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Monotone operators in Banach space and nonlinear partial differential equations
10 Dec 1996-
TL;DR: PDE examples by type linear problems as mentioned in this paper, including nonlinear stationary problems, nonlinear evolution problems, and nonlinear Cauchy problems, can be found in this paper.
Abstract: PDE examples by type Linear problems...An introduction Nonlinear stationary problems Nonlinear evolution problems Accretive operators and nonlinear Cauchy problems Appendix Bibliography Index.
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01 Feb 2003TL;DR: In this article, the Furstenberg-Zimmer structure theorem and host's theorem are derived from the Pinsker algebra, CPE and zero-entropy systems, and the relation between measure and topological entropy is discussed.
Abstract: Introduction General group actions: Topological dynamics Dynamical systems on Lebesgue spaces Ergodicity and mixing properties Invariant measures on topological systems Spectral theory Joinings Some applications of joinings Quasifactors Isometric and weakly mixing extensions The Furstenberg-Zimmer structure theorem Host's theorem Simple systems and their self-joinings Kazhdan's property and the geometry of $M_{\Gamma}(\mathbf{X})$ Entropy theory for $\mathbb{Z}$-systems: Entropy Symbolic representations Constructions The relation between measure and topological entropy The Pinsker algebra, CPE and zero entropy systems Entropy pairs Krieger's and Ornstein's theorems Prerequisite background and theorems Bibliography Index of symbols Index of terms.
765 citations
TL;DR: In the context of canonical sparse estimation problems, it is proved uniform superiority of this method over the minimum l1 solution in that, 1) it can never do worse when implemented with reweighted l1, and 2) for any dictionary and sparsity profile, there will always exist cases where it does better.
Abstract: A variety of practical methods have recently been introduced for finding maximally sparse representations from overcomplete dictionaries, a central computational task in compressive sensing applications as well as numerous others. Many of the underlying algorithms rely on iterative reweighting schemes that produce more focal estimates as optimization progresses. Two such variants are iterative reweighted l1 and l2 minimization; however, some properties related to convergence and sparse estimation, as well as possible generalizations, are still not clearly understood or fully exploited. In this paper, we make the distinction between separable and non-separable iterative reweighting algorithms. The vast majority of existing methods are separable, meaning the weighting of a given coefficient at each iteration is only a function of that individual coefficient from the previous iteration (as opposed to dependency on all coefficients). We examine two such separable reweighting schemes: an l2 method from Chartrand and Yin (2008) and an l1 approach from Cande's (2008), elaborating on convergence results and explicit connections between them. We then explore an interesting non-separable alternative that can be implemented via either l2 or l1 reweighting and maintains several desirable properties relevant to sparse recovery despite a highly non-convex underlying cost function. For example, in the context of canonical sparse estimation problems, we prove uniform superiority of this method over the minimum l1 solution in that, 1) it can never do worse when implemented with reweighted l1, and 2) for any dictionary and sparsity profile, there will always exist cases where it does better. These results challenge the prevailing reliance on strictly convex (and separable) penalty functions for finding sparse solutions. We then derive a new non-separable variant with similar properties that exhibits further performance improvements in empirical tests. Finally, we address natural extensions to group sparsity problems and non-negative sparse coding.
426 citations
30 Nov 2009
TL;DR: Fornasier and Romlau as mentioned in this paper discuss various theoretical and practical topics related to total variation-based image reconstruction, focusing first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the overall variation in a finite-differences setting.
Abstract: These are the lecture notes of a course taught in Linz in Sept., 2009, at the school "summer school on sparsity", organized by Massimo Fornasier and Ronny Romlau. They address various theoretical and practical topics related to Total Variation-based image reconstruction. They focu first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finite-differences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.
413 citations
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08 Aug 2008TL;DR: In this paper, the existence of compact global attractors for evolutions of the second order in time was studied for the Semilinear wave equation with a nonlinear dissipation Von Karman evolutions and other models from continuum mechanics.
Abstract: Introduction Abstract results on global attractors Existence of compact global attractors for evolutions of the second order in time Properties of global attractors for evolutions of the second order in time Semilinear wave equation with a nonlinear dissipation Von Karman evolutions with a nonlinear dissipation Other models from continuum mechanics Bibliography Index
388 citations
Book•
01 Jan 2002
TL;DR: In this article, the Kawasaki Riemann-Roch formula was used to prove the Hamiltonian cobordism invariance of the index of a transversally elliptic operator.
Abstract: Introduction Part 1. Cobordism: Hamiltonian cobordism Abstract moment maps The linearization theorem Reduction and applications Part 2. Quantization: Geometric quantization The quantum version of the linearization theorem Quantization commutes with reduction Part 3. Appendices: Signs and normalization conventions Proper actions of Lie groups Equivariant cohomology Stable complex and Spin$^{\mathrm{c}}$structures Assignments and abstract moment maps Assignment cohomology Non-degenerate abstract moment maps Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization The Kawasaki Riemann-Roch formula Cobordism invariance of the index of a transversally elliptic operator Bibliography Index.
380 citations