| 1. |
|
|
| 2. |
- Andreotti, Eleonora, 1988, et al.
(författare)
-
Measuring stability of spectral clustering
- 2019
-
Ingår i: Linear Algebra and Its Applications. - : Elsevier BV. - 0024-3795. ; 610, s. 673-697
-
Tidskriftsartikel (refereegranskat)abstract
- As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The k-th spectral gap is characterized here as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing kth spectral gap. As a more appropriate measure of stability, the structured distance to ambiguity of the k-clustering is introduced as the minimal distance of the Laplacian to Laplacians of the same graph with weights that are perturbed such that the k-th spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number k of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.
|
|
| 3. |
- Cohen, David, et al.
(författare)
-
Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations
- 2008
-
Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 110:2, s. 113-143
-
Tidskriftsartikel (refereegranskat)abstract
- For classes of symplectic and symmetric time-stepping methods- trigonometric integrators and the Stormer-Verlet or leapfrog method-applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.
|
|
| 4. |
- Cohen, David, et al.
(författare)
-
Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions
- 2015
-
Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 55:3, s. 705-732
-
Tidskriftsartikel (refereegranskat)abstract
- For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies.
|
|
| 5. |
- Cohen, David, et al.
(författare)
-
Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions
- 2008
-
Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 187:2, s. 341-368
-
Tidskriftsartikel (refereegranskat)abstract
- A modulated Fourier expansion in time is used to show long-time near-conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.
|
|
| 6. |
- Cohen, David, et al.
(författare)
-
Modulated Fourier expansions of highly oscillatory differential equations
- 2003
-
Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 3:4, s. 327-345
-
Tidskriftsartikel (refereegranskat)abstract
- Modulated Fourier expansions are developed as a tool for gaining insight into the long-time behavior of Hamiltonian systems with highly oscillatory solutions. Particle systems of Fermi-Pasta-Ulam type with light and heavy masses are considered as an example. It is shown that the harmonic energy of the highly oscillatory part is nearly conserved over times that are exponentially long in the high frequency. Unlike previous approaches to such problems, the technique used here does not employ nonlinear coordinate transforms and can therefore be extended to the analysis of numerical discrelizations.
|
|
| 7. |
- Cohen, David, et al.
(författare)
-
Numerical energy conservation for multi-frequency oscillatory differential equations
- 2005
-
Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 45:2, s. 287-305
-
Tidskriftsartikel (refereegranskat)abstract
- The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the Stormer-Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included.
|
|
| 8. |
- Cohen, David, 1977, et al.
(författare)
-
Numerical integrators for highly oscillatory Hamiltonian systems: a review
- 2006
-
Ingår i: Analysis, Modeling and Simulation of Multiscale Problems. - : Springer Berlin Heidelberg. - 3540356568 - 9783540356561 ; , s. 553-576
-
Konferensbidrag (refereegranskat)abstract
- Numerical methods for oscillatory, multi-scale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time- or state-dependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail.
|
|
| 9. |
|
|
| 10. |
|
|
