Martin J. Gander’s research while affiliated with University of Geneva and other places

... -The Stein equation (obtained for A 1 = I n and B 1 = I m ), also known as the discrete-time Sylvester equation, appears in the analysis of dynamical systems [14] and in the stage equations of implicit Runge-Kutta methods [15]. ...

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Preconditioning techniques for generalized Sylvester matrix equations

... The switching function used in NDSD uses the reduced density gradient of the frozen electron density to detect regions close to the nuclei. Recently, Wesolowski and co-workers improved on this functional by considering a more general kernel of a differential operator instead of the von Weizsäcker functional and by improving the switching function, resulting in the non-decomposable NDCS approximation [66] as well as a symmetrized version thereof [67]. ...

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Subsystem Density-Functional Theory (Update)

... Furthermore, our numerical experiments in higher dimensions show that the 1D results are still giving close to optimal relaxation parameters, even on non-tensor and irregular meshes, which indicates that our 1D analysis captures fundamental diffusion and singularly perturbed reaction diffusion behavior of the underlying operator, not just in 1D and for tensor product meshes. A further illustration of the interest of our detailed 1D analysis is our publication [14] showing that the optimization can be carried as far as to obtain an exact solver from an iterative one, with exact analytical expressions for the relaxation parameters involved. ...

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Optimization of two-level methods for DG discretizations of reaction-diffusion equations

... Specifically, for each face of M n , we use the recorded codes ({γ} and {τ}) to find all the triangles of M n+1 that intersect it, and compute the triangle-to-triangle intersection. For more details of the intersection implementation, refer to [39]. Moreover, in the step (2) and step (4) of the method described above, we can record additional triplets and quadruplets for embedding M n on M n+1 . ...

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Self-Parameterization Based Multi-Resolution Mesh Convolution Networks

... This approach involves making a detailed quantum mechanical treatment of the central molecule or protein while considering an average potential of the surrounding solvent. In particular, we are considering frozen density embedding theory [31][32][33] in which the full density is partitioned into the embedded subsystem and the environment. This is somewhat reminiscent of the ion correlation models [22] where the central ion sphere is embedded in a large correlation sphere which acts as the average environment potential of the plasma. ...

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Embedding theory contributions to average atom models for warm dense matter

... • Krylov acceleration: replacing stationary iterations by Krylov solvers is classical in Schwarz methods [30]. The Dirichlet condition Q F = ∞ preserves some symmetry so that we can derive a conjugate gradient algorithm, see "Conjugate gradient" section. ...

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Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

... However, it is also possible to write an overlapping method, such as Lions' Parallel Schwarz Method (PSM) [28], which is equivalent to RAS [16], in substructured form, even though this approach is much less common in the literature. For a two subdomain decomposition, a substructuring procedure applied to the PSM is carried out in [15,Section 5], [18,Section 3.4] and [10]. In [10,11], the authors introduced a substructured formulation of the PSM at the continuous level for decompositions with many subdomains and crosspoints, and further studied ad hoc spectral and geometric two-level methods. ...

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Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning

... In the free space problem C = n − i and in the waveguide problem C = n (n being the unit outer normal vector). Assume that Ω = ∪ =1 Ω with Ω = ( − −1 , + ) × (0, 1), ± := ± 2 , > 0 and ≥ 0. The optimized Schwarz method iteratively solves (1) restricted to Ω for ≈ | Ω in parallel or in some order of = 1, .., with the transmission conditions From the Sturm-Liouville problem − = 2 in (0, 1), (1) to an ODE forˆ ( , ) for each , and the iteration operator acting on { ∓ := B ∓ at { − −1 , + } × (0, 1)} to a matrix for each ; see [8]. The spectral radius of the iteration matrix as a function of or Re / is called the convergence factor . ...

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Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique

... Here, the fine propagator in time subintervals is only performed sequentially, which can be implemented in parallel. Further studies based on the parareal method include the parallel implicit time-integrator (PITA) [8], ParaExp [10], adaptive parareal method [19,23], etc. Parareal can also be constructed by combining with other techniques, such as the strategies of domain decomposition and waveform relaxation [3,12,20], the diagonalization technique [14], and the application of probabilistic methods to time-parallelization [25]. ...

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Energy-Preserving Parareal-RKN Algorithms for Hamiltonian Systems

... Under the energy norm, we found similar results in the symmetric case as for the Poisson problem. Therefore, we can expect similar convergence behavior for many subdomains as presented in [3]. Furthermore, explicit formulations along with an upper bound are also given for the optimal relaxation parameters with a nonsymmetric decomposition, for which the methods converge still in two iterations in the one-dimensional case. ...

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Dirichlet-Neumann and Neumann-Neumann Methods for Elliptic Control Problems