Frequency-domain full-waveform inversion (FWI) is suitable for long-offset stationary-recording acquisition, since reliable subsurface models can be reconstructed with a few frequencies and attenuation is easily implemented without computational overhead. In the frequency domain, wave modelling is a Helmholtz-type boundary-value problem which requires to solve a large and sparse system of linear equations per frequency with multiple right-hand sides (sources). This system can be solved with direct or iterative methods. While the former are suitable for FWI application on 3D dense OBC acquisitions covering spatial domains of moderate size, the later should be the approach of choice for sparse node acquisitions covering large domains (more than 50 millions of unknowns). Fast convergence of iterative solvers for Helmholtz problems remains however challenging due to the non definiteness of the Helmholtz operator, hence requiring efficient preconditioners. In this study, we use the Krylov subspace GMRES iterative solver combined with a multi-level domain-decomposition preconditioner. Discretization relies on continuous finite elements on unstructured tetrahedral meshes to comply with complex geometries and adapt the size of the elements to the local wavelength ($h$-adaptivity). We assess the convergence and the scalability of our method with the acoustic 3D SEG/EAGE Overthrust model up to a frequency of 20~Hz and discuss its efficiency for multi right-hand side processing.
Discretization of flow in fractured porous media commonly lead to large systems of linear equations that require dedicated solvers. In this work, we develop an efficient linear solver and its practical implementation for mixed-dimensional scalar elliptic problems. We design an effective preconditioner based on approximate block factorization and algebraic multigrid techniques. Numerical results on benchmarks with complex fracture structures demonstrate the effectiveness of the proposed linear solver and its robustness with respect to different physical and discretization parameters.
University of Oxford（牛津大学）；European Centre For Medium Range Weather Forecasts（欧洲中期天气预报中心）；National Center for Atmospheric Research（美国国家大气研究中心）
It is tested whether machine learning methods can be used for preconditioning to increase the performance of the linear solver -- the backbone of the semi-implicit, grid-point model approach for weather and climate models. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding…
The emergence of long-offset sparse stationary-recording surveys carried out with ocean bottom nodes (OBN) makes frequency-domain full waveform inversion (FWI) attractive to manage compact volume of data and perform attenuation imaging. One challenge of frequency-domain FWI is the forward problem, which requires the solution of large and sparse linear systems with multiple right-hand sides. While direct methods are suitable for dense acquisitions and problems involving less than 100 million unknowns, iterative solver are more suitable for large computational domains covered by sparse OBN surve…
Frequency-domain full-waveform inversion (FWI) is suitable for long-offset stationary-recording acquisition, since reliable subsurface models can be reconstructed with a few frequencies and attenuation is easily implemented without computational overhead. In the frequency domain, wave modeling is a Helmholtz-type boundary-value problem which requires to solve a large and sparse system of linear equations per frequency with multiple right-hand sides (sources). This system can be solved with direct or iterative methods. While the former are suitable for FWI application on 3D dense OBC acquisitio…
We present a novel preconditioning technique for Krylov subspace algorithms to solve fluid-structure interaction (FSI) linearized systems arising from finite element discretizations. An outer Krylov subspace solver preconditioned with a geometric multigrid (GMG) algorithm is used, where for the multigrid level sub-solvers, a field-split (FS) preconditioner is proposed. The block structure of the FS preconditioner is derived using the physical variables as splitting strategy. To solve the subsystems originated by the FS preconditioning, an additive Schwarz (AS) block strategy is employed. The p…
Three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy preconditioner. The multigroup Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. The new multigrid in energy preconditioner reduces iteration count for many problem types and takes ad…
The JOREK extended magneto-hydrodynamic (MHD) code is a widely used simulation code for studying the non-linear dynamics of large-scale instabilities in divertor tokamak plasmas. Due to the large scale-separation intrinsic to these phenomena both in space and time, the computational costs for simulations in realistic geometry and with realistic parameters can be very high, motivating the investment of considerable effort for optimization. In this article, a set of developments regarding the JOREK solver and preconditioner is described, which lead to overall significant benefits for large produ…
We propose an alternative implementation of preconditioning techniques for the solution of non-linear problems. Within the framework of Newton-Krylov methods, preconditioning techniques are needed to improve the performance of the solvers. We propose a different implementation approach to re-utilize existing semi-implicit methods to precondition fully implicit non-linear schemes. We propose a predictor-corrector approach where the fully non-linear scheme is the corrector and the pre-existing semi-implicit scheme is the predictor. The advantage of the proposed approach is that it allows to retr…
We study the algorithmic optimization and performance tuning of the Lattice QCD clover-fermion solver for the K computer. We implement the Lüscher's SAP preconditioner with sub-blocking in which the lattice block in a node is further divided to several sub-blocks to extract enough parallelism for the 8-core CPU SPARC64$^{\mathrm{TM}}$ VIIIfx of the K computer. To achieve a better convergence property we use the symmetric successive over-relaxation (SSOR) iteration with {\it locally-lexicographical} ordering for the sub-blocks in obtaining the block inverse. The SAP preconditioner is included i…
Force-free plasmas are a good approximation where the plasma pressure is tiny compared with the magnetic pressure, which is the case during the cold vertical displacement event (VDE) of a major disruption in a tokamak. On time scales long compared with the transit time of Alfven waves, the evolution of a force-free plasma is most efficiently described by the quasi-static magnetohydrodynamic (MHD) model, which ignores the plasma inertia. Here we consider a regularized quasi-static MHD model for force-free plasmas in tokamak disruptions and propose a mimetic finite difference (MFD) algorithm. Th…
Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves, while not suffering from severe restrictions on the timestep size. To propagate the state of the atmosphere forward in time, a non-linear equation for the prognostic variables has to be solved at every timestep. Since the nonlinearity is typically weak, this is done with a small number of Newton- or Picard- iterations, which in turn require the efficient sol…
We present a Bayesian model for multi-resolution CMB component separation based on Wiener filtering and/or computation of constrained realizations, extending a previously developed framework. We also develop an efficient solver for the corresponding linear system for the associated signal amplitudes. The core of this new solver is an efficient preconditioner based on the pseudo-inverse of the coefficient matrix of the linear system. In the full sky coverage case, the method gives a speed-up of 2--3x in compute time compared to a simple diagonal preconditioner, and it is easier to implement in …
