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| 2. |
- Altmann, Robert, et al.
(författare)
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The J-method for the Gross-Pitaevskii eigenvalue problem
- 2021
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 148:3, s. 575-610
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Tidskriftsartikel (refereegranskat)abstract
- This paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross-Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the J-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the J-method's unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.
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| 3. |
- Axelsson, Owe, et al.
(författare)
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Krylov improvements of the Uzawa method for Stokes type operator matrices
- 2021
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 148:3, s. 611-631
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Tidskriftsartikel (refereegranskat)abstract
- The paper is devoted to Krylov type modifications of the Uzawa method on the operator level for the Stokes problem in order to accelerate convergence. First block preconditioners and their effect on convergence are studied. Then it is shown that a Krylov-Uzawa iteration produces superlinear convergence on smooth domains, and estimation is given on its speed.
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| 4. |
- Axelsson, Owe, et al.
(författare)
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Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks
- 2020
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 146:2, s. 335-368
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Tidskriftsartikel (refereegranskat)abstract
- Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.
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| 5. |
- Babuska, I., et al.
(författare)
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Worst case scenario analysis for elliptic problems with uncertainty
- 2005
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 101:2, s. 185-219
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Tidskriftsartikel (refereegranskat)abstract
- This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology.
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| 6. |
- Bensow, Rickard, 1972, et al.
(författare)
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Discontinuous Least-Squares finite element method for the Div-Curl problem
- 2005
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 101:4, s. 601-617
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given data (divergence and curl of the solution) and obtain convergence also on nonconvex domains. In general, this is not possible in the least-squares method with standard continuous elements. We show that our method is stable, derive a priori error estimates, and present numerical examples illustrating the method. © Springer-Verlag 2005.
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| 7. |
- Berman, Robert, 1976
(författare)
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The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Ampere equations
- 2020
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 145, s. 771-836
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Tidskriftsartikel (refereegranskat)abstract
- We show that the discrete Sinkhorn algorithm-as applied in the setting of Optimal Transport on a compact manifold-converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.
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| 8. |
- Betancourt, Fernando, et al.
(författare)
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A random sampling method for a family of Temple-class systems of conservation laws
- 2018
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 138:1, s. 37-73
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Tidskriftsartikel (refereegranskat)abstract
- The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear (Formula presented.) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.
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| 9. |
- Burman, Erik, et al.
(författare)
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A cut finite element method for a model of pressure in fractured media
- 2020
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 146:4, s. 783-818
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Tidskriftsartikel (refereegranskat)abstract
- We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.
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| 10. |
- Burman, Erik, et al.
(författare)
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Cut finite element methods for coupled bulk–surface problems
- 2016
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 133:2, s. 203-231
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Tidskriftsartikel (refereegranskat)abstract
- We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and (Formula presented.) norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.
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