scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

2011 ◽  
Vol 139 (9) ◽  
pp. 2962-2975 ◽  
Author(s):  
William C. Skamarock ◽  
Almut Gassmann
Keyword(s):  
Fourth Order ◽  
Time Integration ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
Diffusion Term ◽  
The Third ◽  
Integration Schemes ◽  
Order Scheme ◽  
Runge Kutta

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

scholarly journals Rapid separable analysis of higher order correlators in large-scale structure

2012 ◽  
Vol 86 (6) ◽  
Author(s):  
J. R. Fergusson ◽  
D. M. Regan ◽  
E. P. S. Shellard
Keyword(s):  
Large Scale ◽  
Large Scale Structure ◽  
Higher Order ◽  
Scale Structure

Zeitstrukturen im Jazz

2014 ◽  
Vol 59 (1) ◽  
pp. 79-92
Author(s):  
Alexander Becker
Keyword(s):  
Large Scale ◽  
Large Scale Structure ◽  
Time Frame ◽  
Scale Structure ◽  
Classical Form ◽  
Jazz Music ◽  
The Moment

Wie erlebt der Hörer Jazz? Bei dieser Frage geht es unter anderem um die Art und Weise, wie Jazz die Zeit des Hörens gestaltet. Ein an klassischer Musik geschultes Ohr erwartet von musikalischer Zeitgestaltung, den zeitlichen Rahmen, der durch Anfang und Ende gesetzt ist, von innen heraus zu strukturieren und neu zu konstituieren. Doch das ist keine Erwartung, die dem Jazz gerecht wird. Im Jazz wird der Moment nicht im Hinblick auf ein Ziel gestaltet, das von einer übergeordneten Struktur bereitgestellt wird, sondern so, dass er den Bewegungsimpuls zum nächsten Moment weiterträgt. Wie wirkt sich dieses Prinzip der Zeitgestaltung auf die musikalische Form im Großen aus? Der Aufsatz untersucht diese Frage anhand von Beispielen, an denen sich der Weg der Transformation von einer klassischen zu einer dem Jazz angemessenen Form gut nachverfolgen lässt.<br><br>How do listeners experience Jazz? This is a question also about how Jazz music organizes the listening time. A classically educated listener expects a piece of music to structure, unify and thereby re-constitute the externally given time frame. Such an expectation is foreign to Jazz music which doesn’t relate the moment to a goal provided by a large scale structure. Rather, one moment is carried on to the next, preserving the stimulus potentially ad infinitum. How does such an organization of time affect the large scale form? The paper tries to answer this question by analyzing two examples which permit to trace the transformation of a classical form into a form germane to Jazz music.

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