An introduction to Multitrace Formulations and Associated Domain Decomposition Solvers

Multitrace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. We introduce in this paper MTFs on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multitrace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderon projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem{\color{black}, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments

Asymptotic and numerical analysis for Holland and Simpson’s thin wire formalism

AbstractIn the context of simulation of electromagnetic propagation, the thin wire formalism of Holland and Simpson allows one to deal with scattering by perfectly conducting thin wires by coupling a standard FDTD method with a discrete 1D wave equation ruling the current inside the wires. This method can be very accurate, but it involves a fitting parameter that requires careful calibration.We propose a consistency analysis and derive a formula for the calibration of this parameter in the case of a simplified 2D analogue of the method of Holland and Simpson. Our proof relies on the observation that this method is actually a hidden version of the singular function method well known in the context of elliptic equations in domains with a singular boundary

Multitrace formulations and Dirichlet-Neumann algorithms

Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [1, 2, 4] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as indicated in [5], but very little is known so far about such associated iterative solvers. The goal of our presentation is to give an elementary introduction to MTFs, and also to establish a natural connection with the more classical Dirichlet-Neumann algorithms that are well understood in the domain decomposition literature, see for example [6, 7]. We present for a model problem a convergence analysis for a naturally arising block iterative method associated with the MTF, and also first numerical results to illustrate what performance one can expect from such an iterative solver

Substructuring the Hiptmair-Xu preconditioner for positive Maxwell problems

Considering positive Maxwell problems, we propose a substructured version of the Hiptmair-Xu preconditioner based on a new formula that expresses the inverse of Schur systems in terms of the inverse matrix of the global volume problem

An introduction to multitrace formulations and associated domain decomposition solvers

Multi-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments

Integral equation methods for acoustic scattering by fractals

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ is uniformly of $d$-dimensional Hausdorff dimension in a sense we make precise (a $d$-set), the operator in our equation is an integral operator on $\Gamma$ with respect to $d$-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code

Non-intersecting squared Bessel paths at a hard-edge tacnode

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of $n$ non-intersecting squared Bessel paths, with all paths starting at the same point $a>0$ at time $t=0$ and ending at the same point $b>0$ at time $t=1$. Our interest lies in the critical regime $ab=1/4$, for which the paths are tangent to the hard edge at the origin at a critical time $t^*\in (0,1)$. The critical behavior of the paths for $n\to\infty$ is studied in a scaling limit with time $t=t^*+O(n^{-1/3})$ and temperature $T=1+O(n^{-2/3})$. This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size $4\times 4$. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlev\'e II equation $q"(x) = xq(x)+2q^3(x)-\nu,$ where $\nu=\alpha+1/2$ with $\alpha>-1$ the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang \cite{DKZ} for the homogeneous case $\nu = 0$.Comment: 54 pages, 13 figures. Corrected error in Theorem 2.

From START to FINISH : the influence of osmotic stress on the cell cycle

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Unusual multisystemic involvement and a novel BAG3 mutation revealed by NGS screening in a large cohort of myofibrillar myopathies

Background: Myofibrillar myopathies (MFM) are a group of phenotypically and genetically heterogeneous neuromuscular disorders, which are characterized by protein aggregations in muscle fibres and can be associated with multisystemic involvement. Methods: We screened a large cohort of 38 index patients with MFM for mutations in the nine thus far known causative genes using Sanger and next generation sequencing (NGS). We studied the clinical and histopathological characteristics in 38 index patients and five additional relatives (n = 43) and particularly focused on the associated multisystemic symptoms. Results: We identified 14 heterozygous mutations (diagnostic yield of 37%), among them the novel p.Pro209Gln mutation in the BAG3 gene, which was associated with onset in adulthood, a mild phenotype and an axonal sensorimotor polyneuropathy, in the absence of giant axons at the nerve biopsy. We revealed several novel clinical phenotypes and unusual multisystemic presentations with previously described mutations: hearing impairment with a FLNC mutation, dysphonia with a mutation in DES and the first patient with a FLNC mutation presenting respiratory insufficiency as the initial symptom. Moreover, we described for the first time respiratory insufficiency occurring in a patient with the p.Gly154Ser mutation in CRYAB. Interestingly, we detected a polyneuropathy in 28% of the MFM patients, including a BAG3 and a MYOT case, and hearing impairment in 13%, including one patient with a FLNC mutation and two with mutations in the DES gene. In four index patients with a mutation in one of the MFM genes, typical histological findings were only identified at the ultrastructural level (29%). Conclusions: We conclude that extraskeletal symptoms frequently occur in MFM, particularly cardiac and respiratory involvement, polyneuropathy and/or deafness. BAG3 mutations should be considered even in cases with a mild phenotype or an adult onset. We identified a genetic defect in one of the known genes in less than half of the MFM patients, indicating that more causative genes are still to be found. Next generation sequencing techniques should be helpful in achieving this aim