Path Following by SVD

Dieci, Luca; Gasparo, M. G.; Papini, Alessandra

doi:10.1007/11758549_92

Cited by 9 publications

(4 citation statements)

References 15 publications

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“…The smooth singular value decomposition would be a useful tool for this project, see [22]. It would also be valuable to try to prove existence of cusp bifurcations in the two dimensional manifold of equilibria of the 2D Cahn-Hilliard model as numerically suggested in [4].…”

Section: Resultsmentioning

confidence: 99%

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Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.

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Section: Resultsmentioning

confidence: 99%

“…To ease the computation of the bound Z = (Z k ) k≥−2 satisfying (12), we consider the expansion (22), with the coefficients z ( ,i,j) k being given in Table 4 in Appendix B. Note that the components Z −2 and Z −1 correspond to the bounds associated to the first two components of F s .…”

Section: Two-dimensional Manifold Of Equilibria Of Cahn-hilliardmentioning

confidence: 99%

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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“…The real case tells us that we should expect a loss of rank already when m = n and A depends on one real parameter (after all, this is detected by the scalar relation det A = 0). This case is fairly understood and already adequately discussed in [2,3], and see also [4] for numerical methods able to detect and bypass the losses of rank of a smooth function A. The complex case is what we will consider in this work when m = n, which has the minimal possible codimension of 2 for a single loss of rank.…”

Section: Introductionmentioning

confidence: 89%

SVD algorithms to approximate spectra of dynamical systems

Dieci

Elia

2008

Mathematics and Computers in Simulation

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“…But being U inversely proportional to the distance between eigenvalues, it can yield prohibitively small values for h in proximity of veering zones, which are routinely encountered in the case of band matrices. In practice, within a veering zone, it is impossible to distinguish a pair of close eigenvalues; to overcome this critical situation, we implemented a strategy (see [6]) in which close eigenvalues are grouped into 2 × 2 blocks, and a smooth block-diagonal eigendecomposition is computed until all eigenvalues are well separated again. A key concern, then, will be how to recover the correct eigendecomposition (smoothly) after we exit the veering zone.…”

Section: Algorithm 21: Predictor-corrector Stepmentioning

confidence: 99%

Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions

2018

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In this work, we develop and implement new numerical methods to locate generic degeneracies (i.e., isolated parameters' values where the eigenvalues coalesce) of banded matrix valued functions. More precisely, our specific interest is in two classes of problems: (i) symmetric, banded, functions A(x) ∈ R n×n , smoothly depending on parameters x ∈ Ω ⊂ R 2 , and (ii) Hermitian, banded, functions A(x) ∈ C n×n , smoothly depending on parameters x ∈ Ω ⊂ R 3 .The computational task of detecting coalescing points of banded parameter dependent matrices is very delicate and challenging, and cannot be handled using existing eigenvalues' continuation approaches. For this reason, we present and justify new techniques that will enable continuing path of eigendecompositions and reliably decide whether or not eigenvalues coalesce, well beyond our ability to numerically distinguish close eigenvalues.As important motivation, and illustration, of our methods, we perform a computational study of the density of coalescing points for random ensembles of banded matrices depending on parameters. Relatively to random matrix models from truncated GOE and GUE ensembles, we will give computational evidence in support of power laws for coalescing points, expressed in terms of the size and bandwidth of the matrices.

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Path Following by SVD

Cited by 9 publications

References 15 publications

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

SVD algorithms to approximate spectra of dynamical systems

Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions

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