High-Dimensional Dependency Structure Learning for Physical Processes

In this paper, we consider the use of structure learning methods for probabilistic graphical models to identify statistical dependencies in high-dimensional physical processes. Such processes are often synthetically characterized using PDEs (partial differential equations) and are observed in a variety of natural phenomena, including geoscience data capturing atmospheric and hydrological phenomena. Classical structure learning approaches such as the PC algorithm and variants are challenging to apply due to their high computational and sample requirements. Modern approaches, often based on sparse regression and variants, do come with finite sample guarantees, but are usually highly sensitive to the choice of hyper-parameters, e.g., parameter $\lambda$ for sparsity inducing constraint or regularization. In this paper, we present ACLIME-ADMM, an efficient two-step algorithm for adaptive structure learning, which estimates an edge specific parameter $\lambda_{ij}$ in the first step, and uses these parameters to learn the structure in the second step. Both steps of our algorithm use (inexact) ADMM to solve suitable linear programs, and all iterations can be done in closed form in an efficient block parallel manner. We compare ACLIME-ADMM with baselines on both synthetic data simulated by partial differential equations (PDEs) that model advection-diffusion processes, and real data (50 years) of daily global geopotential heights to study information flow in the atmosphere. ACLIME-ADMM is shown to be efficient, stable, and competitive, usually better than the baselines especially on difficult problems. On real data, ACLIME-ADMM recovers the underlying structure of global atmospheric circulation, including switches in wind directions at the equator and tropics entirely from the data.Comment: 21 pages, 8 figures, International Conference on Data Mining 201

A method for enhancing the stability and robustness of explicit schemes in astrophysical fluid dynamics

A method for enhancing the stability and robustness of explicit schemes in computational fluid dynamics is presented. The method is based in reformulating explicit schemes in matrix form, which cane modified gradually into semi or strongly-implicit schemes. From the point of view of matrix-algebra, explicit numerical methods are special cases in which the global matrix of coefficients is reduced to the identity matrix $I$. This extreme simplification leads to severer stability range, hence of their robustness. In this paper it is shown that a condition, which is similar to the Courant-Friedrich-Levy (CFL) condition can be obtained from the stability requirement of inversion of the coefficient matrix. This condition is shown to be relax-able, and that a class of methods that range from explicit to strongly implicit methods can be constructed, whose degree of implicitness depends on the number of coefficients used in constructing the corresponding coefficient-matrices. Special attention is given to a simple and tractable semi-explicit method, which is obtained by modifying the coefficient matrix from the identity matrix $I$ into a diagonal-matrix $D$. This method is shown to be stable, robust and it can be applied to search for stationary solutions using large CFL-numbers, though it converges slower than its implicit counterpart. Moreover, the method can be applied to follow the evolution of strongly time-dependent flows, though it is not as efficient as normal explicit methods. In addition, we find that the residual smoothing method accelerates convergene toward steady state solutions considerably and improves the efficiency of the solution procedure.Comment: 33 pages, 15 figure

Fornax: a Flexible Code for Multiphysics Astrophysical Simulations

This paper describes the design and implementation of our new multi-group, multi-dimensional radiation hydrodynamics (RHD) code Fornax and provides a suite of code tests to validate its application in a wide range of physical regimes. Instead of focusing exclusively on tests of neutrino radiation hydrodynamics relevant to the core-collapse supernova problem for which Fornax is primarily intended, we present here classical and rigorous demonstrations of code performance relevant to a broad range of multi-dimensional hydrodynamic and multi-group radiation hydrodynamic problems. Our code solves the comoving-frame radiation moment equations using the M1 closure, utilizes conservative high-order reconstruction, employs semi-explicit matter and radiation transport via a high-order time stepping scheme, and is suitable for application to a wide range of astrophysical problems. To this end, we first describe the philosophy, algorithms, and methodologies of Fornax and then perform numerous stringent code tests, that collectively and vigorously exercise the code, demonstrate the excellent numerical fidelity with which it captures the many physical effects of radiation hydrodynamics, and show excellent strong scaling well above 100k MPI tasks.Comment: Accepted to the Astrophysical Journal Supplement Series; A few more textual and reference updates; As before, one additional code test include

Calculus on surfaces with general closest point functions

The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs