On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints

We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.

A Permeability Estimation Method Based on Elliptical Pore Approximation

Digital rock images may capture more detailed pore structure than the traditional laboratory methods. No explicit function can correlate permeability accurately for flow within the pore space. This has motivated researchers to predict permeability through the application of numerical techniques, e.g., using the finite difference method (FDM). However, in order to get better permeability calculation results, the grid refinement was needed for the traditional FDM and the accuracy of the traditional method decreased in pores with elongated cross sections. The goal of this study is to develop an improved FDM (IFDM) to calculate the permeabilities of digital rock images with complex pore space. An elliptical pore approximation method is invoked to describe the complex pore space. The permeabilities of four types of idealized porous media are calculated by IFDM. The calculated results are in sound agreement with the analytical solutions or semi-empirical solutions. What’s more, the permeabilities of the digital rock images after grid coarsening are calculated by IFDM in three orthogonal directions. These results are compared with the previously validated lattice-Boltzmann method (LBM), which indicates that the predicted permeabilities calculated by IFDM usually agree with permeabilities calculated by LBM. We conclude that the presented IFDM is suitable for complex pore space.

Vertical grid refinement for stratocumulus clouds in the radiation scheme of the global climate model ECHAM6.3-HAM2.3-P3

In this study, we implement a vertical grid refinement scheme in the radiation routine of the global aerosol–climate model ECHAM-HAM, aiming to improve the representation of stratocumulus clouds and address the underestimation of their cloud cover. The scheme is based on a reconstruction of the temperature inversion as a physical constraint for the cloud top. On the refined grid, the boundary layer and the free troposphere are separated and the cloud's layer is made thinner. The cloud cover is recalculated either by conserving the cloud volume (SC-VOLUME) or by using the Sundqvist cloud cover routine on the new grid representation (SC-SUND). In global climate simulations, we find that the SC-VOLUME approach is inadequate, as there is a mismatch, in most cases, between the layer of the inversion and the layer of the stratocumulus cloud, which prevents its application and is itself likely caused by an overly low vertical resolution. Additionally, we find that the occurrence frequency of stratocumulus clouds is underestimated in ECHAM-HAM, limiting a priori the potential benefits of a scheme like SC-VOLUME targeting only cloud amount when present. With the SC-SUND approach, the possibility for new clouds to be formed on the refined grid results in a large increase in mean total cloud cover in stratocumulus regions. In both cases, however, the changes exerted in the radiation routine are too weak to produce a significant improvement in the simulated stratocumulus cloud cover. We investigate and discuss the reasons behind this. The grid refinement scheme could be used more effectively for this purpose if implemented directly in the model's cloud microphysics and cloud cover routines, but other possible ways forward are also discussed.

Comparative Assessment of Methods for Coupling Regional and Local Groundwater Flow Models: A Case Study in the Beijing Plain, China

A coupled regional and local model is required when groundwater flow and solute transport are to be simulated in local areas of interest with a finer grid while regional aquifer boundary and major stresses should be retained with a coarser grid. The coupled model should also maintain interactions between the regional and local flow systems. In the Beijing Plain (China), assessment of managed aquifer recharge (MAR), groundwater pollution caused by rivers, capture zone of well fields, and land subsidence at the cone of depression requires a coupled regional and local model. This study evaluates three methods for coupling regional and local flow models for simulating MAR in the Chaobai River catchment in the Beijing Plain. These methods are the conventional grid refinement (CGR) method, the local grid refinement (LGR) method and the unstructured grid (USG) method. The assessment included the comparison of the complexity of the coupled model construction, the goodness of fit of the computed and observed groundwater heads, the consistency of regional and local groundwater budgets, and the capture zone of a well filed influenced by the MAR site. The results indicated that the CGR method based on MODFLOW-2005 is the easiest to implement the coupled model, capable of reproducing regional and local groundwater heads and budget, and already coupled with density and viscosity dependent model codes for transport simulation. However, the CGR method inherits shortcomings of finite difference grids to create multiple local models with inefficient computing efforts. The USG method based on MODFLOW-USG has the advantage of creating multi-scale models and is flexible to simulate rivers, wells, irregular boundaries, heterogeneities and the MAR site. However, it is more difficult to construct the coupled models with the unstructured grids, therefore, a good graphic user interface is necessary for efficient model construction. The LGR method based on MODFLOW-LGR can be used to create multiple local models in uniform aquifer systems. So far, little effort has been devoted to upgrade the LGR method for complex aquifer structures and develop the coupled transport models.

Assessment of Discretization Uncertainty Estimators Based On Grid Refinement Studies

This paper presents the assessment of the performance of 9 discretization uncertainty estimates based on grid refinement studies including methods that use grid triplets and others that use a largest number of data points, which in the present study was set to five. The uncertainty estimates are performed for the data set proposed for the 2017 ASME Workshop on Estimation of Discretization Errors including functional and local flow quantities from the two-dimensional incompressible flows over a flat plate and the NACA 0012 airfoil. The data were generated with a RANS solver using three eddy-viscosity turbulence models with double precision and sufficiently tight iterative convergence criteria to ensure that the numerical error is dominated by the discretization error. The use of several geometrically similar grid sets with different near-wall cell sizes lead to a wide range of convergence properties for the selected flow quantities. The evaluation of uncertainty estimates is based on the ratio of the estimated uncertainty over the "exact error" that is obtained from an "exact solution" obtained from extra grid sets significantly more refined than those used to generate the Workshop data. Although none of the methods tested fulfilled the goal of bounding the "exact error" 95 times out of 100 that was tested, the results suggest that the methods tested are useful tools for the assessment of the numerical uncertainty of practical numerical simulations even for cases where it is not possible to generate data in the "asymptotic range".

High performance Wannier interpolation of Berry curvature and related quantities with WannierBerri code

Wannier interpolation is a powerful tool for performing Brillouin zone integrals over dense grids ofkpoints, which are essential to evaluate such quantities as the intrinsic anomalous Hall conductivity or Boltzmann transport coefficients. However, more complex physical problems and materials create harder numerical challenges, and computations with the existing codes become very expensive, which often prevents reaching the desired accuracy. In this article, I present a series of methods that boost the speed of Wannier interpolation by several orders of magnitude. They include a combination of fast and slow Fourier transforms, explicit use of symmetries, and recursive adaptive grid refinement among others. The proposed methodology has been implemented in the python code WannierBerri, which also aims to serve as a convenient platform for the future development of interpolation schemes for other phenomena.