There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

/pdf/unified-analysis-of-discontinuous-galerkin-methods-for-9numgtph62.pdf

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We propose a multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input. We prove an error versus work analysis which allows a multilevel f...

/pdf/improved-efficiency-of-a-multi-index-fem-for-computational-307dulclp3.pdf

Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $\Omega \subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $\Omega$. In $\Omega\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(\Omega)$. 
We also prove that tensorization of a $P_1$-FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_\Omega$, the number of degrees of freedom in the domain $\Omega$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $\Omega$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_\Omega$. Finally, under the stronger assumption that the data is analytic in $\overline{\Omega}$, and without compatibility at $\partial \Omega$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

Tensor FEM for spectral fractional diffusion

We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphy-based argument allows a unified, ``differentiation-free'' proof of sparsity (expressed in terms of $\ell^p$-summability or weighted $\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding analyticity and sparsity results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various \emph{constructive} high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both forward and inverse computational uncertainty quantification.

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

/pdf/hp-dgfem-for-second-order-mixed-elliptic-problems-in-2922wv5yqk.pdf

hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra: II: Exponential Convergence

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic partial differential equation (PDE). We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\mathbb{N}$, and that the PDE operator may depend nonaffinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [A. Cohen, R. DeVore, and Ch. Schwab, Found. Comput. Math., 10 (2010), pp. 615--646; Anal. Appl., 9 (2011), pp. 1--37], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both state and control in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\geq 1}$ of the random inputs, and we prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the...

Christoph Schwab

Papers

Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

Tensor FEM for spectral fractional diffusion

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra: II: Exponential Convergence

Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs