| 1. |
- Adriani, Andrea, et al.
(författare)
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Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II : Numerical applications
- 2024
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 437
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Tidskriftsartikel (refereegranskat)abstract
- In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Ω ⊂ Rd , d ≥ 1. When Ω = [0, 1], such graphs include the standard Toeplitz graphs and, for Ω = [0,1]d, the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.
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| 2. |
- Arjmand, D., et al.
(författare)
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Efficient low rank approximations for parabolic control problems with unknown heat source
- 2024
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier B.V.. - 0377-0427 .- 1879-1778. ; 450
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Tidskriftsartikel (refereegranskat)abstract
- An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions.
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| 3. |
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| 4. |
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| 5. |
- Engström, Christian, et al.
(författare)
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Higher Order Composite DG approximations of Gross–Pitaevskii ground state : benchmark results and experiments
- 2022
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 400
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Tidskriftsartikel (refereegranskat)abstract
- Discontinuous Galerkin composite finite element methods (DGCFEM) are designed totackle approximation problems on complicated domains. Partial differential equationsposed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In thispaper, an optical lattice will be used to illustrate the performance of the approximationalgorithm for the ground state computation of a Gross–Pitaevskii equation, which isan eigenvalue problem with eigenvector nonlinearity. We will adapt the convergenceresults of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penaltyhp-adaptive algorithm against the performance of the hp-DGCFEM.
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| 6. |
- Engström, Christian, et al.
(författare)
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Numerical solution of distributed-order time-fractional diffusion-wave equations using Laplace transforms
- 2023
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 425
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider the numerical inverse Laplace transform for distributed order time-fractional equations, where a discontinuous Galerkin scheme is used to discretize the problem in space. The success of Talbot’s approach for the computation of the inverse Laplace transform depends critically on the problem’s spectral properties and we present a method to numerically enclose the spectrum and compute resolvent estimates independent of the problem size. The new results are applied to time-fractional wave and diffusion-wave equations of distributed order.
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| 7. |
- Giani, Stefano, et al.
(författare)
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khp-adaptive spectral projection based discontinuous Galerkin method for the numerical solution of wave equations with memory
- 2023
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 429
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (-adaptivity), but it is also applied to the size of the computed spectrum (-adaptivity). The meshes are refined using a residual based error estimator, while the size of the computed spectrum is adapted using the norm of the error of the projected data. We show that the approach can be very efficient and accurate.
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| 8. |
- Hanke, Michael, et al.
(författare)
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A reliable direct numerical treatment of differential–algebraic equations by overdetermined collocation : An operator approach
- 2021
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 387
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Tidskriftsartikel (refereegranskat)abstract
- Recently reported experiments and theoretical contributions concerning overdetermined polynomial collocation applied to higher-index differential–algebraic equations give rise to the conjecture that next to the existing derivative-array based methods there is further potential toward a reliable direct numerical treatment of DAEs. By analyzing first-order differential–algebraic operators and their special approximations in detail, we contribute to justify the overdetermined polynomial collocation applied to first-order higher-index differential–algebraic equations and fill the hitherto existing gap between the theoretical convergence results and its practical realization. Moreover, we shortly touch related questions for higher-order DAEs. We discuss several practical aspects of higher-order differential–algebraic operators and the associated equations which may be important for the application of collocation methods.
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| 9. |
- Hanke, Michael, et al.
(författare)
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Convergence analysis of least-squares collocation methods for nonlinear higher-index differential–algebraic equations
- 2021
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 387
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Tidskriftsartikel (refereegranskat)abstract
- We approach a direct numerical treatment of nonlinear higher-index differential–algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard collocation methods for regular ordinary differential equations and the numerical experiments show promising results. Nevertheless, the theoretical basic concept turns out to be considerably challenging. So far, quite recently, convergence proofs have been published for linear problems. In the present paper we come up with a first basic qualitative convergence result for nonlinear problems.
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| 10. |
- Korotov, Sergey, et al.
(författare)
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Divergence-free finite element spaces for stress tensors
- 2024
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Ingår i: Journal of Computational and Applied Mathematics. - 0377-0427 .- 1879-1778. ; 438, s. 115537-115537
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Tidskriftsartikel (refereegranskat)abstract
- We construct finite element spaces of symmetric stress tensors that are exactly divergence-free. Moreover, their basis functions can be chosen so that they have small supports. These properties are highly desired in a number of important applications. Approximation properties of finite element spaces of divergence-free tensor functions are derived from properties of C1 finite elements.
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