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We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and; (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we propose solutions for problems (i) and (ii), backed up by theory and numerical experiments. Problem (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Problem (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the space of valid coefficients vectors. The coefficients vectors can then be selected using either a least squares approach or column subset selection.

This paper studies time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.

Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of a coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.

Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between a pair of probability distributions or quantum states, typically, the total variation distance or trace distance. However, if an additional distance measure between the probability distributions or states is known, then the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality proven in [I. Sason, IEEE Trans. Inf. Th., 59, 7118 (2013)]. We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. The bound is tight when the quotient of the trace distance by the operator norm distance is an integer. We then apply our results to compute upper bounds on the quantum- and private classical capacities of channels. We begin by refining the concept of approximate degradable channels, namely, $\varepsilon$-degradable channels, which are, by definition, $\varepsilon$-close in diamond norm to their complementary channel when composed with a degrading channel. To this end, we introduce the notion of $(\varepsilon,\nu)$-degradable channels; these are $\varepsilon$-degradable channels that are, in addition, $\nu$-close in completely bounded spectral norm to their complementary channel, when composed with the same degrading channel. This allows us to derive improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. We illustrate our results by obtaining a new upper bound on the quantum capacity of the qubit depolarizing channel.

In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schr\"odinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove $L^2$-error bounds and observable error bounds for the approximating Gaussian wave packet.

Ferromagnetic materials in indoor environments give rise to disturbances in the ambient magnetic field. Maps of these magnetic disturbances can be used for indoor localisation. A Gaussian process can be used to learn the spatially varying magnitude of the magnetic field using magnetometer measurements and information about the position of the magnetometer. The position of the magnetometer, however, is frequently only approximately known. This negatively affects the quality of the magnetic field map. In this paper, we investigate how an array of magnetometers can be used to improve the quality of the magnetic field map. The position of the array is approximately known, but the relative locations of the magnetometers on the array are known. We include this information in a novel method to make a map of the ambient magnetic field. We study the properties of our method in simulation and show that our method improves the map quality. We also demonstrate the efficacy of our method with experimental data for the mapping of the magnetic field using an array of 30 magnetometers.

We develop a novel discontinuous Galerkin method for solving the rotating thermal shallow water equations (TRSW) on a curvilinear mesh. Our method is provably entropy stable, conserves mass, buoyancy and vorticity, while also semi-discretely conserving energy. This is achieved by using novel numerical fluxes and splitting the pressure and convection operators. We implement our method on a cubed sphere mesh and numerically verify our theoretical results. Our experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilization.

For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving (SSP) temporal discretizations, we show that the simple bound-preserving limiter in [Li H., Xie S., Zhang X., SIAM J. Numer. Anal., 56 (2018)]. can enforce the strict bounds of the vorticity, if the velocity field satisfies a discrete divergence free constraint. For reducing oscillations, a modified TVB limiter adapted from [Cockburn B., Shu CW., SIAM J. Numer. Anal., 31 (1994)] is constructed without affecting the bound-preserving property. This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.

In this paper, we study two graph convexity parameters: iteration time and general position number. The iteration time was defined in 1981 in the geodesic convexity, but its computational complexity was so far open. The general position number was defined in the geodesic convexity and proved NP-hard in 2018. We extend these parameters to any graph convexity and prove that the iteration number is NP-hard in the P3 convexity. We use this result to prove that the iteration time is also NP-hard in the geodesic convexity even in graphs with diameter two, a long standing open question. These results are also important since they are the last two missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities. We also prove that the general position number of the monophonic convexity is W[1]-hard (parameterized by the size of the solution) and $n^{1-\varepsilon}$-inapproximable in polynomial time for any $\varepsilon>0$ unless P=NP, even in graphs with diameter two. Finally, we also obtain FPT results on the general position number in the P3 convexity and we prove that it is W[1]-hard (parameterized by the size of the solution).