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).
An edge state is a time-harmonic mode of a conservative wave system, e.g. Schroedinger, Maxwell, which is propagating (plane-wave-like) parallel to and localized, transverse to a line-defect or ``edge”. Topologically protected edge states are edge states which are immune to local scattering impurities. First studied in the context of the quantum Hall effect, protected edge states have attracted great interest recently due to their role in the field of topological insulators. Such states are potential vehicles for robust energy-transfer in the presence of strong localized defects and random imperfections. They are therefore considered ideal for use in nano-scale devices. The theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. After an introduction to the spectral properties of continuous honeycomb structures and their novel properties such as Dirac points, we introduce a rich family of continuum PDE models and discuss regimes (``phases") where topologically protected edge states exist along a ``zig-zag edge", and regimes where edge states may exist but are not protected. These results follow from a general theorem on the bifurcation of edge states from Dirac points of the background honeycomb structure. This bifurcation is seeded by the zero mode of an effective Dirac equation. The key to applying the general theorem is the verification of a spectral no-fold condition along the zig-zag edge. This is joint work with C.L. Fefferman and J.P. Lee-Thorp.
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.
The Airy beam (AiB) has attracted a lot of attention recently because of its intriguing features. We have previously provided a cogent physical explanation for these properties by showing that the AiB is, in fact, a caustic of rays emanating from the tail of the Airy function aperture distribution. We have also introduced a class of ultra wide band (UWB) Airy pulsed beams (AiPB), where a key step has been the use of a proper frequency scaling of the initial aperture field that ensures that all the frequency components propagate along the same curved trajectory so that the wavepacket of the AiPB does not disperse. An exact closed form solution for the AiPB has been derived using the spectral theory of transients (STT) which is an extension of the well know Cagniard deHoop (CdH) method. In this paper discuss the properties of the AiB and AiPB, and then use the present setting of the AiPB to discuss the basic principles of the CdH and STT techniques and to demonstrate how the STT circumvents the inherent limitations of the CdH method.
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.)
Recently, kinetic-type models have raised a lot of interest in multiscale modelling of collective motion and dispersal evolution. For instance, kinetic models are a very good option for modelling concentration waves of chemotactic bacteria in a micro-channel. Another example is the propagation of some invasive species with a high heterogeneity in dispersal capability among individuals. A minimal reaction-diffusion model, very similar to a kinetic equation, has recently been proposed by Benichou et al. I will present some recent progresses about the existence (and non-existence) of travelling waves for these two analogous models.
It is well-known that the asymptotic complexity of matrix-matrix product and matrix inversion is given by the rank of a 3-tensor, recently shown to be at most O(n^2.3728639) by Le Gall. This approach is attractive as a rank decomposition of that 3-tensor gives an explicit algorithm that is guaranteed to be fastest possible and its tensor nuclear norm (according to Grothendieck and Schatten, not the mean of matrix nuclear norms obtained from flattening the tensor into matrices) quantifies the optimal numerical stability. There is also an alternative approach due to Cohn and Umans that relies on embedding matrices into group algebras. We will see that the tensor decomposition and group algebra approaches, when combined, allow one to systematically discover fast(est) algorithms. We will determine the exact (as opposed to asymptotic) tensor ranks, and correspondingly the fastest algorithms, for products of Circulant, Toeplitz, Hankel, and other structured matrices. This is joint work with Ke Ye (Chicago).
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