We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as @math , where @math is the number of volume unknowns, and @math is the number of processors, as long as @math grows at most like a small fractional power of @math . The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations.
proposed the multi-trace formulation @cite_84 , which involves solving an integral equation posed on the interfaces of a decomposed domain, which is naturally well suited for operator preconditionning in the case of piece-wise constant medium;
@cite_84
2083690159
We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in "subdomains") that directly lends itself to operator preconditioning via Calderon projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise @math -distributions over each associated interface. Through the use of interior Calderon projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning. We study a high order Discontinuous Galerkin Time Domain (DGTD) method for solving the system of Maxwell equations. This method is formulated on a non-conforming multi-element mesh mixing an unstructured triangulation for the discretization of irregularly shaped objects with a structured (Cartesian) quadrangulation for the rest of the computational domain. Within each element, the electromagnetic field components are approximated by a high order nodal polynomial. The DG method proposed here makes use of a centered scheme for the definition of the numerical traces of the electric and magnetic fields at element interfaces, associated to a second order or fourth order leap-frog scheme for the time integration of the associated semi-discrete equations. We formulate this DGTD method in 3D and study its theoretical properties. In particular, an a priori convergence estimation is elaborated. Finally, we present numerical results for the application of the method to 2D test problems. We propose and study a domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The convergence of the method is proved for the case of constant wave number based on the analysis of the fundamental solution of the PML equation. The method can be used as an efficient preconditioner in the preconditioned GMRES method for solving discrete Helmholtz equations with constant and heterogeneous wave numbers. Numerical examples are included. A preconditioned iterative solver for the 3D frequency-domain wave equation applied to seismic problems is evaluated. The preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the (undamped) wave equation. The approximate inverse is computed with one multigrid cycle. Numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength. To evaluate the relevance of this iterative solver, the number of floating-point operations required for two imaging problems are roughly evaluated. This rough estimate indicates that the time-domain migration approach is more than one order of magnitude faster. The full-wave-form tomography, based on a least-squares formulation and a scale separation approach, has the same complexity in both domains. In this paper we present a formal framework and an experimental software supporting the user in the task of formulating a precise query ‐ which best captures their information needs ‐ even in the case of complete ignorance of the vocabulary of the underlying information system holding the data. Our intelligent interface is driven by means of appropriate automated reasoning techniques over an ontology describing the domain of the data in the information system. We will define what a query is and how it is internally represented, which operations are available to the user in order to modify the query and how contextual feedback is provided about it presenting only relevant pieces of information. We will then describe the elements that constitute the query interface available to the user, providing visual access to the underlying reasoning services and operations for query manipulation. Lastly, we will define a suitable representation in “linear form”, starting from which the query can be more easily expressed in natural language. Author Keywords The fused lasso was proposed recently to enable recovery of high-dimensional patterns which are piece-wise constant on a graph, by penalizing the ‘1-norm of dierences of measurements at vertices that share an edge. While there have been some attempts at coming up with ecient algorithms for solving the fused lasso optimization, a theoretical analysis of its performance is mostly lacking except for the simple linear graph topology. In this paper, we investigate sparsistency of fused lasso for general graph structures, i.e. its ability to correctly recover the exact support of piece-wise constant graphstructured patterns asymptotically (for largescale graphs). To emphasize this distinction over previous work, we will refer to it as Edge Lasso. We focus on the (structured) normal means setting, and our results provide necessary and sucient conditions on the graph properties as well as the signal-to-noise ratio needed to ensure sparsistency. We examplify our results using simple graph-structured patterns, and demonstrate that in some cases fused lasso is sparsistent at very weak signal-to-noise ratios (scaling as p (logn)=jAj, where n is the number of vertices in the graph and A is the smallest set of vertices with constant activation). In other cases, it performs no better than thresholding the dierence This paper presents a solution to panoramic image stitching of two images with coinciding optical centers, but unknown focal length and radial distortion. The algorithm operates with a minimal set of corresponding points (three) which means that it is well suited for use in any RANSAC style algorithm for simultaneous estimation of geometry and outlier rejection. Compared to a previous method for this problem, we are able to guarantee that the right solution is found in all cases. The solution is obtained by solving a small system of polynomial equations. The proposed algorithm has been integrated in a complete multi image stitching system and we evaluate its performance on real images with lens distortion. We demonstrate both quantitative and qualitative improvements compared to state of the art methods. Making use of a perturbed Lagrangian formulation, a finite element procedure for contact problems is developed for the general case in which node-to-node contact no longer holds. The proposed procedure leads naturally to a discretization of the contact interface into contact segments. Within the context of a bilinear interpolation for the displacement field, a mixed finite element approximation is introduced by assuming discont naous contact pressure, constant on the contact segment. Because of this piece-wise constant approximation, the gap function enters into the formulation in an ‘average’ sense instead of through a point-wise definition. Numerical examples are presented that illustrate the performance of the proposed procedure. Current finite element formulations for contact problems based on either the classical Lagrange parameter procedure [l-3, 12, 203 or the penalty-function method [4-6,11], are characterized by a point-wise enforcement of the contact-constraint condition, in the sense that penetration of the bodies is established on a nodal basis. Moreover, in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure. Within the framework of classical Lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables (Lagrange multipliers). These extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms. Penalty methods, on the other hand, enable one to transform the constrained problem into an unconstrained one without introducing additional variables. The constraint condition is now satisfied only approximately for finite values of the penalty parameter. The main difhculty associated with these methods, however, lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition. This is a well-understood phenomenon, particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics (e.g. see [l&22,25] for a review). Recently, augmented Lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and ‘regularize’ the penalty formulation (e.g. see the survey in [7,8]). Within the framework of Zinearited e u c , it is possible to restrict the finite element formulation of contact problems by assuming that node-to-node contact occurs. This is in fact the case often considered in the literature [l, 2, 4, 6, 12, 20, 211. In the general context of fully The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be devastating. In this paper, we present a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples. Parameterizing three degree-of-freedom (DOF) rotations is difficult to do well. Many graphics applications demand that we be able to compute and differentiate positions and orientations of articulated figures with respect to their rotational (and other) parameters, as well as integrate differential equations, optimize rotation parameters, and interpolate orientations. Widely used parameterizations such as Euler angles and quaternions are well suited to only a few of these operations. The exponential map maps a vector in R 3 describing the axis and magnitude of a three-DOF rotation to the corresponding rotation. Several graphics researchers have applied it with limited success to interpolation of orientations, but it has been virtually ignored with respect to the other operations mentioned above. In this paper we present formulae for computing, differentiating, and integrating three-DOF rotations with the exponential map. We show that our formulation is numerically stable in the face of machine precision issues, and that for most applications all singularities in the map can be avoided through a simple technique of dynamic reparameterization. We demonstrate how to use the exponential map to solve both the "freely rotating body" problem, and the important ball-and-socket joint required to accurately model shoulder and hip joints in articulated figures. Examining several common graphics applications, we explain the benefits of our formulation of the exponential map over Euler angles and quaternions, including robustness, small state vectors, lack of explicit constraints, good modeling capabilities, simplicity of solving ordinary differential equations, and good interpolation behavior. This contribution is the numerically oriented companion article of the work [E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Cances, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problems. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings. A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well-known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble. We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units. We present an energy-stable low-storage high order discontinuous Galerkin method for the wave equation.We describe algorithms for efficient implementation on Graphics Processing Units (GPUs).We derive order-dependent global and local timestep restrictions for DG on hybrid meshes.We verify high order convergence and computational efficiency with numerical experiments. We propose a novel partitioning method for block-structured adaptive meshes utilizing the meshless Lagrangian particle concept. With the observation that an optimum partitioning has high analogy to the relaxation of a multi-phase fluid to steady state, physically motivated model equations are developed to characterize the background mesh topology and are solved by multi-phase smoothed-particle hydrodynamics. In contrast to well established partitioning approaches, all optimization objectives are implicitly incorporated and achieved during the particle relaxation to stationary state. Distinct partitioning sub-domains are represented by colored particles and separated by a sharp interface with a surface tension model. In order to obtain the particle relaxation, special viscous and skin friction models, coupled with a tailored time integration algorithm are proposed. Numerical experiments show that the present method has several important properties: generation of approximately equal-sized partitions without dependence on the mesh-element type, optimized interface communication between distinct partitioning sub-domains, continuous domain decomposition which is physically localized and implicitly incremental. Therefore it is particularly suitable for load-balancing of high-performance CFD simulations. We study integral methods applied to the resolution of the Maxwell equations where the linear system is solved using an iterative method which requires only matrix?vector products. The fast multipole method (FMM) is one of the most efficient methods used to perform matrix?vector products and accelerate the resolution of the linear system. A problem involving N degrees of freedom may be solved in CNiterNlogN floating operations, where C is a constant depending on the implementation of the method. In this article several techniques allowing one to reduce the constant C are analyzed. This reduction implies a lower total CPU time and a larger range of application of the FMM. In particular, new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms. Several numerical tests are also described. These confirm the efficiency and the theoretical complexity of the FMM. In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called "shifted Laplace" preconditioners of the form Δφ-αk2φ with α ∈ C. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR. Given that natural image segmentation is well-known as an ill-posed problem, then how can we design an algorithm to obtain good performance as human subjects? A choice is to learn from human segmentations as MCG [3] which obtains the state-of-the-art performance and fairly high computational efficiency. Then a question arises here: what should we learn? The way all existing literatures exploit human segmentations ignores a basic fact that human segmentations are produced by human operations. The human segmentation process would inevitably fulfill the human behavior’s principles including the least effort principle (LEP): a human will strive to solve his problem in such a way as to minimize the total work that he must expend [1]. The principle gives us a new insight into our problem. Suppose you are a human subject in the BSDS segmentation experiment, and are required to segment images into pieces under the ending instruction: each piece contains only one single distinguished thing. Then your total effort F of segmenting an image should including understanding images (U) to guide following operations, and tracing boundaries (T ) by hands with a mouse: F (I,S) =U (I)+T (S). Then from the viewpoint of LEP, human subjects would like to find an acceptable segmentation S by minimizing their effort, or formally, The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially with the number of dimensions, making the use of classic iterative solvers infeasible. During the last few years, low-rank tensor approaches have been developed that allow one to mitigate this curse of dimensionality by exploiting the underlying structure of the linear operator. In this work, we focus on tensors represented in the Tucker and tensor train formats. We propose two preconditioned gradient methods on the corresponding low-rank tensor manifolds: a Riemannian version of the preconditioned Richardson method as well as an approximate Newton scheme based on the Riemannian Hessian. For the latter, considerable attention is given to the efficient solution of the resulting Newton equation. In numerical experiments, we compare the efficiency of our Riemannian algorit... Abstract Efficient algorithms for the solution of partial differential equations on parallel computers are often based on domain decomposition methods. Schwarz preconditioners combined with standard Krylov space solvers are widely used in this context, and such a combination is shown here to perform very well in the case of the Wilson–Dirac equation in lattice QCD. In particular, with respect to even-odd preconditioned solvers, the communication overhead is significantly reduced, which allows the computational work to be distributed over a large number of processors with only small parallelization losses. Abstract Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial differential equations. The large size and the condition of many technical or physical applications in this area result in the need for efficient parallelization and preconditioning techniques of the CG method, in particular on massively parallel machines. Here, the data distribution and the communication scheme for the sparse matrix operations of the preconditioned CG are based on the analysis of the indices of the non-zero elements. Polynomial preconditioning is shown to reduce global synchronizations considerably, and a fully local incomplete Cholesky preconditioner is presented. On a PARAGON XP S 10 with 138 processors, the developed parallel methods outperform diagonally scaled CG markedly with respect to both scaling behavior and execution time for many matrices from real finite element applications.