Many of the high tech devices we use today possess components that are made from polycrystalline materials. From interconnects in microprocessors with higher conductivity to materials that resist fracture, the engineering of these microstructures is crucial to our technological advancement. These macroscopic properties are affected by mesoscale characteristics like grain size, texture, and arrangement of the grains in their topological network of boundaries. A crucial problem in understanding this area of materials science is the modeling and simulation of the evolution of these networks. Simulation of these large, metastable networks involves solution of an average of 50,000 nonlinear partial differential equations. This approach to microstructural evolution can be replaced at a larger scale by a system of master equations for the evolution of grain size distribution functions. An earlier known phenomenological system of this type can be found in the paper of Fradkov et al [1]. Modeling of these equations, however, must be based on accommodating critical events into the deterministic rules for grain growth in order to obtain network properties. Such a model should be built empirically, from large scale simulation. And it must reproduce statistics of the simulation.These properties are the foundation of the master equations proposed by the authors BKLT [2]. In their work, they propose a system of equations that is internally consistent, and suggest that the reassignment of sides from deleted grains is random in nature.In the work we present in this seminar, we will show that these master equations do in fact have a consistent probabilistic interpretation. It follows that the empirical equations posed by BKLT have a life of their own as the 'upscale' equations of a stochastic process that describes the coarsening of the grain network . Finally, we show that the existence theory for solutions to these highly nonlinear integro-partial differential equations is symbiotically coupled to an existence theory for this stochastic process.
- [1] V.E. Fradkov and D. Udler, Two-dimensional normal grain growth: topological aspects, Adv. Physics 43, 739 (1994)
- [2] K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta'asan, Remarks on a Multiscale Approach to Grain Growth in Polycrystals, Progress in Nonlinear Differential Equations and Their Applications, 68 (2006), pp.1-11.
We discuss a class of complex-valued functions in the plane whose complex partial derivatives can be multiplied by functions of unit modulus such that these rotated derivatives then are the end results of two martingales that are martingale transforms of each other. The motivation is that whenever this can be done, known inequalities for martingale transforms due to Burkholder then imply inequalities for various quantities associated with the partial derivatives, including their p-norms. The question originally arose from an attempt to verify a still open conjecture concerning the p-norm of the Beurling-Ahlfors transformation.Standard approximation results show that to get such inequalities, it suffices to consider continuous piecewise affine mappings of compact support. The problem becomes how to construct martingales by "combining" affine maps into new ones. We present a proposed algorithm for doing this and describe cases where this algorithm can be carried out to the end to achieve the desired result. The question of whether the algorithm always works remains open. An affirmative answer to this would imply affirmative answers to a number of open conjectures.
In this talk, we give precise rates of convergence in the strong invariance principle for the partial sums associated to a class of weakly dependent sequences included strong mixing sequences. We shall give the ideas of proofs which are based on an explicit construction of the approximating sequence of iid Gaussian random variables with the help of conditional quantile transformations. This talk is based on a joint work with E. Rio.
The classical parabolic nonlinear evolution equations such as Burgers, and Navier-Stokes equations have, in a sense, balanced influence of the quadratic nonlinearity and the Brownian Motion, local diffusion over the long-time shape of the solutions. This balance can be destroyed if one considers other nonlinearities and nonlocal diffusion, including those related to the Levy processes which feature long-range jumps. The issue will be discussed in the case of the so-called fractional conservation laws. Interacting particle systems approximations for such evolution problems, with numerical implications, will also be presented.
The QR-algorithm is a standard method of diagonalizing a matrix. It has an intimate relationship with the Toda lattice equations (a classic example of a coupled system of nonlinear differential equations). Integer evaluations of a Toda lattice evolution with a non-standard choice of Hamiltonian give the iterates of the QR-algorithm.The SR-algorithm is a generalization of the QR-algorithm which computes the eigenvalues of a symplectic matrix while preserving the symplectic form of the matrix. We will show that this algorithm is equivalent to a member of the Pfaff lattice hierarchy. The Pfaff lattice hierarchy was introduced by Adler and van Moerbeke to describe the partition functions of GOE and GSE random matrices. This lattice is presented as a system of evolution equations on a matrix variable and is based on the SR-factorization. We will show that the even members of Pfaff lattice hierarchy give alternative algorithms for diagonalizing a symplectic matrix which preserve the symplectic form of the matrix.
This is joint work with Yuji Kodama.
The small dispersion or semiclassical limit of the sine-Gordon equation models magnetic flux propagation in long Josephson junctions. In principle, any well-posed initial-value problem for the sine-Gordon equation can be solved using the inverse-scattering transform. However, in practice the solution can be computed explicitly for only a handful of initial conditions. We first present a recent spectral confinement result, giving conditions when the spectrum of the system must lie in certain regions. Secondly, we discuss ongoing work on the asymptotic behavior of general soliton ensembles in the semiclassical limit. Parts of this work are joint with Peter Miller.
Free probability is maybe the most popular part of non-commutative probability theory. This young field, at the intersection of probability theory, operator algebra, complex analysis and combinatorics is also connected to other sciences, from physics to psychology and engineering. But freeness is not the only notion of non-commutative independence. Under certain universality assumptions one gets 2,3, or 4 types of independence and various interpolation models. In this talk I will give an introduction to non-commutative probability , some of its specific techniques and problems it has tackled I will emphasize the combinatorial part, where root systems, partitions, crossings, embracings, planar and rooted trees aggregate in an effective machinery.
I will talk about statistical inference of trends in mean non-stationary models, and mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The Simultaneous confidence bands are useful for model specification problems in nonlinear time series. The results are applied to environmental and financial time series.
I will discuss estimation of covariance matrices of stationary processes. Under a short-range dependence condition for a wide class of nonlinear processes, I will show that the banded covariance matrix estimates converge in operator norm to the true covariance matrix with explicit rates of convergence. I will also consider the consistency of the estimate of the inverse covariance matrix. These results are applied to a prediction problem, and error bounds for the finite predictor coefficients are obtained. The work is joint with Mohsen Pourahmadi of TAMU.
Harnack inequalities describe, in quantitative ways, behavior of solutions to partial differential equations. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the theory of harmonic functions and, more generally, partial differential equations. The purpose of my talk is to explain major ideas behind Harnack inequalities in different cases. The emphasis is in the qualitative behavior of solutions. Moreover, I will introduce a few consequences that may be deduced from Harnack inequalities, motivating the study of them. They are both deep and powerful. The understanding of Harnack inequalities for solutions to a general class of nonlinear parabolic PDEs has risen significantly recently. I will explain the history of the problem, reviewing fundamental works of De Giorgi and Moser in the linear case, and then introducing new results for a general class of equations with degenerate structure. I will also very briefly introduce main techniques to prove Harnack inequalities in different cases.
Abstract: I will present an asymptotic expansion for probabilities of moderate deviations for iid random variables and for stationary processes. The sharpness of moment conditions will be discussed. The dependence measures are easily verifiable (cf W. B. Wu (2005), Nonlinear system theory: Another look at dependence. Proc Natl Acad Sci USA. 102)
I will discuss Fourier and wavelet transforms of stationary, causal processes. Under mild conditions, Fourier transforms are shown to be asymptotically independent complex Gaussian at different frequencies. To this end, I will apply Carleson's Theorem, a very deep result in harmonic analysis.
It is shown that the McKay (1993) and Jones, McKay and Hu (1994) modifications of Abramson's (1982) variable bandwidth kernel density estimator satisfies optimal asymptotic properties for estimating densities with four or six uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from zero.
Abstract: I will present an asymptotic expansion for probabilities of moderate deviations for iid random The sharpness of moment conditions will be discussed. The dependence measures are easily verifiable (cf W. B. Wu (2005), Nonlinear system theory: Another look at dependence. Proc Natl Acad Sci USA. 102)
The Tracy-Widom functions describe the limiting distribution of a variety of statistical quantities, including the largest eigenvalue of a random matrix drawn from the Gaussian orthogonal, symplectic, or unitary ensembles (GOE, GSE, or GUE), the longest increasing subsequence of a random permutation, and the outermost particle in a sea of non-intersecting Brownian particles. We obtain new formulas for the Tracy-Widom functions in terms of integrals of Painleve functions. Using these new formulas we find the complete asymptotic expansion of the left-hand tail of the GOE and GSE Tracy-Widom functions for the first time, as well as a second proof of the recently obtained result for the GUE case. We conclude by discussing progress on a new family of "incomplete" Tracy-Widom distributions corresponding to the largest observed eigenvalue if each eigenvalue has a fixed probability of being observed. This is joint work with Jinho Baik and Jeffery DiFranco.
We introduce a measure on strict plane partitions that is an analog of the uniform measure on plane partitions. We describe this measure in terms of a Pfaffian point process and compute its bulk limit when partitions become large.The above measure is a special case of the shifted Schur process, which generalizes the shifted Schur measure introduced by Tracy and Widom and is an analog of the Schur process introduced by Okounkov and Reshetikhin. We use the Fock space formalism to prove that the shifted Schur process is a Pfaffian point process and calculate its correlation kernel.
We also obtain a generalization of MacMahon's formula for the generating function of plane partitions. We give a 2-parameter generalization related to Macdonald's symmetric functions. The formula is especially simple in the Hall-Littlewood case.
Limit theorems for a hyperbolic or partially hyperbolic dynamical system are usually proved by means of a clever partitioning the phase space of the system. This should lead to the creation of a family of sigma-filelds with customary mixing properties when the machinery of weak dependence is applicable. We are going to consider an alternative approach when no cutting of the phase space is performed. Instead, by means of probabilistic tools an extension of the original dynamical system is constructed supplied with a family of sigma-field enjoying nice mixing properties. We will discuss advantages and drawbacks of this approach and consider ergodic toral automorphisms as examples where this approach goes smoothly and leads to new conclusions.
An interesting estimation problem, arising in many dynamical systems, is that of filtering; namely, one wishes to estimate a trajectory of a signal process (which is not observed) from a given path of an observation process, where the latter is a nonlinear functional of the signal plus noise.In the classical mathematical framework, the stochastic processes are parameterized by a single parameter (interpreted as ``time''), the observation noise is a martingale (say, a Brownian motion), and the best mean-square estimate of the signal, called the optimal filter, has a number of useful representations and satisfies the well-known Kushner-FKK and Duncan-Mortensen-Zakai stochastic partial differential equations.
However, there are many applications, arising, for example, in connection with denoising and filtering of images and video-streams, where the parameter space has to be multidimensional. Another level of difficulty is added if the observation noise has a long-memory structure, which leads to nonstandard filtering evolution equations. Each of the two features (multidimensional parameter space and long-memory observation noise) does not permit the use of the classical theory of filtering and the combination of the two has not been previously explored in mathematical literature on stochastic filtering.
This talk focuses on nonlinear filtering of a signal in the presence of long-memory fractional Gaussian noise. We will start by introducing first the evolution equations and integral representations of the optimal filter in the one-parameter case, when the noise driving the observation is represented by a fractional Brownian motion. Next, using fractional calculus and multiparameter martingale theory, the case of spatial nonlinear filtering of a random field observed in the presence of a persistent fractional Brownian sheet will be explored.
Abstract: Permanents of random matrices with iid entries converge to lognormal or normal variables. For generalized permanents the limit is described in terms of the multiple Ito-Wiener integral of elements of Hoeffding decomposition. This theory parallels the one which has been developed for U-statistics. Applications for counting problems for perfect matchings in bi-partite graphs include for instance counting monochromatic matchings or counting matchings with a given color structure of edges. The talk is based on Ch. 5 of a little book "Symmetric Functionals on Random Matrices and Random Matching Problems" (Springer, 2008) co-authored by Grzegorz Rempala and myself.
This talk will survey some distances between two probability laws and the relations between them. They are used to obtain various rates of convergence in the central limit theorem for sums of independent and dependent random variables.
The logarithmic Sobolev inequality, discovered by L. Gross, is a weak form of a Sobolev inequality: it gives control of a function through average information about its derivatives. It has become a ubiquitous tool in global analysis and probability theory, with important applications in stochastic analysis, large deviations, spectral theory, non-linear PDE, geometric analysis, non-commutative geometry, and more. As a stunning example, the logarithmic Sobolev inequality inspired Perelman's proof of the Poincar\'e conjecture.In this talk, I will give an introduction to the area of logarithmic Sobolev inequalities. I will then discuss new insights on such inequalities in the context of regular function spaces, such as holomorphic and subharmonic functions. In recent work with P. Graczyk, J. Loeb, and T. Zak, we have discovered a new and very general logarithmic Sobolev inequality for logarithmically-subharmonic functions. I will discuss these results and give clues about their significance.
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Stochastic Calculus study grouporganized by B. Zhang and T. Oraby will study An Introduction to Stochastic Differential Equations (version 1.2) by L. C. Evans.Tuesdays 5-7 in the math launge
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Abstract: The Law of Large Numbers, the Central Limit Theorem, and the Law of the Iterated Logarithm for independent and identically distributed sequences of random variables are three central, perhaps dominant, results of classical probability theory. The Ergodic Theorem provides a complete extension of the Law of Large Numbers to sequences that are dependent, but stationary. The Central Limit Theorem and Law of the Iterated Logarithm do not extend as completely, but only under additional conditions that effectively limit the amount of dependence. During the past decade there has been some progress on understanding the Central Limit Theorem for stationary processes, resulting in conditions that are sufficient and nearly necessary, at least for the conditional version of the Central Limit Theorem. The talk will present recent efforts to modify the arguments leading to the Central Limit Theorem to obtain a Law of the Iterated Logarithm. It will begin with some background material on the Law of the Iterated Logarithm and a selective review of recent work on the Central Limit Theorem for stationary sequences. It will then describe the modifications necessary to obtain the Law of the Iterated Logarithm.
Abstract: In this joint work with J. Dedecker and E. Rio, we obtain convergence rates in the central limit theorem for stationary sequences in Lp for Wasserstein distances of order r, for p in ]2,3] and r in ]p-2,p]. The conditions are expressed in terms of projective criteria. The results apply in particular to non-adapted sequences.
Informally, the simple exclusion process follows a collection of random walks which interact in that they are not allowed to jump onto each other. In this talk, we consider the motion of a distinguished, or tagged, particle in this particle system. We review some of the past results and discuss some new contributions.
| Functional Analysis SeminarWednesday 3-4 pm, Feb 21, Feb 28, March 7 Seminar room (OC 807) Victor Kaftal, Majorization theory for infinite sequencesMajorization for finite sequences is linked to doubly stochastic matrices, convexity, the diagonals of selfadjoint matrices (the Schur-Horn Theorem) and more, and so it has been of interest to researchers in several areas of math. Little was know until recently about majorization for infinite sequences but we have now some new results. |
We revisit some cumulant methods for Independent Component Analysis - an unsupervised learning method with increasing popularity and applicability in number of disciplines. A rigorous justification of identifiability of the linear ICA method by kurtosis maximization is given by a simple lemma from optimization theory, which gives a basis for a generalization of the famous Fixed Point Algorithm for ICA for high order cumulants. We propose a measure for independence of group of random variables, given by a sum of cross-cumulants of a given order n. Similar measure was known for the case of four order cross-cumulants from the JADE algorithm for ICA. We derive a formula for its calculation using cumulant tensors. In the case n=4 our formula allows efficient calculation of this measure, using cumulant matrices. Much attention is devoted to the case of six order cross-cumulants, aiming to show that this measure can be calculated using again cumulant matrices. We provide a simple proof of the main ICA theorem concerning identifiability of the linear ICA model using the properties of the cross-cumulants instead of the Darmois-Skitovitch theorem from statistics, used for this purpose in the literature on ICA. Various ICA algorithms are demonstrated.
I shall present some results about the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to the given sigma-algebra, as done in Gordin (1969) and Heyde (1974.) These conditions are well-adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.
The "zero-range" system is an (infinite) collection of dependent random walks on Zd which models various types of traffic. Informally, the interaction is in that a particle jumps with a rate depending on the number of particles at its vertex, but to where it jumps is selected independently. In this talk, we consider a distinguished, or "tagged," particle in this system and discuss its asymptotic behavior including some recent diffusive estimates in "equilibrium." In particular cases, we also discuss approximation of the tagged position by a Brownian motion with parameters depending on the form of the interaction and the density of particles.
This is a "practice talk for a conference". The topic is classificiatioon of rpocesses with linear regressions and quadratic conditional variances.
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Related Activities
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Statistics SeminarWednesdays 3-4, Room 825 |
Abstract We survey the role played by orthogonal polynomials in the analysis of transision probabilities of birth and death processes and associated random walks. Several examples will be mentioned.
Abstract Stieltjes and Hilbert computed discriminants of Jacobi polynomials. Stieltjes studied the electrostatics equilibrium problem of n-unit charged particles restricted to (-1,1) under the external field of charges (a+1)/2 and (b+1)/2 at ±1. The potential is a logarithmic potential. Stieljes showed that the equilibrium position of the particle is at the zeros of the Jacobi polynomial Pn(a, b)(x). We discuss the recent developments on this problem, its extension to general orthogonal polynomials and the role discriminants play in the solution of the problem. We also mention the more recent work where similar techniques are used to solve the Bethe Ansatz equations for the XXZ and XXX models.
Abstract We assume that real-world financial markets are partially hedgable, therefore they are fundamentally incomplete [3]. It has been known for some time that there is no unique price for financial contracts in incomplete market. Recently, by the help of reduced Monge-Ampere equations, [1] introduces a new method for pricing and hedging of financial instruments in in/complete market. Financial contracts still have non-unique prices in incomplete setting, but the prices depend on only g, so-called relative risk aversion.
- [1] S.D. Stojanovic: Higher dimensional fair option pricing and hedging under Hara and Cara utilities. (Preprint, August 2005)
- [2] S.D. Stojanovic: Pde methods in financial modeling. 25th Annual Searcde in Dayton
- [3] S.D. Stojanovic: Actuaries vs. Financial Engineers in regard to valuation: the truth is now found to be in between. Garp Risk Review, Sept/Oct 2005
Schedule
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FA Seminar We 2-3 Room 708
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