arXiv subjects

Krzysztof Gonciarz

Publications and source records attributed to Krzysztof Gonciarz.

4 recordsLinked to original sources

arXiv2025

Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.

math.NA↗

arXiv2024

Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

We extend Newton and Lagrange interpolation to arbitrary dimensions. The core contribution that enables this is a generalized notion of non-tensorial unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. By validation, we reach the optimal exponential Trefethen rates for a class of analytic functions, we term Trefethen functions. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on these results, we propose an algorithm to efficiently and numerically stably solve arbitrary-dimensional interpolation problems, with at most quadratic runtime and linear memory requirement.

math.NA↗

arXiv2021

Parallel Discrete Convolutions on Adaptive Particle Representations of Images

We present data structures and algorithms for native implementations of discrete convolution operators over Adaptive Particle Representations (APR) of images on parallel computer architectures. The APR is a content-adaptive image representation that locally adapts the sampling resolution to the image signal. It has been developed as an alternative to pixel representations for large, sparse images as they typically occur in fluorescence microscopy. It has been shown to reduce the memory and runtime costs of storing, visualizing, and processing such images. This, however, requires that image processing natively operates on APRs, without intermediately reverting to pixels. Designing efficient and scalable APR-native image processing primitives, however, is complicated by the APR's irregular memory structure. Here, we provide the algorithmic building blocks required to efficiently and natively process APR images using a wide range of algorithms that can be formulated in terms of discrete convolutions. We show that APR convolution naturally leads to scale-adaptive algorithms that efficiently parallelize on multi-core CPU and GPU architectures. We quantify the speedups in comparison to pixel-based algorithms and convolutions on evenly sampled data. We achieve pixel-equivalent throughputs of up to 1 TB/s on a single Nvidia GeForce RTX 2080 gaming GPU, requiring up to two orders of magnitude less memory than a pixel-based implementation.

cs.CV↗

arXiv2020

Tight Localizations of Feedback Sets

The classical NP-hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs $\varepsilon \subseteq E$ or vertices $ν\subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus ν$ makes a given multi-digraph $G=(V,E)$ acyclic, respectively. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new $\mathcal{O}(|V||E|^4)$-heuristic for the directed FASP. While a ratio of $r \approx 1.3606$ is known to be a lower bound for the APX-hardness, at least by empirical validation we achieve an approximation of $r \leq 2$. The most relevant applications, such as circuit testing, ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds due to our approach.

cs.DM↗