Related papers: Non-intrusive uncertainty quantification using red…
For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using only samples that describe the probability distribution of the uncertain parameters.…
Many applications require multi-dimensional numerical integration, often in the form of a cubature formula. These cubature formulas are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions).…
Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by…
Let $d$ and $k$ be positive integers. Let $\mu$ be a positive Borel measure on $\mathbb{R}^2$ possessing finite moments up to degree $2d-1$. If the support of $\mu$ is contained in an algebraic curve of degree $k$, then we show that there…
This paper deals with some of the methodologies used to construct polynomial surrogate models based on generalized polynomial chaos (gPC) expansions for applications to uncertainty quantification (UQ) in aerodynamic computations. A core…
We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields…
Gau{\ss} cubature (multidimensional numerical integration) rules are the natural generalisation of the 1D Gau{\ss} rules. They are optimal in the sense that they exactly integrate polynomials of as high a degree as possible for a particular…
A novel mathematical framework is derived for the addition of nodes to univariate and interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde matrix describing the relation between the…
Uncertainty principle is an inherent nature of quantum system that undermines the precise measurement of incompatible observables and hence the applications of quantum theory. Entanglement, another unique feature of quantum physics, was…
We present a construction for improving numerical cubature formulas with equal weights and a convolution structure, in particular equal-weight product formulas, using linear error-correcting codes. The construction is most effective in low…
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate…
Efficient and systematic numerical methods for robust control design are crucial in quantum systems due to inevitable uncertainties or disturbances. We propose a novel approach that models uncertainties as random variables and quantifies…
The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many…
Quantum resources, such as entanglement, can decrease the uncertainty of a parameter-estimation procedure beyond what is classically possible. This phenomenon is well described for noiseless systems with asymptotically many measurement…
The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded payoffs. We first introduce a recently described flexible functional analytic…
First principles approaches have revolutionized our ability in using computers to predict, explore and design materials. A major advantage commonly associated with these approaches is that they are fully parameter free. However, numerically…
73 new cubature rules are found for three standard multidimensional integrals with spherically symmetric regions and weights, using direct search with a numerical zero-finder. All but four of the new rules have fewer integration points than…
We provide explicit expressions for quadrature rules on the space of $C^1$ cubic splines with non-uniform, symmetrically stretched knot sequences. The quadrature nodes and weights are derived via an explicit recursion that avoids an…
Calibrating model parameters to measured data by minimizing loss functions is an important step in obtaining realistic predictions from model-based approaches, e.g., for process optimization. This is applicable to both knowledge-driven and…
We present a simple and robust strategy for the selection of sampling points in Uncertainty Quantification. The goal is to achieve the fastest possible convergence in the cumulative distribution function of a stochastic output of interest.…