Tikhonov regularization is studied in the case of linear pseudodifferential operator as forward map and additive white Gaussian noise measurement error. The model for an unknown function $u(x)$ \begin{eqnarray*} m(x) = Au(x) + \delta\hspace{.2mm}\varepsilon(x), \end{eqnarray*} where $\delta>0$ magnitude. If $\varepsilon$ was $L^2$-function, gives estimate T_\alpha(m) \text{argmin}_{u\in H^r}\big\{\|A u-m\|_{L^2}^2+ \alpha\|u\|_{H^r}^2 \big\}\end{eqnarray*} $u$ $\alpha=\alpha(\delta)$...

Bayesian approach to inverse problems is studied in the case where forward map a linear hypoelliptic pseudodifferential operator and measurement error additive white Gaussian noise.The model for an unknown random variable U (x, ω) iswhere A finitely many orders smoothing δ > 0 noise magnitude.The covariance CU of order 2r, self-adjoint, injective elliptic operator.If E was taking values L 2 then solving conditional mean (and maximum posteriori) estimate linked minimisation problemHowever,...

We consider the statistical inverse problem of recovering an unknown function $f$ from a linear measurement corrupted by additive Gaussian white noise. employ nonparametric Bayesian approach with standard priors, for which posterior-based reconstruction corresponds to Tikhonov regularizer $\bar f$ reproducing kernel Hilbert space norm penalty. prove semiparametric Bernstein--von Mises theorem large collection functionals $f$, implying that posterior estimation and uncertainty quantification...

Abstract We consider the statistical non-linear inverse problem of recovering absorption term f &gt; 0 in heat equation <?CDATA \begin{equation*}\begin{cases}{\partial }_{t}u-\frac{1}{2}{\Delta}u+fu=0\quad \hfill & \quad \text{on}\;\mathcal{O}\times (0,\mathbf{T})\hfill \\ u=g\quad \text{on}\;\partial \mathcal{O}\times u(\cdot ,0)={u}_{0}\quad \text{on}\;\mathcal{O},\hfill \end{cases}\end{equation*}?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll">...

Abstract In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded the image space of forward operator. We introduce a Banach setting allows to define reasonable notion solutions more larger provided one has sufficient mapping properties operators. A key observation, which guides us through subsequent analysis, such model can be understood same as approximate source conditions (while standard bounded related directly...

We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is construct priors have similar good edge-preserving properties as total variation or Mumford-Shah but correspond well-defined infinite-dimensional random variables, and can be approximated by finite-dimensional variables. introduce a new wavelet-based model, where non-zero coefficients chosen systematic way so draws certain fractal behaviour. show realisations this...

One of the disadvantages oceanographic models is that they can be very computationally expensive. When combined with data assimilation, dynamical approaches such as EnKF become expensive need a large number ensemble members and thus model runs. In this work we investigate use Multi-Fidelity Ensemble Kalman Filter (MF-EnKF), where lower fidelity machine learned surrogate high original full model. The idea behind to an few but runs, many cheap less accurate way reach similar or increased...

Recent advancements in Markov chain Monte Carlo (MCMC) sampling and surrogate modelling have significantly enhanced the feasibility of Bayesian analysis across engineering fields. However, selection integration models cutting-edge MCMC algorithms, often depend on ad-hoc decisions. A systematic assessment their combined influence analytical accuracy efficiency is notably lacking. The present work offers a comprehensive comparative study, employing scalable case study computational mechanics...

We consider the statistical inverse problem of recovering an unknown function $f$ from a linear measurement corrupted by additive Gaussian white noise. employ nonparametric Bayesian approach with standard priors, for which posterior-based reconstruction corresponds to Tikhonov regulariser $\bar f$ reproducing kernel Hilbert space norm penalty. prove semiparametric Bernstein-von Mises theorem large collection functionals $f$, implying that posterior estimation and uncertainty quantification...

In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded the image space of forward operator. We introduce a Banach setting allows to define reasonable notion solutions more larger provided one has sufficient mapping properties operators. A key observation, which guides us through subsequent analysis, such model can be understood same as approximate source conditions (while standard bounded related directly classical...

Minimal surfaces can be though as a mathematical generalisation of formed by soap films. We consider Bour's minimal $\mathcal{B}_m$ that are intrinsically revolution. show how to generate crochet patterns for using basic trigonometric identities calculate required arc lengths. Three special cases considered in more detail, namely Enneper's, Richmond's, and $\mathcal{B}_3$ surfaces, we provide exact instructions the classical Enneper's surface.

Building blocks and tiles are an excellent way of learning about geometry mathematics in general. There several versions that either snapped together or connected with magnets can be used to introduce topics like volume, tessellations, Platonic solids. However, since these made hard plastic, they not very suitable for creating hyperbolic surfaces shapes where the need bend. Curvagons flexible regular polygon building allow you quickly build anything from tori dinosaurs shoes. They...

How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry students the general public using physical models, including chalksphere, crocheted hyperbolic surfaces, curved folding, polygon tilings. Spherical offers a simple yet surprising introduction topic, whereas is an entirely new exciting concept most. Non-Euclidean...

We propose alternatives to Bayesian a priori distributions that are frequently used in the study of inverse problems. Our aim is construct priors have similar good edge-preserving properties as total variation or Mumford-Shah but correspond well defined infinite-dimensional random variables, and can be approximated by finite-dimensional variables. introduce new wavelet-based model, where non zero coefficient chosen systematic way so prior draws certain fractal behaviour. show realisations...