相关论文: Bringing errors into focus

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel…

A recent study investigated the propagation of error in a Velocimetry-based Pressure (V-Pressure) field reconstruction problem by directly analyzing the properties of the pressure Poisson equation (Pan et al., 2016). In the present work, we…

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when…

In his 1986 book, Aitchison explains that compositional data is regularly mishandled in statistical analyses, a pattern that continues to this day. The Dirichlet Type I distribution is a multivariate distribution commonly used to model a…

We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit…

Back-propagation has been the workhorse of recent successes of deep learning but it relies on infinitesimal effects (partial derivatives) in order to perform credit assignment. This could become a serious issue as one considers deeper and…

A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…

In this article, we present a solution to the problem: "Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator $-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension…

In the second paper of this series we extend our Bayesian reanalysis of the evidence for a cosmic variation of the fine structure constant to the semi-parametric modelling regime. By adopting a mixture of Dirichlet processes prior for the…

Using diffusion priors to solve inverse problems in imaging have significantly matured over the years. In this chapter, we review the various different approaches that were proposed over the years. We categorize the approaches into the more…

In the present article, we assume that the first approximation of the scattering operator is given and that it has the logarithmic divergence. This first approximation allows us to construct the so called deviation factor. Using the…

The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying…

This lecture notes cover a Part III (first year graduate) course that was given at Cambridge University over several years on pseudo-differential operators. The calculus on manifolds is developed and applied to prove propagation of…

Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational…

Dirichlet distribution and Dirichlet process as its infinite dimensional generalization are primarily used conjugate prior of categorical and multinomial distributions in Bayesian statistics. Extensions have been proposed to broaden…

These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads…

We study the properties of the ``Rigid Laplacian'' operator, that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the…

A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems,…

A convenient way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. The general goal of this paper is to describe how to approximate Fliess operators with iterated…

Orbifolding two-dimensional quantum field theories by a symmetry group can involve a choice of discrete torsion. We apply the general formalism of `orbifolding defects' to study and elucidate discrete torsion for topological field theories.…