Computing Discrepancies Related to Spaces of Smooth Periodic Functions

A notion of discrepancy is introduced, which represents the integration error on spaces of $r$-smooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical $L_2$-discrepancy as weil as $r$-smooth versions of it introduced recently by Paskov [Pas93]. Based on previous work [FH96], we develop an efficient algorithm for computing periodic discrepancies for quadrature formulas possessing certain tensor product structures, in particular, for Smolyak quadrature rules (also called sparse grid methods). Furthermore, fast algorithms of computing periodic discrepancies for lattice rules can easily be derived from well-known properties of lattices. On this basis we carry out numerical comparisons of discrepancies between Smolyak and lattice rules

Loaded Dice in Monte Carlo: importance sampling in phase space integration and probability distributions for discrepancies

Discrepancies play an important role in the study of uniformity properties of point sets. Their probability distributions are a help in the analysis of the efficiency of the Quasi Monte Carlo method of numerical integration, which uses point sets that are distributed more uniformly than sets of independently uniformly distributed random points. In this thesis, generating functions of probability distributions of quadratic discrepancies are calculated using techniques borrowed from quantum field theory. The second part of this manuscript deals with the application of the Monte Carlo method to phase space integration, and in particular with an explicit example of importance sampling. It concerns the integration of differential cross sections of multi-parton QCD-processes, which contain the so-called kinematical antenna pole structures. The algorithm is presented and compared with RAMBO, showing a substantial reduction in computing time. In behalf of completeness of the thesis, short introductions to probability theory, Feynman diagrams and the Monte Carlo method of numerical integration are included.Comment: PhD thesis, uses times and eule

Approximation methods in geometry and topology: learning, coarsening, and sampling

Data materialize in many different forms and formats. These can be continuous or discrete, from algebraic expressions to unstructured pointclouds and highly structured graphs and simplicial complexes. Their sheer volume and plethora of different modalities used to manipulate and understand them highlight the need for expressive abstractions and approximations, enabling novel insights and efficiency. Geometry and topology provide powerful and intuitive frameworks for modelling structure, form, and connectivity. Acting as a multi-focal lens, they enable inspection and manipulation at different levels of detail, from global discriminant features to local intricate characteristics. However, these fundamentally algebraic theories do not scale well in the digital world. Adjusting topology and geometry to the computational setting is a non-trivial task, adhering to the “no free lunch” adage. The necessary discretizations can be inaccurate, the underlying combinatorial structures can grow unmanageably in size, and computing salient topological and geometric features can become computationally taxing. Approximations are a necessity when theory cannot accommodate for efficient algorithms. This thesis explores different approaches to simplifying computations pertaining to geometry and topology via approximations. Our methods contribute to the approximation of topological features on discrete domains, and employ geometry and topology to efficiently guide discretizations and approximations. This line of work fits un der the umbrella of Topological Data Analysis (TDA) and Discrete Geometry, which aim to bridge the continuous algebraic mindset with the discrete. We construct topological and geometric approximation methods operating on three different levels. We approximate topological features on discrete combinatorial spaces; we approximate the combinatorial spaces themselves; and we guide processes that allow us to discretize domains via sampling. With our Dist2Cycle model we learn geometric manifestations of topological features, the “optimal” homology generating cycles. This is achieved by a novel simplicial complex neural network that exploits the kernel of Hodge Laplacian operators to localize concise homology generators. Compression of meshes and arbitrary simplicial complexes is made possible by our general spectral coarsening strategy. Functional and structural properties are preserved by optimizing for important eigenspaces of general differential operators, the Hodge Laplacians, at multiple dimensions. Finally, we offer a geometry-driven sampling strategy for data accumulation and stochastic integration. By employing the kd-tree geometric partitioning algorithm we construct a sample set with provable equidistribution guarantees. Our findings are contextualized within prior and recent work, and our methods are thoroughly discussed and evaluated on diverse settings. Ultimately, we are making a claim towards the usefulness of examining the ever-present topological and geometric properties of data, not only in terms of feature discovery, but also as informed generation, manipulation, and simplification tools