| 1. |
- Massing, Andre, et al.
(författare)
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Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions
- 2013
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Ingår i: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197 .- 1095-7200. ; 35:1, s. C23-C47
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Tidskriftsartikel (refereegranskat)abstract
- In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes. Examples of such methods are the fictitious domain method, the extended finite element method, and Nitsche's method. In all these methods, integrals must be computed over cut cells or subsimplices, which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsche's method, which we apply to Poisson's equation and a linear elastic problem. © 2013 Society for Industrial and Applied Mathematics.
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| 2. |
- Abdulle, Assyr, et al.
(författare)
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High weak order methods for stochastic differential equations based on modified equations
- 2012
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 34:3, s. A1800-A1823
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Tidskriftsartikel (refereegranskat)abstract
- Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic firstintegrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
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| 3. |
- Adlerborn, Björn, et al.
(författare)
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A parallel QZ algorithm for distributed memory HPC systems
- 2014
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM publications. - 1064-8275 .- 1095-7197. ; 36:5, s. C480-C503
-
Tidskriftsartikel (refereegranskat)abstract
- Appearing frequently in applications, generalized eigenvalue problems represent one of the core problems in numerical linear algebra. The QZ algorithm of Moler and Stewart is the most widely used algorithm for addressing such problems. Despite its importance, little attention has been paid to the parallelization of the QZ algorithm. The purpose of this work is to fill this gap. We propose a parallelization of the QZ algorithm that incorporates all modern ingredients of dense eigensolvers, such as multishift and aggressive early deflation techniques. To deal with (possibly many) infinite eigenvalues, a new parallel deflation strategy is developed. Numerical experiments for several random and application examples demonstrate the effectiveness of our algorithm on two different distributed memory HPC systems.
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| 4. |
- Adlerborn, Björn, et al.
(författare)
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Distributed one-stage Hessenberg-triangular reduction with wavefront scheduling
- 2018
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 40:2, s. C157-C180
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Tidskriftsartikel (refereegranskat)abstract
- A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented. The formulation is based on a sequential cacheblocked algorithm by K degrees agstrom et al. [BIT, 48 (2008), pp. 563 584]. A static scheduling algorithm is proposed that addresses the problem of underutilized processes caused by two-sided updates of matrix pairs based on sequences of rotations. Experiments using up to 961 processes demonstrate that the new formulation is an improvement of the state of the art and also identify factors that limit its scalability.
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| 5. |
- af Klinteberg, Ludvig, et al.
(författare)
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Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions
- 2018
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 40:3, s. A1225-A1249
-
Tidskriftsartikel (refereegranskat)abstract
- When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX—adaptive quadrature by expansion—which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss–Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard.
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| 6. |
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| 7. |
- Almquist, M., et al.
(författare)
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ORDER-PRESERVING INTERPOLATION FOR SUMMATION-BY-PARTS OPERATORS AT NONCONFORMING GRID INTERFACES
- 2019
-
Ingår i: Siam Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 41:2
-
Tidskriftsartikel (refereegranskat)abstract
- We study nonconforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the nonconforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order-preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order-preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.
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| 8. |
- Almquist, Martin, et al.
(författare)
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Order-preserving interpolation for summation-by-parts operators at nonconforming grid interfaces
- 2019
-
Ingår i: SIAM Journal of Scientific Computing. - : Society for Industrial and Applied Mathematics Publications. - 1064-8275 .- 1095-7197. ; 41:2, s. A1201-A1227
-
Tidskriftsartikel (refereegranskat)abstract
- We study nonconforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the nonconforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order-preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order-preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.
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| 9. |
- Amsallem, David, et al.
(författare)
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Energy Stable Model Reduction of Neurons by Non-negative Discrete Empirical Interpolation
- 2016
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM publishing. - 1064-8275 .- 1095-7197. ; 38:2, s. B297-B326
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Tidskriftsartikel (refereegranskat)abstract
- The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reduced-order basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups.
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| 10. |
- Andersson, Anders
(författare)
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Schwarz–Christoffel Mappings for Nonpolygonal Regions
- 2008
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Ingår i: SIAM Journal on Scientific Computing. - Philadelphia : SIAM, Philadelphia, USA. - 1064-8275 .- 1095-7197. ; 31:1, s. 94-111
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Tidskriftsartikel (refereegranskat)abstract
- An approximate conformal mapping for an arbitrary region $\varOmega$ bounded by a smooth curve $\varGamma$ is constructed using the Schwarz–Christoffel mapping for a polygonal region in which $\varOmega$ is embedded. An algorithm for finding this so-called outer polygon is presented. The resulting function is a conformal mapping from the upper half-plane or the unit disk to a region $R$, approximately equal to $\varOmega$. $R$ is bounded by a $C^\infty$ curve, and since the mapping function originates from the Schwarz–Christoffel mapping and tangent polygons are used to determine it, important properties of $\Gamma$ such as direction, linear asymptotes, and inflexion points are preserved in the boundary of $R$. The method makes extensive use of existing Schwarz–Christoffel software in both the determination of outer polygons and the calculation of function values. By the use suggested here, the capabilities of such well-written software are extended.
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| 11. |
- Andersson, Kristoffer, et al.
(författare)
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Convergence of a robust deep FBSDE method for stochastic control
- 2023
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Ingår i: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197. ; 45:1, s. A226-A255
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.
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| 12. |
- Andersson, K., et al.
(författare)
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CONVERGENCE OF A ROBUST DEEP FBSDE METHOD FOR STOCHASTIC CONTROL
- 2023
-
Ingår i: Siam Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 45:1
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.
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| 13. |
- Andersson, Lars-Erik, 1942-, et al.
(författare)
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Error analysis for operations in solid modeling in the presence of uncertainty
- 2008
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 29:2, s. 811-826
-
Tidskriftsartikel (refereegranskat)abstract
- The problem of maintaining consistent representations of solids in computer-aided design and of giving rigorous proofs of error bounds for operations such as regularized Boolean intersection has been widely studied for at least two decades. One of the major difficulties is that the representations used in practice not only are in error but are fundamentally inconsistent. Such inconsistency is one of the main bottlenecks in downstream applications. This paper provides a framework for error analysis in the context of solid modeling, in the case where the data is represented using the standard representational method, and where the data may be uncertain. Included are discussions of ill-condition, error measurement, stability of algorithms, inconsistency of defining data, and the question of when we should invoke methods outside the scope of numerical analysis. A solution to the inconsistency problem is proposed and supported by theorems: it is based on the use of Whitney extension to define sets, called Quasi-NURBS sets, which are viewed as realizations of the inconsistent data provided to the numerical method. A detailed example illustrating the problem of regularized Boolean intersection is also given.
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| 14. |
- Aoki, Yasunori, 1982-, et al.
(författare)
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Cluster Newton Method for Sampling Multiple Solutions of Underdetermined Inverse Problems : Application to a Parameter Identification Problem in Pharmacokinetics
- 2014
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 36:1, s. B14-B44
-
Tidskriftsartikel (refereegranskat)abstract
- A new algorithm is proposed for simultaneously finding multiple solutions of an underdetermined inverse problem. The algorithm was developed for an ODE parameter identification problem in pharmacokinetics for which multiple solutions are of interest. The algorithm proceeds by computing a cluster of solutions simultaneously, and is more efficient than algorithms that compute multiple solutions one-by-one because it fits the Jacobian in a collective way using a least squares approach. It is demonstrated numerically that the algorithm finds accurate solutions that are suitably distributed, guided by a priori information on which part of the solution set is of interest, and that it does so much more efficiently than a baseline Levenberg-Marquardt method that computes solutions one-by-one. It is also demonstrated that the algorithm benefits from improved robustness due to an inherent smoothing provided by the least-squares fitting.
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| 15. |
- Appelo, Daniel, et al.
(författare)
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Waveholtz : Iterative solution of the helmholtz equation via the wave equation
- 2020
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 42:4, s. A1950-A1983
-
Tidskriftsartikel (refereegranskat)abstract
- A new iterative method, the WaveHoltz iteration, for solution of the Helmholtz equation is presented. WaveHoltz is a fixed point iteration that filters the solution to the solution of a wave equation with time periodic forcing and boundary data. The WaveHoltz iteration corresponds to a linear and coercive operator which, after discretization, can be recast as a positive definite linear system of equations. The solution to this system of equations approximates the Helmholtz solution and can be accelerated by Krylov subspace techniques. Analysis of the continuous and discrete cases is presented, as are numerical experiments.
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| 16. |
- Ariel, G., et al.
(författare)
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Parareal multiscale methods for highly oscillatory dynamical systems
- 2016
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics Publications. - 1064-8275 .- 1095-7197. ; 38:6, s. A3540-A3564
-
Tidskriftsartikel (refereegranskat)abstract
- We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solutions, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.
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| 17. |
- Artebrant, Robert, et al.
(författare)
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Limiter-free third order logarithmic reconstruction
- 2006
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 28:1, s. 359-381
-
Tidskriftsartikel (refereegranskat)abstract
- A third order conservative reconstruction, in the context of finite volume schemes for hyperbolic conservation laws, is constructed based on logarithmic functions. This logarithmic method reconstructs without the use of a limiter, any preprocessing of input data, special treatments for local extrema, or shock solutions. Also the method is local in the sense that data from only the nearest neighbors are required. We test the new reconstruction method in several numerical experiments, including nonlinear systems in one and two space dimensions.
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| 18. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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Discrete-Ordinates and Streamline Diffusion Methods for a Flow Described by BGK Model
- 2014
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 36:4
-
Tidskriftsartikel (refereegranskat)abstract
- A rarefied gas flow through a channel with arbitrary cross section is studied based on the linearized Bhatnagar-Gross-Krook model. The discrete velocity and streamline diffusion finite element methods are combined to yield a numerical scheme. For this scheme we derive stability and optimal convergence rates in the L-2-type norms. The optimality is due to the maximal available regularity of the exact solution for the corresponding hyperbolic PDE. The potential of the proposed, combined methods is illustrated with some numerical examples.
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| 19. |
- Barnett, Alexander H., et al.
(författare)
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A Parallel Nonuniform Fast Fourier Transform Library Based on an “Exponential of Semicircle" Kernel
- 2019
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 41:5, s. C479-C504
-
Tidskriftsartikel (refereegranskat)abstract
- The nonuniform fast Fourier transform (NUFFT) generalizes the FFT to off-grid data. Its many applications include image reconstruction, data analysis, and the numerical solution of differential equations. We present FINUFFT, an efficient parallel library for type 1 (nonuniform to uniform), type 2 (uniform to nonuniform), or type 3 (nonuniform to nonuniform) transforms, in dimensions 1, 2, or 3. It uses minimal RAM, requires no precomputation or plan steps, and has a simple interface to several languages. We perform the expensive spreading/interpolation between nonuniform points and the fine grid via a simple new kernel-the ""exponential of semicircle"" e\beta \surd 1 - x 2 in x \in [ - 1, 1]-in a cache-aware load-balanced multithreaded implementation. The deconvolution step requires the Fourier transform of the kernel, for which we propose efficient numerical quadrature. For types 1 and 2, rigorous error bounds asymptotic in the kernel width approach the fastest known exponential rate, namely that of the Kaiser-Bessel kernel. We benchmark against several popular CPU-based libraries, showing favorable speed and memory footprint, especially in three dimensions when high accuracy and/or clustered point distributions are desired.
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| 20. |
- Bayer, Christian, et al.
(författare)
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On nonasymptotic optimal stopping criteria in monte carlo simulations
- 2014
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 36:2, s. A869-A885
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.
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| 21. |
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| 22. |
- Bernland, Anders, et al.
(författare)
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SHAPE OPTIMIZATION OF A COMPRESSION DRIVER PHASE PLUG
- 2019
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM PUBLICATIONS. - 1064-8275 .- 1095-7197. ; 41:1, s. B181-B204
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Tidskriftsartikel (refereegranskat)abstract
- A compression driver is an electro-acoustic transducer with considerably higher efficiency than direct radiating loudspeakers, thanks to the increased radiation resistance caused by a large vibrating diaphragm placed in a compression chamber with small openings. The transition section between compression chamber and output waveguide, the phase plug, must be carefully designed to avoid irregularities in the output sound pressure level (SPL) as a function of frequency. Here we present a shape optimization method based on an implicit level-set description and adjoint sensitivity analysis, which enables a large number of design parameters and vast design freedom. The CutFEM approach, a fictitious domain finite element method, removes the need for mesh updates and makes the method robust and computationally inexpensive. Numerical experiments for a generic annular diaphragm compression driver are presented, with optimized designs showing only minor frequency irregularities. Two different objective functions are considered: one for maximum SPL and one where the SPL is fitted to that of a hypothetical ideal design; the latter approach is found to be more effective in reducing irregularities. Visco-thermal boundary-layer losses are included in a post-processing step, and, though the influence of losses is clearly noticeable, the overall performance is similar and the optimized designs still outperform the original design.
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| 23. |
- Bonito, Andrea, et al.
(författare)
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Numerical Simulations of Surface Quasi-Geostrophic Flows on Periodic Domains
- 2021
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 43:2, s. B405-B430
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Tidskriftsartikel (refereegranskat)abstract
- We propose a novel algorithm for the approximation of surface quasi-geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed, and the algorithm does not have restrictions on the mesh structure or on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.
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| 24. |
- Boon, Wietse M., et al.
(författare)
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Robust Monolithic Solvers For The Stokes-Darcy Problem With The Darcy Equation In Primal Form
- 2022
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 44:4, s. B1148-B1174
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Tidskriftsartikel (refereegranskat)abstract
- We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and nonconforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.
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| 25. |
- Boon, Wietse M., et al.
(författare)
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Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure
- 2021
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 43:4, s. B961-B983
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Tidskriftsartikel (refereegranskat)abstract
- We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure, and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properties, is demonstrated through several numerical experiments.
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| 26. |
- Bosner, Nela, et al.
(författare)
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Parallel and Heterogeneous $m$-Hessenberg-Triangular-Triangular Reduction
- 2017
-
Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 39:1, s. C29-C47
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Tidskriftsartikel (refereegranskat)abstract
- The m-Hessenberg-triangular-triangular (mHTT) reduction is a simultaneous orthogonal reduction of three matrices to condensed form. It has applications, for example, in solving shifted linear systems arising in various control theory problems. A new heterogeneous CPU/GPU implementation of the mHTT reduction is presented and evaluated against an existing CPU implementation. The algorithm offloads the compute-intensive matrix-matrix multiplications to the GPU and keeps the inner loop, which is memory intensive and has a complicated control flow, on the CPU. Experiments demonstrate that the heterogeneous implementation can be superior to the existing CPU implementation on a system with 2 x 8 CPU cores and one GPU. Future development should focus on improving the scalability of the CPU computations.
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| 27. |
- Bozorgnia, Farid
(författare)
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Numerical algorithm for spatial segregation of competitive systems
- 2009
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 31:5, s. 3946-3958
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Tidskriftsartikel (refereegranskat)abstract
- Two novel iterative methods for a class of population models of competitive type are considered. This numerical solution is related to the positive solution as the competitive rate tends to infinity. Furthermore, the idea first is applied to an optimal partition problem.
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| 28. |
- Budisa, Ana, et al.
(författare)
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Mixed-dimensional auxiliary space preconditioners
- 2020
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 42:5, s. A3367-A3396
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Tidskriftsartikel (refereegranskat)abstract
- This work introduces nodal auxiliary space preconditioners for discretizations of mixed-dimensional partial differential equations. We first consider the continuous setting and generalize the regular decomposition to this setting. With the use of conforming mixed finite element spaces, we then expand these results to the discrete case and obtain a decomposition in terms of nodal Lagrange elements. In turn, nodal preconditioners are proposed analogous to the auxiliary space preconditioners of Hiptmair and Xu [SIAM T. Numer. Anal., 45 (2007), pp. 2483-2509]. Numerical experiments show the performance of this preconditioner in the context of flow in fractured porous media.
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| 29. |
- Burman, Erik, et al.
(författare)
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A Stabilized Cut Finite Element Method for the Three Field Stokes Problem
- 2015
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 37:4, s. A1705-A1726
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Tidskriftsartikel (refereegranskat)abstract
- We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure, and extra-stress tensor are discretized on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings numerically.
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| 30. |
- Burman, Erik, et al.
(författare)
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Explicit time stepping for the wave equation using cutFEM with discrete extension
- 2022
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 44:3, s. A1254-A1289
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.
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| 31. |
- Burman, Erik, et al.
(författare)
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Hybridized CutFEM for Elliptic Interface Problems
- 2019
-
Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 41:5, s. A3354-A3380
-
Tidskriftsartikel (refereegranskat)abstract
- We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.
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| 32. |
- Carey, V, et al.
(författare)
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Blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions
- 2010
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 32:4, s. 2121-2145
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Tidskriftsartikel (refereegranskat)abstract
- We describe and test an adaptive algorithm for evolution problems that employs a sequence of "blocks" consisting of fixed, though non-uniform, space meshes. This approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing, re-meshing, matrix reassembly, and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes. A major issue whith a blockadaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy. We describe several strategies to achieve this goal using adjoint-based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples.
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| 33. |
- Censor, Yair, et al.
(författare)
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On Diagonally Relaxed Orthogonal Projection Methods
- 2008
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Ingår i: SIAM Journal on Scientific Computing. - Philadelphia, PA, United States : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 30:1, s. 473-504
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Tidskriftsartikel (refereegranskat)abstract
- We propose and studya block-iterative projection method for solving linear equations and/or inequalities.The method allows diagonal componentwise relaxation in conjunction with orthogonalprojections onto the individual hyperplanes of the system, and isthus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation hasproven useful in accelerating the initial convergence of simultaneous andblock-iterative projection algorithms, but until now it was available onlyin conjunction with generalized oblique projections in which there isa special relation between the weighting and the oblique projections.DROP has been used by practitioners, and in this papera contribution to its convergence theory is provided. The mathematicalanalysis is complemented by some experiments in image reconstruction fromprojections which illustrate the performance of DROP.
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| 34. |
- Cohen, David, et al.
(författare)
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MultiSymplectic discretisation of wave map equations
- 2016
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 38:2, s. A953-A972
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Tidskriftsartikel (refereegranskat)abstract
- We present a new multisymplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multisymplectic discretization based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint, and can be seen as a generalization of the \smaller SHAKE algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions.
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| 35. |
- Correnty, Siobhán, et al.
(författare)
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Preconditioned Infinite GMRES for Parameterized Linear Systems
- 2024
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 46:2, s. S120-S141
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Tidskriftsartikel (refereegranskat)abstract
- We are interested in obtaining solutions to parameterized linear systems of the form A(mu)x(mu) = b for many values of the parameter mu. Here A(mu) is large, sparse, and nonsingular with a nonlinear, analytic dependence on mu. Our approach approximates the solution to a linearized system in a flexible GMRES setting [Y. Saad, SIAM J. Sci. Comput., 14 (1993), pp. 461-469], where the linearization is based on a companion matrix similar to the operator in the infinite Arnoldi method [E. Jarlebring, W. Michiels, and K. Meerbergen, Numer. Math., 122 (2012), pp. 169-195]. This novel approach applies the action of a preconditioner inexactly, providing performance improvement over the method infinite GMRES [Jarlebring and Correnty, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 1382-1405] without a loss of accuracy in general. The method returns a function (x) over tilde(mu) which is cheap to evaluate for different mu. We show that the error of our method is estimated based on the magnitude of the parameter mu, the inexactness of the preconditioning, and the spectrum of the companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The software used in the simulations is publicly available online, and all the experiments are reproducible.
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| 36. |
- Dahne, Joel, et al.
(författare)
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Computation of Tight Enclosures for Laplacian Eigenvalues
- 2020
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 42:5, s. A3210-A3232
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Tidskriftsartikel (refereegranskat)abstract
- Recently, there has been interest in high precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the three-dimensional Kreweras model that has been the object of previous efforts.
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| 37. |
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| 38. |
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| 39. |
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| 40. |
- Elden, Lars, et al.
(författare)
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Wavelet and Fourier methods for solving the sideways heat equation
- 2000
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 21:6, s. 2187-2205
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Tidskriftsartikel (refereegranskat)abstract
- We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.
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| 41. |
- Elfving, Tommy, et al.
(författare)
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Semiconvergence and Relaxation Parameters for Projected SIRT Algorithms
- 2012
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 34:4, s. A2000-A2017
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Tidskriftsartikel (refereegranskat)abstract
- We give a detailed study of the semiconvergence behavior of projected nonstationary simultaneous iterative reconstruction technique (SIRT) algorithms, including the projected Landweber algorithm. We also consider the use of a relaxation parameter strategy, proposed recently for the standard algorithms, for controlling the semiconvergence of the projected algorithms. We demonstrate the semiconvergence and the performance of our strategies by examples taken from tomographic imaging.
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| 42. |
- Elfving, Tommy, et al.
(författare)
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UNMATCHED PROJECTOR/BACKPROJECTOR PAIRS: PERTURBATION AND CONVERGENCE ANALYSIS
- 2018
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM PUBLICATIONS. - 1064-8275 .- 1095-7197. ; 40:1, s. A573-A591
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Tidskriftsartikel (refereegranskat)abstract
- In tomographic reconstruction problems it is not uncommon that there are errors in the implementation of the forward projector and/or the backprojector, and hence we encounter a so-called unmatched projektor/backprojector pair. Consequently, the matrices that represent the two projectors are not each others transpose. Surprisingly, the influence of such errors in algebraic iterative reconstruction methods has received little attention in the literature. The goal of this paper is to perform a rigorous first-order perturbation analysis of the minimization problems underlying the algebraic methods in order to understand the role played by the nonmatch of the matrices. We also study the convergence properties of linear stationary iterations based on unmatched matrix pairs, leading to insight into the behavior of some important row-and column-oriented algebraic iterative methods. We conclude with numerical examples that illustrate the perturbation and convergence results.
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| 43. |
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| 44. |
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| 45. |
- Engquist, Björn, et al.
(författare)
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Fast directional multilevel algorithms for oscillatory kernels
- 2007
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 29:4, s. 1710-1737
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Tidskriftsartikel (refereegranskat)abstract
- This paper introduces a new directional multilevel algorithm for solving N-body or N-point problems with highly oscillatory kernels. These systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non-uniformity of the particle distribution. We address the problem by first proving that the interaction between a ball of radius r and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of O(1/r) and is at a distance which is at least O(r(2)) away from from the ball. We then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated, low rank representation. Based on the resulting representations, our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency. The algorithm performs well on a large group of highly oscillatory kernels. Our algorithm is proved to have O(N log N) computational complexity for any given accuracy when the points are sampled from a two dimensional surface. We also provide numerical results to demonstrate these properties.
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| 46. |
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| 47. |
- Engquist, Björn, et al.
(författare)
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Fast wavelet based algorithms for linear evolution equations
- 1994
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 15:4, s. 755-775
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Tidskriftsartikel (refereegranskat)abstract
- The authors devise a class of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin [Comm. Pure Appl. Math., 44 (1991), pp. 141-1841, which they applied to general Calderon-Zygmund type integral operators. The authors apply a modification of their idea to linear hyperbolic and parabolic equations, with spatially varying coefficients. The complexity for hyperbolic equations in one dimension is reduced from O(N2) to O(N log3 N). There are somewhat better gains for parabolic equations in multidimensions
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| 48. |
- Eriksson, Kenneth, 1952, et al.
(författare)
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Explicit time-stepping for stiff ODES
- 2003
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 25:4, s. 1142-1157
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Tidskriftsartikel (refereegranskat)abstract
- We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of well-known stiff test problems.
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| 49. |
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| 50. |
- Fasi, Massimiliano, 1989-, et al.
(författare)
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Generating extreme-scale matrices with specified singular values or condition numbers
- 2021
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 43:1, s. A663-A684
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Tidskriftsartikel (refereegranskat)abstract
- A widely used form of test matrix is the randsvd matrix constructed as the product A = U Sigma V*, where U and V are random orthogonal or unitary matrices from the Haar distribution and Sigma is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution. The cost of forming an m x n randsvd matrix is m(3) + n(3) flops, which is prohibitively expensive at extreme scale; moreover, the randsvd construction requires a significant amount of communication, making it unsuitable for distributed memory environments. By dropping the requirement that U and V be Haar distributed and that both be random, we derive new algorithms for forming A that have cost linear in the number of matrix elements and require a low amount of communication and synchronization. We specialize these algorithms to generating matrices with a specified 2-norm condition number. Numerical experiments show that the algorithms have excellent efficiency and scalability.
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