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 novel class of sparse tensor algorithms for the numerical solution of stochastic elliptic PDEs. The methods are based on a hierarchic discretization in both, physical and probability space. The discretization spaces are then intertwined in a sparse tensor product fashion, leading to algorithms of log-linear complexity. We will present this idea in the framework of the stochastic Galerkin method. Theoretical results as well as numerical examples indicate the superiority of this sparse tensor product algorithms, compared to the full tensor product approaches used so far.

Convergence Rates of Sparse Tensor GPC FEM for Elliptic sPDEs

Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Ele- ment (FE) discretizations of elliptic partial differentia l equations (PDEs) with a ran- dom coefficient. The representation of the random coefficien t is assumed to require a countably infinite number of terms. The multi-level FE discretizations are combined with families of QMC meth- ods (specifically, randomly shifted lattice rules) to estim ate expected values of linear functionals of the solution, as in (17, 18, 23) in the single- level setting. Here, the ex- pected value is considered as an infinite-dimensional integ ral in the parameter space corresponding to the randomness induced by the random coeffi cient. In this paper we study the same model as in (23). The error analysis of (23) is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level, and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error of the expected value of the functionals of the solution (i.e., the root-mean-square error averaged over all shifts ) is of order O(h 2 ), where h is the finest FE mesh width, or O(N −1+δ ) for arbitrary δ > 0, where N denotes the maximal number of QMC sampling points in the parameter space. For these scenar- ios, the total work for all PDE solves in the multi-level QMC-FE method is shown to be essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension d ≥ 2 with linear elements. The analysis exploits regularity of the parametric solutio n with respect to both the physical variables (the variables in the physical domai n) and the parametric vari-

Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients

We construct several classes of neural networks with ReLU and BiSU (Binary Step Unit) activations, which exactly emulate the lowest order Finite Element (FE) spaces on regular, simplicial partitions of polygonal and polyhedral domains Ω ⊂ R, d = 2, 3. For continuous, piecewise linear (CPwL) functions, our constructions generalize previous results in that arbitrary, regular simplicial partitions of Ω are admitted, also in arbitrary dimension d ≥ 2. Vector-valued elements emulated include the classical Raviart-Thomas and the first family of Nédélec edge elements on triangles and tetrahedra. Neural Networks emulating these FE spaces are required in the correct approximation of boundary value problems of electromagnetism in nonconvex polyhedra Ω ⊂ R, thereby constituting an essential ingredient in the application of e.g. the methodology of “physics-informed NNs” or “deep Ritz methods” to electromagnetic field simulation via deep learning techniques. They satisfy exact (De Rham) sequence properties, and also spawn discrete boundary complexes on ∂Ω which satisfy exact sequence properties for the surface divergence and curl operators divΓ and curlΓ, respectively, thereby enabling “neural boundary elements” for computational electromagnetism. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations in particular the Crouzeix-Raviart elements and Hybridized, Higher Order (HHO) methods.

De Rham compatible Deep Neural Networks

We prove exponential convergence in the energy norm of hp ﬁnite element discretizations for the integral fractional diﬀusion operator of order 2 s ∈ (0 , 2) subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains Ω ⊂ R 2 . Key ingredient in the analysis are the weighted analytic regularity from [15] and meshes that feature anisotropic geometric reﬁnement towards ∂Ω . Mathematics Subject Classiﬁcation (2020) 35R11 · 65N12 · 65N30. Science Fund (FWF) project F

Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

The Helmholtz equation in a three-dimensional plate is approximated by a hierarchy of two-dimensional models. The necessity of including, besides polynomials, a certain number of trigonometric director functions into the Ansatz, in order to prevent pollution effects at high wave numbers is demonstrated for exponentially weighted norms.

Christoph Schwab

Papers

Convergence Rates of Sparse Tensor GPC FEM for Elliptic sPDEs

Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients

De Rham compatible Deep Neural Networks

Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

Hierarchic Models of Vibration Problems on Thin Domains