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

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Session Overview
MS 15a: p- and hp-Galerkin methods and approximation of singularities
Wednesday, 14/July/2021:
2:00pm - 4:00pm

Session Chair: Zhaonan Dong
Session Chair: Lorenzo Mascotto
Virtual location: Zoom 6

Session Abstract

The mini-symposium focuses on recent developments in p- and hp- Galerkin methods,

most notably for the approximation of singularities. Both theoretical advancements and more

engineering-oriented applications are tackled.

In particular, the mini-symposium will cover the following topics: a priori and a posteriori

error analysis, approximation of isotropic and anisotropic singularities, high-frequency problems,

fractional diffusion, kinetic equations, efficient solvers and geometric/algebraic preconditioners,

boundary elements, and discontinuous Galerkin methods.

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2:00pm - 2:30pm

Exponential convergence for nonlinear problems in quantum chemistry

Carlo Marcati

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 analytic-type weighted Sobolev spaces, with isolated point singularities.

In this talk, I will present some weighted analyticity results for the solutions of Schrödinger-type 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

Hp-FEM and POD applied to object characterisation for metal detection and security screening

Paul D. Ledger1, Ben A. Wilson2, William R.B. Lionheart3

1Keele University, United Kingdom; 2Swansea University, United Kingdom; 3The 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 non-threat objects can be computed accurately using hp-finite 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 threat-object identification using machine learning classification algorithms trained using this dictionary.


[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), 3520-3533, 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), 830-860, 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), 78-113, 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), 1940-1963, 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

hp-methods for time dependent fractional diffusion

Jens Markus Melenk1, Alexander Rieder2

1TU Wien, Austria; 2University 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 hp-Finite 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 Riesz-Dunford functional calculus. We prove that, when combined with hp-FEM in space, the resulting scheme delivers exponential convergence towards the exact solution without compatibility conditions on the data. By including a double-exponential coordinate transformation, the convergence of the scheme can be further improved.

We also sketch an alternative approach to apply hp-FEM techniques for such a non-local problem. Namely, one can extend the spacial dimension by one using the so-called Caffarelli-Silvestre extension. This localizes the fractional laplacian, leading to a degenerate, elliptic problem with dynamic boundary condition. This problem can then be treated using standard hp-FEM tools.

3:30pm - 4:00pm

mortar coupling of hp-FEM and hp-BEM for the Helmholtz equation

Jens Markus Melenk1, Lorenzo Mascotto2, Ilaria Perugia2, Alexander Rieder2, Christoph Erath3

1TU Wien; 2University of Vienna; 3PH Vorarlberg

We present a FEM-BEM coupling strategy for time-harmonic 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. Quasi-optimal convergence is shown for sufficiently fine discretizations. We consider both conforming and DG discretizations.

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