy clustering, CMB lensing and weak lensing data written in Julia. Using Julia’s auto-differentiation ecosystem, LimberJack.jl can obtain gradients for its outputs an order of magnitude faster than traditional finite difference methods. This makes LimberJack.jl greatly synergistic with gradient-based sampling methods, such as Hamiltonian Monte Carlo, capable of efficiently exploring parameter spaces with hundreds of dimensions. We first prove LimberJack.jl’s reliability by reanalysing the DES Y1 3×2-point data. We then showcase its capabilities by using a O(100) parameters Gaussian Process to reconstruct the cosmic growth from a combination of DES Y1 galaxy clustering and weak lensing data, eBOSS QSO’s, CMB lensing and redshift-space distortions. Our Gaussian process reconstruction of the growth factor is statistically consistent with the ΛCDM Planck 2018 prediction at all redshifts. Moreover, we show that the addition of RSD data is extremely beneficial to this type of analysis, reducing the uncertainty in the reconstructed growth factor by 20% on average across redshift. LimberJack.jl is a fully open-source project available on Julia’s general repository of packages and GitHub