Wednesdays at 4:15 PM in 380C.
This talk is motivated by a recent experiment in optics called ghost imaging. In ghost imaging an image of an object is produced by correlating the intensities measured by two detectors, a high-resolution detector that does not view the object and a low-resolution (one-pixel) detector that does view the object. We analyze this imaging method when the medium through which the waves propagate is random. A stochastic multi-scale approach allows for the computation of second and fourth-order moments of the wave fields in the paraxial regime. We can then discuss the role of the partial coherence of the source and we study how scattering affects the resolution properties of the ghost imaging function.
In a famous paper of the 70s, Pierre Gilles de Gennes called "the ant in the labyrinth" the random walker on percolation clusters. It is probably the central example of the possibly anomalous behavior of diffusion (and transport). Rigorous mathematical progress has been initially rather slow, but recent progress has been steady (due mainly to Martin Barlow, Takashi Kumagai, Remco Van der Hofstadt, Gady Kozma, Assaf Nachmias among others). We will survey some of the recent results on this central topic of statistical mechanics in disordered media, and very recent work with Alexander Fribergh and Manuel Cabezas.
Operator splitting schemes organize the monotone operators, such as subgradient, projection, and proximal operators, to form first-order algorithms for optimization. They break a complicated problem consisting of multiple operators, cost functions, and constraints into simple steps. The resulting algorithms are short, easy to code, and often have (nearly) state-of-the-art performance. They have found applications in widely many areas. Given optimization problems, this talk uses simple diagrams to illustrate how a number of old and new first-order algorithms can be naturally derived. They include some distributed and decentralized algorithms. Assisted with the diagrams, we also deduce their basic convergence properties and in some cases the rates of convergence.
In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase. This means that we need to solve quadratic equations — this is notoriously NP hard — as opposed to linear ones. The focus of the talk is on a novel non-convex algorithms, which is provably exact for solving such problems. This algorithm, dubbed the Wirtinger flow algorithm, finds the solution to randomized quadratic systems from a number of equations (samples) and flops that are both nearly optimal. A a high level, the algorithm can be interpreted as a sort of stochastic gradient scheme, starting from a guess obtained by means of a spectral method. We demonstrate that the Wirtinger flow reconstruction degrades gracefully as the signal-to-noise ratio decreases. The empirical performance shall be demonstrated on phase retrieval problems, which is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes. This is joint work with Xiaodong Li and Mahdi Soltanolkotabi.
The motivation for this work is to lay the mathematical foundations of a formal theory introduced by P. Degond and C. Ringhofer in 2003 about quantum hydrodynamics. Their idea is to transpose to the quantum setting Levermore's closure strategy by entropy minimization for the derivation of hydrodynamical models. We will present two different types of results: the first ones concern the resolution of the so-called quantum moment problem, which is the first brick of the theory and a transposition to operators of the classical moment problem for measures. The second ones concern the dynamics of quantum states and their convergence to statistical equilibria. This is jointwork with F. Mehats (University of Rennes, France).
Abstract: We develop a scattering theory for a class of eternal solutions of the Boltzmann equation posed over all space. In three spatial dimensions each of these solutions has thirteen conserved quantities. The Boltzmann entropy has a unique minimizer with the same thirteen conserved values. This minimizer is a local Maxwellian that is also a global solution of the Boltzmann equation --- a so-called global Maxwellian. We show that each of our eternal solutions has a streaming asymptotic state as time goes to minus or plus infinity. However it does not converge to the associated global Maxwellian as time goes to infinity unless it is that global Maxwellian. The Boltzmann entropy decreases as time increases, but does not decrease to its minimum as time goes to infinity. Said another way, the final step in the traditional argument for the heat death of the universe is not valid.
We study on-diagonal heat kernel estimates and exit time estimates for continuous time random walks (CTRWs) among i.i.d. random conductances with a power-law tail near zero. For two types of natural CTRWs, we give optimal exponents of the tail such that the behaviors are ‘standard’ (i.e. similar to the random walk on the Euclidean space) above the exponents. We then establish the local CLT for the CTRWs. We will also compare our results to the recent results by Andres-Deuschel-Slowik. This talk is a joint work with O. Boukhadra (Constantine) and P. Mathieu (Marseille).
Gunnar Carlsson has made the provocative suggestion that topology can be a useful tool in the understanding of high dimensional and noisy data sets. In this talk, I’d like to explain several developments in topology that can be viewed as being directly inspired by this quest. The basic problems we will discuss is the inference topological structure, or failing that, the computation of topological invariants from (perhaps noisy) samples. We will study the theoretical limits of structure finding (joint with Dranishnikov and Ferry), the (sample, Kolmogorov, and computational) complexity of some inference problems -- suggesting a theory of “sampleable invariants”. It turns out that these are connected (by ideas of Benjamini and Schramm, Abert et al and others) to ideas that occur in the study of the physical properties of quasicrystals and other disordered solids. (Some of this last part is based on joint discussions with Belissard and Ulgen-Yildirim.)
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