Abstract Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.
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FLAME: A Fast Large-scale Almost Matching Exactly Approach to Causal Inference
A classical problem in causal inference is that of matching, where treatment units need to be matched to control units based on covariate information. In this work, we propose a method that computes high quality almost-exact matches for high-dimensional categorical datasets. This method, called FLAME (Fast Large-scale Almost Matching Exactly), learns a distance metric for matching using a hold-out training data set. In order to perform matching efficiently for large datasets, FLAME leverages techniques that are natural for query processing in the area of database management, and two implementations of FLAME are provided: the first uses SQL queries and the second uses bit-vector techniques. The algorithm starts by constructing matches of the highest quality (exact matches on all covariates), and successively eliminates variables in order to match exactly on as many variables as possible, while still maintaining interpretable high-quality matches and balance between treatment and control groups. We leverage these high quality matches to estimate conditional average treatment effects (CATEs). Our experiments show that FLAME scales to huge datasets with millions of observations where existing state-of-the-art methods fail, and that it achieves significantly better performance than other matching methods.
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- Award ID(s):
- 1703431
- NSF-PAR ID:
- 10291692
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 22
- Issue:
- 31
- ISSN:
- 1533-7928
- Page Range / eLocation ID:
- 1-41
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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