arXiv subjects

Wim Vanroose

Publications and source records attributed to Wim Vanroose.

At least 37 records · Page 2Linked to original sources

arXiv2016

On the optimality of shifted Laplacian in the class of expansion preconditioners for the Helmholtz equation

This paper introduces and explores the class of expansion preconditioners EX(m) that forms a direct generalization to the classic complex shifted Laplace (CSL) preconditioner for Helmholtz problems. The construction of the EX(m) preconditioner is based upon a truncated Taylor series expansion of the original Helmholtz operator inverse. The expansion preconditioner is shown to significantly improve Krylov solver convergence rates for the Helmholtz problem for growing values of the number of series terms m. However, the addition of multiple terms in the expansion also increases the computational cost of applying the preconditioner. A thorough cost-benefit analysis of the addition of extra terms in the EX(m) preconditioner proves that the CSL or EX(1) preconditioner is the practically most efficient member of the expansion preconditioner class. Additionally, possible extensions to the expansion preconditioner class that further increase preconditioner efficiency are suggested.

math.AP↗

arXiv2015

A Generalized Bidiagonal-Tikhonov Method Applied To Differential Phase Contrast Tomography

Phase contrast tomography is an alternative to classic absorption contrast tomography that leads to higher contrast reconstructions in many applications. We review how phase contrast data can be acquired by using a combination of phase and absorption gratings. Using algebraic reconstruction techniques the object can be reconstructed from the measured data. In order to solve the resulting linear system we propose the Generalized Bidiagonal Tikhonov (GBiT) method, an adaptation of the generalized Arnoldi-Tikhonov method that uses the bidiagonal decomposition of the matrix instead of the Arnoldi decomposition. We also study the effect of the finite difference operator in the model by examining the reconstructions with either a forward difference or a central difference approximation. We validate our conclusions with simulated and experimental data.

math.NA↗

arXiv2015

A fast and robust computational method for the ionization cross sections of the driven Schroedinger equation using an O(N) multigrid-based scheme

This paper improves the convergence and robustness of a multigrid-based solver for the cross sections of the driven Schroedinger equation. Adding an Coupled Channel Correction Step (CCCS) after each multigrid (MG) V-cycle efficiently removes the errors that remain after the V-cycle sweep. The combined iterative solution scheme (MG-CCCS) is shown to feature significantly improved convergence rates over the classical MG method at energies where bound states dominate the solution, resulting in a fast and scalable solution method for the complex-valued Schroedinger break-up problem for any energy regime. The proposed solver displays optimal scaling; a solution is found in a time that is linear in the number of unknowns. The method is validated on a 2D Temkin-Poet model problem, and convergence results both as a solver and preconditioner are provided to support the O(N) scalability of the method. This paper extends the applicability of the complex contour approach for far field map computation [S. Cools, B. Reps, W. Vanroose, An Efficient Multigrid Calculation of the Far Field Map for Helmholtz and Schroedinger Equations, SIAM J. Sci. Comp. 36(3) B367--B395, 2014].

physics.comp-ph↗

arXiv2015

A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.

math.NA↗

arXiv2015

Localised auxin peaks in concentration-based transport models of the shoot apical meristem

We study the formation of auxin peaks in a generic class of concentration-based auxin transport models, posed on static plant tissues. Using standard asymptotic analysis we prove that, on bounded domains, auxin peaks are not formed via a Turing instability in the active transport parameter, but via simple corrections to the homogeneous steady state. When the active transport is small, the geometry of the tissue encodes the peaks' amplitude and location: peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We test our theory and perform numerical bifurcation analysis on two models which are known to generate auxin patterns for biologically plausible parameter values. In the same parameter regimes, we find that realistic tissues are capable of generating a multitude of stationary patterns, with a variable number of auxin peaks, that can be selected by different initial conditions or by quasi-static changes in the active transport parameter. The competition between active transport and production rate determines whether peaks remain localised or cover the entire domain. We relate the occurrence of localised patterns to a snaking bifurcation structure, which is known to arise in a wide variety of nonlinear media but has not yet been reported in plant models.

math.DS↗

arXiv2014

Constrained Runs algorithm as a lifting operator for the Boltzmann equation

Lifting operators play an important role in starting a kinetic Boltzmann model from given macroscopic information. The macroscopic variables need to be mapped to the distribution functions, mesoscopic variables of the Boltzmann model. A well-known numerical method for the initialization of Boltzmann models is the Constrained Runs algorithm. This algorithm is used in literature for the initialization of lattice Boltzmann models, special discretizations of the Boltzmann equation. It is based on the attraction of the dynamics toward the slow manifold and uses lattice Boltzmann steps to converge to the desired dynamics on the slow manifold. We focus on applying the Constrained Runs algorithm to map density, average flow velocity, and temperature, the macroscopic variables, to distribution functions. Furthermore, we do not consider only lattice Boltzmann models. We want to perform the algorithm for different discretizations of the Boltzmann equation and consider a standard finite volume discretization.

cs.CE↗

arXiv2014

Initialization of lattice Boltzmann models with the help of the numerical Chapman-Enskog expansion

We extend the applicability of the numerical Chapman-Enskog expansion as a lifting operator for lattice Boltzmann models to map density and momentum to distribution functions. In earlier work [Vanderhoydonc et al. Multiscale Model. Simul. 10(3): 766-791, 2012] such an expansion was constructed in the context of lifting only the zeroth order velocity moment, namely the density. A lifting operator is necessary to convert information from the macroscopic to the mesoscopic scale. This operator is used for the initialization of lattice Boltzmann models. Given only density and momentum, the goal is to initialize the distribution functions of lattice Boltzmann models. For this initialization, the numerical Chapman-Enskog expansion is used in this paper.

cs.CE↗

arXiv2014

An efficient multigrid calculation of the far field map for Helmholtz and Schrödinger equations

In this paper we present a new highly efficient calculation method for the far field amplitude pattern that arises from scattering problems governed by the d-dimensional Helmholtz equation and, by extension, Schrödinger's equation. The new technique is based upon a reformulation of the classical real-valued Green's function integral for the far field amplitude to an equivalent integral over a complex domain. It is shown that the scattered wave, which is essential for the calculation of the far field integral, can be computed very efficiently along this complex contour (or manifold, in multiple dimensions). Using the iterative multigrid method as a solver for the discretized damped scattered wave system, the proposed approach results in a fast and scalable calculation method for the far field map. The complex contour method is successfully validated on Helmholtz and Schrödinger model problems in two and three spatial dimensions, and multigrid convergence results are provided to substantiate the wavenumber scalability and overall performance of the method.

math.NA↗

arXiv2013

A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems

In this paper we construct and analyse a level-dependent coarsegrid correction scheme for indefinite Helmholtz problems. This adapted multigrid method is capable of solving the Helmholtz equation on the finest grid using a series of multigrid cycles with a grid-dependent complex shift, leading to a stable correction scheme on all levels. It is rigourously shown that the adaptation of the complex shift throughout the multigrid cycle maintains the functionality of the two-grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state-of-the-art multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or BiCGStab.

math.NA↗

arXiv2013

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.

math.AP↗

arXiv2013

GMRES-based multigrid for the complex scaled preconditoner for the indefinite Helmholtz equation

Multigrid preconditioners and solvers for the indefinite Helmholtz equation suffer from non-stability of the stationary smoothers due to the indefinite spectrum of the operator. In this paper we explore GMRES as a replacement for the stationary smoothers of the standard multigrid method. This results in a robust and efficient solver for a complex shifted or stretched Helmholtz problem that can be used as a preconditioner. Very few GMRES iterations are required on each level to build a good multigrid method. The convergence behavior is compared to a theoretically derived stable polynomial smoother. We test this method on some benchmark problems and report on the observed convergence behavior.

math.NA↗

arXiv2013

Exploration of stable and unstable vortex patterns in a superconductor under a magnetic disc

The stable and unstable solutions of a square 2D extreme type-II superconductor are studied in the field of a magnetic disc. We use a preconditioned Newton-Krylov solver to find the solutions and use numerical continuation to track the solutions as the field strength varies. For a disc with a small radius, we have identified generic scenarios through which the state loses and regains its stability.

cond-mat.supr-con↗

arXiv2012

Efficient determination of the energy landscape of nonlinear Schrödinger-type equations

We describe a systematic approach for the efficient numerical solution of nonlinear Schrödinger-type partial differential equations of the form $(K +V + g|ψ|^2)ψ=0$, with an energy operator $K$, a scalar potential $V$, and a scalar parameter $g$. Instrumental to the approach are developments in numerical linear and nonlinear algebra, specifically numerical parameter continuation. We demonstrate how a continuous sequence of solutions can be obtained regardless of their stability, so that finally the spectrum of stable and unstable solutions in the specified parameter range is fully revealed. The method is demonstrated for the GL equation in a three-dimensional superconducting domain with an inhomogeneous magnetic field, a numerically demanding problem known to have an involved solution landscape.

physics.comp-ph↗

arXiv2012

An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n).

physics.comp-ph↗

arXiv2012

Improved convergence of scattering calculations in the oscillator representation

The Schrödinger equation for two and tree-body problems is solved for scattering states in a hybrid representation where solutions are expanded in the eigenstates of the harmonic oscillator in the interaction region and on a finite difference grid in the near-- and far--field. The two representations are coupled through a high--order asymptotic formula that takes into account the function values and the third derivative in the classical turning points. For various examples the convergence is analyzed for various physics problems that use an expansion in a large number of oscillator states. The results show significant improvement over the JM-ECS method [Bidasyuk et al, Phys. Rev. C 82, 064603 (2010)].

physics.comp-ph↗

arXiv2012

Numerical bifurcation analysis of pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al (2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.

math.DS↗

arXiv2012

Numerical extraction of a macroscopic pde and a lifting operator from a lattice Boltzmann model

Lifting operators play an important role in starting a lattice Boltzmann model from a given initial density. The density, a macroscopic variable, needs to be mapped to the distribution functions, mesoscopic variables, of the lattice Boltzmann model. Several methods proposed as lifting operators have been tested and discussed in the literature. The most famous methods are an analytically found lifting operator, like the Chapman-Enskog expansion, and a numerical method, like the Constrained Runs algorithm, to arrive at an implicit expression for the unknown distribution functions with the help of the density. This paper proposes a lifting operator that alleviates several drawbacks of these existing methods. In particular, we focus on the computational expense and the analytical work that needs to be done. The proposed lifting operator, a numerical Chapman-Enskog expansion, obtains the coefficients of the Chapman-Enskog expansion numerically. Another important feature of the use of lifting operators is found in hybrid models. There the lattice Boltzmann model is spatially coupled with a model based on a more macroscopic description, for example an advection-diffusion-reaction equation. In one part of the domain, the lattice Boltzmann model is used, while in another part, the more macroscopic model. Such a hybrid coupling results in missing data at the interfaces between the different models. A lifting operator is then an important tool since the lattice Boltzmann model is typically described by more variables than a model based on a macroscopic partial differential equation.

cs.CE↗

arXiv2011

Numerical bifurcation study of superconducting patterns on a square

This paper considers the extreme type-II Ginzburg-Landau equations that model vortex patterns in superconductors. The nonlinear PDEs are solved using Newton's method, and properties of the Jacobian operator are highlighted. Specifically, it is illustrated how the operator can be regularized using an appropriate phase condition. For a two-dimensional square sample, the numerical results are based on a finite-difference discretization with link variables that preserves the gauge invariance. For two exemplary sample sizes, a thorough bifurcation analysis is performed using the strength of the applied magnetic field as a bifurcation parameter and focusing on the symmetries of this system. The analysis gives new insight in the transitions between stable and unstable states, as well as the connections between stable solution branches.

math.DS↗