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Analysis and its Applications  | enregistrements trouvés : 61

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The most important works of the young Lagrange were two very learned memoirs on sound and its propagation. In a tour de force of mathematical analysis, he solved the relevant partial differential equations in a novel manner and he applied the solutions to a number of acoustic problems. Although Euler and d'Alembert may have been the only contemporaries who fully appreciated these memoirs, their contents anticipated much more of Fourier analysis than is usually believed. On the physical side, Lagrange properly explained the functioning of string and air-column instruments, although he did not accept harmonic analysis as we now understand it.
Lagrange - acoustics - propagation of sound - harmonic analysis - Fourier analysis - vibrating strings - organ pipes
The most important works of the young Lagrange were two very learned memoirs on sound and its propagation. In a tour de force of mathematical analysis, he solved the relevant partial differential equations in a novel manner and he applied the solutions to a number of acoustic problems. Although Euler and d'Alembert may have been the only contemporaries who fully appreciated these memoirs, their contents anticipated much more of Fourier analysis ...

01A50 ; 35-03 ; 40-03 ; 76-03

Retrieving an arbitrary signal from the magnitudes of its inner products with the elements of a frame is not possible in infinite dimensions. Under certain conditions, signals can be retrieved satisfactorily however.

42C15 ; 46C05 ; 94A12 ; 94A15 ; 94A20

This talk sketches connections between Whitney problems and e.g. the problem of deciding whether a given rational function on $\mathbb{R}^n$ belongs to $C^m$.

26Bxx ; 46E10 ; 58A20 ; 14Qxx

Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

15-XX ; 41-XX ; 42-XX ; 46-XX

The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, Poissonian, and multiplicative, and with several types of regularizers (TV, frame-based analysis, synthesis, or combinations thereof). I will then describe more recent work on using ADMM for other problems, namely blind deconvolution and image segmentation, as well as very recent work where ADMM is used with plug-in learned denoisers to achieve state-of-the-art results in class-specific image deconvolution. Finally, on the theoretical front, I will describe very recent work on tackling the infamous problem of how to adjust the penalty parameter of ADMM. The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, ...

65J22 ; 65K10 ; 65T60 ; 94A08

Post-edited  A universal hypercyclic representation
Glasner, Eli (Auteur de la Conférence) | CIRM (Editeur )

For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about the development of this idea and its applications as expounded in a subsequent work of Sophie Grivaux. For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about ...

37A15 ; 37A05 ; 37A25 ; 37A30 ; 47A16 ; 47A67 ; 47D03

I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions. I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close ...

28A80 ; 37A10 ; 03D99 ; 54H20

Post-edited  25+ years of wavelets for PDEs
Kunoth, Angela (Auteur de la Conférence) | CIRM (Editeur )

Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality and optimality of approximations and related concepts from approximation the- ory. We will see that wavelet bases can serve as a basic ingredient, both for the theory as well as for algorithmic realizations. Particularly for situations where solutions exhibit singularities, wavelet concepts enable adaptive appproximations for which convergence and optimal algorithmic complexity can be established. I will describe corresponding implementations based on biorthogonal spline-wavelets.
Moreover, wavelet-related concepts have triggered new developments for efficiently solving complex systems of PDEs, as they arise from optimization problems with PDEs.
Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality ...

65T60 ; 94A08 ; 65N12 ; 65N30 ; 49J20

Starting with a personal tribute to Victor Havin (1933-2015), I discuss a dozen achievements of Great Havin's Analysis Seminar, as well as some challenging still unsolved problems.
The Havin publications list is available in the PDF file at the bottom of the page.

30H10 ; 01A60 ; 01A70

In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine learning.
imaging - image processing - sparsity - convex optimization - inverse problem - super-resolution
In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine ...

62H35 ; 65D18 ; 94A08 ; 68U10 ; 90C31 ; 80M50 ; 47N10

Nigel Kalton played a prominent role in the development of a holomorphic functional calculus for unbounded sectorial operators. He showed, in particular, that such a calculus is highly unstable under perturbation: given an operator $D$ with a bounded functional calculus, fairly stringent conditions have to be imposed on a perturbation $B$ for $DB$ to also have a bounded functional calculus. Nigel, however, often mentioned that, while these results give a fairly complete picture of what is true at a pure operator theoretic level, more should be true for special classes of differential operators. In this talk, I will briefly review Nigel's general results before focusing on differential operators with perturbed coefficients acting on $L_p(\mathbb{R}^{n})$. I will present, in particular, recent joint work with $D$. Frey and A. McIntosh that demonstrates how stable the functional calculus is in this case. The emphasis will be on trying, as suggested by Nigel, to understand what makes differential operators so special from an operator theoretic point of view. Nigel Kalton played a prominent role in the development of a holomorphic functional calculus for unbounded sectorial operators. He showed, in particular, that such a calculus is highly unstable under perturbation: given an operator $D$ with a bounded functional calculus, fairly stringent conditions have to be imposed on a perturbation $B$ for $DB$ to also have a bounded functional calculus. Nigel, however, often mentioned that, while these ...

47F05 ; 47A60 ; 42B30 ; 42B37

Two important examples of the determinantal point processes associated with the Hilbert spaces of holomorphic functions are the Ginibre point process and the set of zeros of the Gaussian Analytic Functions on the unit disk. In this talk, I will talk such class of determinantal point processes in greater generality. The main topics concerned are the equivalence of the reduced Palm measures and the quasi-invariance of these point processes under certain natural group action of the group of compactly supported diffeomorphisms of the phase space. This talk is based partly on the joint works with Alexander I. Bufetov and partly on a more recent joint work with Alexander I. Bufetov and Shilei Fan. Two important examples of the determinantal point processes associated with the Hilbert spaces of holomorphic functions are the Ginibre point process and the set of zeros of the Gaussian Analytic Functions on the unit disk. In this talk, I will talk such class of determinantal point processes in greater generality. The main topics concerned are the equivalence of the reduced Palm measures and the quasi-invariance of these point processes under ...

60G55 ; 46E20 ; 30H20

I will give a survey of the operator theory that is currently evolving on Hardy spaces of Dirichlet series. We will consider recent results about multiplicative Hankel operators as introduced and studied by Helson and developments building on the Gordon-Hedenmalm theorem on bounded composition operators on the $H^2$ space of Dirichlet series.

47B35 ; 30B50 ; 30H10

I present a joint work with S. Popa and D. Shlyakhtenko introducing a cohomology theory for quasi-regular inclusions of von Neumann algebras. In particular, we define $L^2$-cohomology and $L^2$-Betti numbers for such inclusions. Applying this to the symmetric enveloping inclusion of a finite index subfactor, we get a cohomology theory and a definition of $L^2$-Betti numbers for finite index subfactors, as well as for arbitrary rigid $C^*$-tensor categories. For the inclusion of a Cartan subalgebra in a $II_1$ factor, we recover Gaboriau's $L^2$-Betti numbers for equivalence relations. I present a joint work with S. Popa and D. Shlyakhtenko introducing a cohomology theory for quasi-regular inclusions of von Neumann algebras. In particular, we define $L^2$-cohomology and $L^2$-Betti numbers for such inclusions. Applying this to the symmetric enveloping inclusion of a finite index subfactor, we get a cohomology theory and a definition of $L^2$-Betti numbers for finite index subfactors, as well as for arbitrary rigid $C^*$-tensor ...

46L37 ; 46L10

définition of the quotient norm - basic properties - existence of minimal liftings: von Neumann algebras - finite dimensional cases - non-uniqueness results - counter-examples: the unitary Fredholm group

We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their duality structure, we introduce two natural partial inner product spaces and formulate a number of open questions.

Keywords: continuous frames - semi-frames - frame duality - reproducing pairs - partial inner product spaces
We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their ...

42C15 ; 42C40 ; 46C50 ; 65T60

We combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on the fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

Keywords: generalized multiscale finite element method - nonlinear PDEs - heterogeneous porous media - discrete empirical interpolation - proper orthogonal decomposition
We combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply ...

35-XX ; 65-XX ; 86-XX

I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not.
In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then $\mathbb{Z}^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space, and one gets from here that there are quite natural nontrivial twisted sums $0\to Z_2\to\mathbb{Z}^{(4)}\to Z_2 \to0$ of $Z_2$ with itself. The twisted sum space $\mathbb{Z}^{(4)}$ does not embeds in, or is a quotient of, a twisted Hilbert space and does not contain $\ell_2$ complemented. We will also construct another nontrivial twisted sum of $Z_2$ with itself that contains $\ell_2$ complemented. These results have some connection with the nowadays called Kalton calculus [3, 4], and thus several recent advances [2] in this theory that combines twisted sums and interpolation theory will be shown.

Banach space - twisted sum - complex interpolation - Hilbert space
I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not.
In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then ...

46M18 ; 46B70 ; 46B20

The unitary extension principle (UEP) by Ron & Shen yields a convenient way of constructing tight wavelet frames in L2(R). Since its publication in 1997 several generalizations and reformulations have been obtained, and it has been proved that the UEP has important applications within image processing. In the talk we will present a recent extension of the UEP to the setting of generalized shift-invariant systems on R (or more generally, on any locally compact abelian group). For example, this generalization immediately leads to a discrete version of the UEP.
(The results are joint work with Say Song Goh).
The unitary extension principle (UEP) by Ron & Shen yields a convenient way of constructing tight wavelet frames in L2(R). Since its publication in 1997 several generalizations and reformulations have been obtained, and it has been proved that the UEP has important applications within image processing. In the talk we will present a recent extension of the UEP to the setting of generalized shift-invariant systems on R (or more generally, on any ...

42C15 ; 42C40 ; 65T60

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