ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 28th Nov 2022, 08:43:59pm CET

Session Overview 
Session  
MS 15a: p and hpGalerkin methods and approximation of singularities
 
Session Abstract  
The minisymposium focuses on recent developments in p and hp Galerkin methods, most notably for the approximation of singularities. Both theoretical advancements and more engineeringoriented applications are tackled. In particular, the minisymposium will cover the following topics: a priori and a posteriori error analysis, approximation of isotropic and anisotropic singularities, highfrequency problems, fractional diffusion, kinetic equations, efficient solvers and geometric/algebraic preconditioners, boundary elements, and discontinuous Galerkin methods.  
Presentations  
2:00pm  2:30pm
Exponential convergence for nonlinear problems in quantum chemistry ETH Zürich, Switzerland Many problems in physics, chemistry, and engineering can be described by models whose solution is not regular in a classical Sobolev sense. This is the case, for example, of the computation of electronic wave functions in quantum chemistry —where the Coulomb interaction between charged particles gives rise to cusps in the solution. We can, nonetheless, show that these functions belong to a class of analytictype weighted Sobolev spaces, with isolated point singularities. In this talk, I will present some weighted analyticity results for the solutions of Schrödingertype nonlinear eigenvalue problems and I will discuss their numerical solution by hp finite element methods. By combining the regularity analysis and the a priori estimates, we obtain exponentially convergent numerical schemes. Furthermore, I will show how these estimates can be used to analyze the approximation of those problems by tensor compression methods and by neural networks. Exploiting the hp approximation results we can, indeed, prove convergence rates for the representation of the electronic wave functions by tensor trains and by neural networks. 2:30pm  3:00pm
HpFEM and POD applied to object characterisation for metal detection and security screening ^{1}Keele University, United Kingdom; ^{2}Swansea University, United Kingdom; ^{3}The University of Manchester, United Kingdom The location and identification of hidden conducting security threats in metal detection is an important, yet challenging, inverse problem. Applications in security and defence include early detection of terrorist threats, finding landmines and locating unexploded ordnance. Current approaches to metal detection use simple thresholding and are incapable of determining an object's size, shape and material properties from the measurements of the perturbed magnetic field. Instead, in our approach, an asymptotic expansion of the perturbed magnetic field separates shape and material parameter dependence from inclusion position for small objects. The former is characterised by a small number of parameters through a magnetic polarizability tensor (MPT) [1,2]. This approach provides a model reduction of the metal detection inverse problem by reducing it to, separately, identifying an object's location, which can be done using a MUSIC algorithm, and then identifying information about the shape and material properties of the hidden object according to the measured MPT coefficients. In this context, exploiting the the MPT spectral signature [3] of objects, by taking measurements over a range of frequencies, is of particular interest. In this talk, we will describe how a dictionary containing the spectral behaviour of the invariants of the MPT for different realistic threat and nonthreat objects can be computed accurately using hpfinite elements, accelerated by a reduced order model [4,5]. We will describe how knowledge of the scaling of the MPT coefficients under parameter changes can be used to further accelerate this computation. We will also include our first results for threatobject identification using machine learning classification algorithms trained using this dictionary. References [1] P. D. Ledger and W.R.B. Lionheart, An explicit formula for the magnetic polarizability tensor for object characterization, IEEE Transactions on Geoscience and Remote Sensing, 56(6), 35203533, 2018. [2] P.D. Ledger, W.R.B. Lionheart and A.A.S. Amad, Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors, Mathematical Methods in the Applied Sciences, 42(3), 830860, 2019. [3] P.D. Ledger and W.R.B. Lionheart, The spectral properties of the magnetic polarizability tensor for metallic object characterisation, Mathematical Methods in the Applied Sciences, 43(1), 78113, 2020. [4] B.A. Wilson and P.D. Ledger, Efficient computation of the magnetic polarizability tensor spectral signature using proper orthogonal decomposition, International Journal for Numerical Methods in Engineering, 122(8), 19401963, 2021. [5] P.D. Ledger, B.A. Wilson, A.A.S. Amad and W.R.B. Lionheart Identification of metallic objects using spectral MPT signatures: object characterisation and invariants, International Journal for Numerical Methods in Engineering Accepted (2021). 3:00pm  3:30pm
hpmethods for time dependent fractional diffusion ^{1}TU Wien, Austria; ^{2}University of Vienna, Austria In this talk, for parameters $\gamma,\beta \in (0,1]$, we consider time dependent fractional diffusion problems of the form $$ \partial_t^{\gamma}u + L^{\beta} u =f,$$ posed on a smooth 2d domain, where $L$ is a symmetric, elliptic operator of second order with homogeneous Dirichlet boundary conditions. If the data u_0 and f do not satisfy the boundary conditions, singularities form at the boundary. We present how, by employing hpFinite Element based discretizations, these difficulties can be resolved and one can obtain exponentially convergent numerical schemes. One possibility of discretizing fractional operators, following a paper by Bonito, Lei and Pasciak from 2017, is based on a representation formula for the solution using the RieszDunford functional calculus. We prove that, when combined with hpFEM in space, the resulting scheme delivers exponential convergence towards the exact solution without compatibility conditions on the data. By including a doubleexponential coordinate transformation, the convergence of the scheme can be further improved. We also sketch an alternative approach to apply hpFEM techniques for such a nonlocal problem. Namely, one can extend the spacial dimension by one using the socalled CaffarelliSilvestre extension. This localizes the fractional laplacian, leading to a degenerate, elliptic problem with dynamic boundary condition. This problem can then be treated using standard hpFEM tools. 3:30pm  4:00pm
mortar coupling of hpFEM and hpBEM for the Helmholtz equation ^{1}TU Wien; ^{2}University of Vienna; ^{3}PH Vorarlberg http://www.math.tuwien.ac.at We present a FEMBEM coupling strategy for timeharmonic acoustic scattering in media with variable sound speed. The coupling is realized with the aid of a mortar variable that is an impedance trace on the coupling boundary. The resulting sesquilinear form is shown to satisfy a Garding inequality. Quasioptimal convergence is shown for sufficiently fine discretizations. We consider both conforming and DG discretizations. 
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