An optimal adaptive tensor product wavelet solver of a space-time FOSLS formulation of parabolic evolution problems

In this work, we construct a well-posed first-order system least squares (FOSLS) simultaneously space-time formulation of parabolic PDEs. Using an adaptive wavelet solver, this problem is solved with the best possible rate in linear complexity. Thanks to the use of a basis that consists of tensor products of wavelets in space and time, this rate is equal to that when solving the corresponding stationary problem. Our findings are illustrated by numerical results

Querying and Learning in Probabilistic Databases

Abstract. Probabilistic Databases (PDBs) lie at the expressive inter-section of databases, first-order logic, and probability theory. PDBs em-ploy logical deduction rules to process Select-Project-Join (SPJ) queries, which form the basis for a variety of declarative query languages such as Datalog, Relational Algebra, and SQL. They employ logical consistency constraints to resolve data inconsistencies, and they represent query an-swers via logical lineage formulas (aka.“data provenance”) to trace the dependencies between these answers and the input tuples that led to their derivation. While the literature on PDBs dates back to more than 25 years of research, only fairly recently the key role of lineage for es-tablishing a closed and complete representation model of relational op-erations over this kind of probabilistic data was discovered. Although PDBs benefit from their efficient and scalable database infrastructures for data storage and indexing, they couple the data computation with probabilistic inference, the latter of which remains a #P-hard problem also in the context of PDBs. In this chapter, we provide a review on the key concepts of PDBs with a particular focus on our own recent research results related to this field. We highlight a number of ongoing research challenges related to PDBs, and we keep referring to an information extraction (IE) scenario as a running application to manage uncertain and temporal facts obtained from IE techniques directly inside a PDB setting