Relaxation (iterative method)

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Abstract. We propose a novel algorithm based on inexact GMRES methods for linear response calculations in density functional theory. Such calculations require iteratively solving a nested linear problem [Formula: see text] to obtain the variation of the electron density [Formula: see text]. Notably each application of the dielectric operator [Formula: see text] in turn requires the iterative solution of multiple linear systems, the Sternheimer equations. We develop computable bounds to estimate the accuracy

Simulating turbulence in the tokamak boundary, which is crucial for understanding and optimizing future fusion power plants performance, is computationally challenging due to the complex, multi-physics, and multi-scale nature of plasma dynamics. In the boundary region, plasma interacts with neutral particles recycled from the walls, leading to nonlinear phenomena that control confinement and heat exhaustion. Accurately capturing these coupled processes requires solving large-scale, stiff, and non-local pro

Mathematical programming establishes a rigorous foundation for solving complex problems in fields such as engineering, operations research, and finance. However, deploying these methods in real-time control applications, particularly Model Predictive Control (MPC), presents significant challenges regarding computational resources, solve times, and scalability. This thesis addresses these hurdles by progressing from theoretical verification methods to numerical algorithm design and experimental deployment.

Kinetic effects of bulk and fast particles in tokamaks can significantly impact the stability predictions of Magnetohydrodynamic (MHD) modes. Hybrid kinetic-MHD models are typically used to estimate these effects. Many such existing codes compute the kinetic effects by solving a reduced drift-kinetic or gyrokinetic equation semi-analytically, which often involves approximations limiting the type of orbits that can be accurately reproduced. Others solve particle orbits as a time evolution problem, requiring

Because of their robustness, efficiency and non-intrusiveness, Monte Carlo methods are probably the most popular approach in uncertainty quantification to computing expected values of quantities of interest (QoIs). Multilevel Monte Carlo (MLMC) methods significantly reduce the computational cost by distributing the sampling across a hierarchy of discretizations and allocating most samples to the coarser grids. For time dependent problems, spatial coarsening typically entails an increased time-step. Geometri

In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeith

The ever-increasing requirements for reliability and quality of supply suggest to enable the self-healing features of modern distribution networks. Within the context of Active Distribution Networks (ADNs), new, fast and efficient restoration strategies ca ...

We analyze the numerical performance of a preconditioning technique recently proposed in [4] for the ecient solution of parametrized linear systems arising from the finite element (FE) discretization of parameter-dependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver,

A novel method for approximating structured singular values (also known as

\mu

- values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the ma

Equilibrium reconstruction consists in identifying, from experimental measurements, a distribution of the plasma current density that satisfies the pressure balance constraint. The LIUQE code adopts a computationally efficient method to solve this problem, ...

Isogeometric Analysis (IGA) is a computational methodology for the numerical approximation of Partial Differential Equations (PDEs). IGA is based on the isogeometric concept, for which the same basis functions, usually Non-Uniform Rational B-Splines (NURBS), are used both to represent the geometry and to approximate the unknown solutions of PDEs. Compared to the standard Finite Element method, NURBS-based IGA offers several advantages: ideally a direct interface with CAD tools, exact geometrical representat

In a previous work we developed a convex infinite dimensional linear programming (LP) approach to approximating the region of attraction (ROA) of polynomial dynamical systems subject to compact basic semialgebraic state constraints. Finite dimensional relaxations to the infinite-dimensional LP lead to a truncated moment problem in the primal and a polynomial sum-of-squares problem in the dual. This primal-dual linear matrix inequality (LMI) problem can be solved numerically with standard semidefinite progra

In this paper, the differential geometry of the novel hierarchical Tucker format for tensors is derived. The set HT,k of tensors with fixed tree T and hierarchical rank k is shown to be a smooth quotient manifold, namely the set of orbits of a Lie group action corresponding to the non-unique basis representation of these hierarchical tensors. Explicit characterizations of the quotient manifold, its tangent space and the tangent space of HT,k are derived, suitable for high-dimensional problems. The usefulnes

The Open Support Platform for Environmental Research (OSPER) has been launched to take forward key data management components developed under the Swiss Experiment platform project to achieve improved usability and a wider scope. With this project, we aim to connect users to data and their context, an area identified during SwissEx as having the greatest potential impact on the research community. OSPER has a clear focus on providing the technology for data storage, management and exploitation with a special

Large-eddy simulation LES is performed of a curved open-channel flow over topography based on the laboratory experiment by Blanckaert “Topographic steering, flow circulation, velocity redistribution and bed topography in sharp meander bends,” Water Resour. Res., doi:10.1029/ 2009WR008303 in press. In the experiment, the large-scale bed topography had developed to a more or less stationary shape which was prescribed in the LES model as boundary conditions neglecting the small-scale dune forms by means

The quantity of MEMS manufacturers has considerably been increasing these last years. This doesn't mean that the introduction of such products on the market has become easier. A packaging operation is needed to obtain a usable product out of a microsystem. ...

Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions. As such, numerical m

Relaxation (iterative method)

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Efficient Krylov Methods for Linear Response in Plane-Wave Electronic Structure Calculations

Acceleration of Solvers for Plasma Simulation in the Tokamak Boundary

Numerical Methods for Optimization and Control: From Verification to GPU-Acceleration

A new kinetic-MHD code based on the Case-Van Kampen approach for studying magnetic fusion relevant instabilities

MATHICSE Technical Report : Uncertainty Quantification by MLMC and Local Time-stepping For Wave Propagation

MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations

Service Restoration in DG-Integrated Distribution Networks Using an Exact Convex OPF Model

MATHICSE Technical Report : A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion

MATHICSE Technical Report : A novel iterative method to approximate structured singular values

Tokamak equilibrium reconstruction code LIUQE and its real time implementation

Reduced Basis method for Isogeometric Analysis: application to structural problems

Inner approximations of the region of attraction for polynomial dynamical systems

The geometry of algorithms using hierarchical tensors

Open Support Platform for Environmental Research (OSPER)-tools for the discovery and exploitation of environmental data

Large-eddy simulation of a curved open-channel flow over topography

Encapsulation hermétique pour systèmes hydro- et thermo-sensibles

Numerical Mathematics