Showing papers in "Communications on Pure and Applied Mathematics in 2006"

Stable signal recovery from incomplete and inaccurate measurements

Abstract: Suppose we wish to recover a vector x_0 Є R^m (e.g., a digital signal or image) from incomplete and contaminated observations y = Ax_0 + e; A is an n by m matrix with far fewer rows than columns (n « m) and e is an error term. Is it possible to recover x_0 accurately based on the data y? To recover x_0, we consider the solution x^# to the l_(1-)regularization problem min ‖x‖l_1 subject to ‖Ax - y‖l(2) ≤ Є, where Є is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level ‖x^# - x_0‖l_2 ≤ C Є. As a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x_0; then stable recovery occurs for almost any set of n coefficients provided that the number of nonzeros is of the order of n/[log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.

For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution

Abstract: We consider linear equations y = Φx where y is a given vector in ℝn and Φ is a given n × m matrix with n 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx0by a coefficient vector x0 ∈ ℝmwith fewer than ρ · n nonzeros, the solution x1of the 1-minimization problem is unique and equal to x0. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost-spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.

For most large underdetermined systems of equations, the minimal 1-norm near-solution approximates the sparsest near-solution

Abstract: We consider inexact linear equations y ≈ Φx where y is a given vector in R n , Φ is a given n x m matrix, and we wish to find x 0,∈ as sparse as possible while obeying ∥y - Φx 0,∈ ∥ 2 ≤ ∈. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the l 1 -minimization problem min ∥x∥ 1 subject to ∥y - Φx∥ 2 ≤ e is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x 0,∈ is sufficiently sparse, then the solution x 1,∈ of the l 1 -minimization problem is a good approximation to x 0,∈ . We suppose that the columns of Φ are normalized to the unit l 2 -norm, and we place uniform measure on such Φ. We study the underdetermined case where m ∼ τn and τ > 1, and prove the existence of p = p(r) > 0 and C = C(p, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ∥y - Φx 0 ∥ 2 ≤ ∈ by a coefficient vector x 0 e R m with fewer than ρ · n nonzeros, ∥x 1,∈ - x 0 ∥ 2 ≤ C ≤ ∈. This has two implications. First, for most Φ, whenever the combinatorial optimization result x 0,∈ would be very sparse, x 1,∈ is a good approximation to x 0,∈ . Second, suppose we are given noisy data obeying y = Φx 0 + z where the unknown x 0 is known to be sparse and the noise ∥z∥ 2 ≤ ∈. For most Φ, noise-tolerant l 1 -minimization will stably recover x 0 from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost-spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices.

Classification of solutions for an integral equation

Abstract: Let n be a positive integer and let 0 < α < n. Consider the integral equation We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn. This solves an open problem posed by Lieb 12. The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well-known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.

On the Floer homology of cotangent bundles

Abstract: This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.

Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate

Abstract: Consider a system of N bosons on the three-dimensional unit torus interacting via a pair potential N2V(N(xi − xj)) where x = (x1, …, xN) denotes the positions of the particles. Suppose that the initial data ψN, 0 satisfies the condition where HN is the Hamiltonian of the Bose system. This condition is satisfied if ψN, 0 = WNϕN, 0 where WN is an approximate ground state to HN and ϕN, 0 is regular. Let ψN, t denote the solution to the Schrodinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrodinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices ⊗k|ut〉 〈ut| solves the GP hierarchy. We prove that as N ∞ the limit points of the k-particle density matrices of ψN, t are solutions of the GP hierarchy. Our analysis requires that the N-boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n. © 2006 Wiley Periodicals, Inc.

Dual‐primal FETI methods for linear elasticity

Abstract: Dual-primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. These methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them. The algorithms are described in terms of a change of basis that has proven to be quite robust in practice. A theoretical analysis is provided for the case of linear elasticity, and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain. The strategies have already proven quite successful in large-scale implementations of these iterative methods. © 2006 Wiley Periodicals, Inc.

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws I: Quasi-Triviality of Bi-Hamiltonian Perturbations

Abstract: We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs vt +[ φ(v)]x = 0. Under certain genericity assumptions it is proved that any bi-Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so-called quasiMiura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi-Hamiltonian structure of hydrodynamic type (the quasi-triviality theorem). We also describe, following [35], the invariants ofsuchbi-Hamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives. c

Boltzmann diffusive limit beyond the navier-stokes approximation

Abstract: Given a normalized Maxwellian μ and n ≥ 1, we establish the global-in-time validity of a diffusive expansion (0.1) F e (t,x,v)=μ+√μ{ef 1 (t,x,v)+e 2 f 2 (t,x,v)+···+e n f e n (t,x,v)}. for a solution F e to the rescaled Boltzmann equation (diffusive scaling) (0.2) e∂ 1 F e +v·∇ x Fe=1 eQ(F e ,F e ) inside a periodic box T 3 . We assume that in the initial expansion (0.1) at t = 0, the fluid parts of these f m (0,x,v) have arbitrary divergence-free velocity fields as well as temperature fields for all 1 ≤ m ≤ n while f 1 (0,x,v) has small amplitude in H 2 . For m ≥ 2, these f m (t,x,v) are determined by a sequence of linear Navier-Stokes-Fourier systems iteratively. More importantly, the remainder f e n (t,x,v) is proven to decay in time uniformly in e via a unified nonlinear energy method. In particular, our results lead to an error estimate for f 1 (t,x,v), the well-known Navier-Stokes-Fourier approximation, and beyond. The collision kernel Q includes hard-sphere, the cutoff inverse-power, as well as the Coulomb interactions.

A deterministic‐control‐based approach motion by curvature

Abstract: The level-set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. For a convex domain, the boundary's “first arrival time” solves a degenerate elliptic equation; this corresponds, in our game-theoretic setting, to a minimum-exit-time problem. For a nonconvex domain the two-person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the “positive part of the curvature.” These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first-order Hamilton-Jacobi equation. Our situation is different because the usual first-order calculation is singular. © 2005 Wiley Periodicals, Inc.

Stochastic homogenization of Hamilton‐Jacobi‐Bellman equations

Abstract: We study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment. c � 2006 Wiley Periodicals, Inc.

Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions

Abstract: In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for $SU(N+1)$ We give a much more precise bubbling behavior of solutions and study its existence in some critical cases

Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance

Abstract: The singularly perturbed two-well problem in the theory of solid-solid phase transitions takes the form I e [u] = ∫ 1 eW(∇u) + e|∇ 2 u| 2 , Ω where u: Ω ⊂ R n → R n is the deformation, and W vanishes for all matrices in K = SO(n)A U SO(n)B. We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp-interface limit for I e . The proof is based on a rigidity estimate for low-energy functions. Our rigidity argument also gives an optimal two-well Liouville estimate: if ∇u has a small BV norm (compared to the diameter of the domain), then, in the L 1 sense, either the distance of ∇u from SO(2)A or the one from SO(2)B is controlled by the distance of Vu from K. This implies that the oscillation of Vu in weak L1 is controlled by the L1 norm of the distance of ∇u to K.

Universality of the double scaling limit in random matrix models

Abstract: We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift-Zhou steepest-descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. (c) 2005 Wiley Periodicals, Inc.

Global solutions for a semilinear, two‐dimensional Klein‐Gordon equation with exponential‐type nonlinearity

Abstract: We prove the existence and uniqueness of global solutions for a Cauchy problem associated to a semilinear Klein-Gordon equation in two space dimensions. Our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method. © 2006 Wiley Periodicals, Inc.

A shallow-water approximation to the full water wave problem

Abstract: We demonstrate that the system of the Green-Naghdi equations as a two-directional, nonlinearly dispersive wave model is a close approximation to the two-dimensional full water wave problem. Based on the energy estimates and the proof of the well-posedness for the Green-Naghdi equations and the water wave problem, we compare solutions of the two systems, showing that without restrictions on the wave amplitude, any two solutions of the two systems remain close, at least in some finite time within the shallow-water regime, provided that their initial data are close in the Banach space Hs × Hs+1 for some s > . As a consequence, we show that if the depth of the water compared with the wavelength is sufficiently small, the two solutions exist for the same finite time using the uniformly bounded energies defined in the paper. © 2006 Wiley Periodicals, Inc.

Mathematical analysis of vortex sheets

Abstract: We consider the motion of the interface separating two domains of the same fluid that moves with different velocities along the tangential direction of the interface. The evolution of the interface (the vortex sheet) is governed by the Birkhoff-Rott (BR) equations. We consider the question of the weakest possible assumptions such that the Birkhoff-Rott equation makes sense. This leads us to introduce chord-arc curves to this problem. We present three results. The first can be stated as the following: Assume that the Birkhoff-Rott equation has a solution in a weak sense and that the vortex strength is bounded away from 0 and ∞. Moreover, assume that the solution gives rise to a vortex sheet curve that is chord-arc. Then the curve is automatically smooth, in fact analytic, for fixed time. The second and third results demonstrate that the Birkhoff-Rott equation can be solved if and only if only half the initial data is given. c � 2005 Wiley Periodicals, Inc.

Malliavin calculus for the stochastic 2D Navier Stokes equation

Abstract: We consider the incompressible, two-dimensional Navier-Stokes equation with periodic boundary conditions under the effect of an additive, white-in-time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite-dimensional projection of the solution possesses a smooth, strictly positive density with respect to Lebesgue measure. In particular, our conditions are viscosity independent. We are mainly interested in forcing that excites a very small number of modes. All of the results rely on proving the nondegeneracy of the infinite-dimensional Malliavin matrix. c � 2006 Wiley Periodicals, Inc.

Stability of a Mach configuration

Abstract: When the incident angle constructed by the incident shock and the surface of the wall is greater than a critical value, the regular shock reflection could not occur, and the Mach shock reflection will occur instead [1, 2, 3, 4, 5, 6, 7] In this chapter we are going to study the occurrence of Mach reflection and its structure Mach configuration is a structure of nonlinear waves, including three shocks starting from a point and a contact discontinuity Besides, we will also give a classification on this structure according to the characteristic feature of the flow field, and particularly study the stability of Mach configuration The main references are [2, 8]

Asymptotic rate of quantum ergodicity in chaotic euclidean billiards

Abstract: The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak is that every eigenfunction ϕn of the Laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue En ∞); that is, “strong scars” are absent. We study numerically the rate of equidistribution for a uniformly hyperbolic, Sinai-type, planar Euclidean billiard with Dirichlet boundary condition (the “drum problem”) at unprecedented high E and statistical accuracy, via the matrix elements 〈ϕn, Âϕm〉 of a piecewise-constant test function A. By collecting 30,000 diagonal elements (up to level n ≈ 7 × 105) we find that their variance decays with eigenvalue as a power 0.48 ± 0.01, close to the semiclassical estimate ½ of Feingold and Peres. This contrasts with the results of existing studies, which have been limited to En a factor 102 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance as a function of distance from the diagonal, against Feingold-Peres (or spectral measure) at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions. © 2006 Wiley Periodicals, Inc.

Lusternik‐Schnirelmann category and systolic category of low‐dimensional manifolds

Abstract: We show that the geometry of a Riemannian manifold (M, ) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M. It is denoted catsys(M) and defined in terms of the existence of systolic inequalities satisfied by every metric , as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality catsysM ≤ catLSM is satisfied, which typically turns out to be an equality, e.g., in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality and that both categories are sensitive to Massey products. The comparison with the value of catLS(M) leads us to prove or conjecture new systolic inequalities on M. © 2006 Wiley Periodicals, Inc.

Decay of correlations and the central limit theorem for meromorphic maps

Abstract: Let f be a dominant meromorphic self-map of large topological degree on a compact Kahler manifold. We give a new construction of the equilibrium measure μ of f and prove that μ is exponentially mixing. As a consequence, we get the central limit theorem in particular for Holder-continuous observables, but also for noncontinuous observables. © 2005 Wiley Periodicals, Inc.

Hyperbolic limit of the Jin-Xin relaxation model

Abstract: We consider the special Jin-Xin relaxation model (0.1) u i +A(u)u x = ∈(u xx - u tt ). We assume that the initial data (u 0 ,∈u 0,t ) are sufficiently smooth and close to (u,0) in L ∞ and have small total variation. Then we prove that there exists a solution (u ∈ (t), ∈u ∈ t (t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz-continuously in the L 1 norm with respect to time and the initial data. Letting ∈ → 0, the solution u ∈ converges to a unique limit, providing a relaxation limit solution to the quasi-linear, nonconservative system (0.2) u t + A(u)u x = 0. These limit solutions generate a Lipschitz semigroup S on a domain D containing the functions with small total variation and close to u. This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1).

Existence of an asymptotic velocity and implications for the asymptotic behaviour in the direction of the singularity in T3-Gowdy

Abstract: This is the first of two papers that together prove strong cosmic censorship in T-3-Gowdy space-times In the end, we prove that there is a set of initial data, open with respect to the C-2 X C-1 topology and dense with respect to the C-infinity topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, ie, the Riemann tensor contracted with itself, blows up in the incomplete direction In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions In this paper, we shall, however, focus on the concept of asymptotic velocity Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint The target of the wave map is the hyperbolic plane There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy We define the asymptotic velocity v(infinity) to be the nonnegative square root of the limit of the kinetic energy density The asymptotic velocity has some very important properties In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v(infinity) It also has properties such that if 0 1 and v(infinity) is continuous at theta(0), then v(infinity) is smooth in a neighborhood of theta(0) Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to the C-2 X C-1 topology on initial data

Homogenization and flame propagation in periodic excitable media: The asymptotic speed of propagation

Abstract: We study the effect of homogenization on flame propagation in periodic excitable media when the width of the flame is much smaller than the characteristic size of the heterogeneities. © 2005 Wiley Periodicals, Inc.

Form boundedness of the general second-order differential Operator

Abstract: We give explicit necessary and sufficient conditions for the boundedness of the general second-order differential operator ℒ = ∑ i,j=ln aij∂i∂j + ∑j=1nbj∂j+c with real- or complex-valued distributional coefficients aij, bj, and c, acting from the Sobolev space W1, 2(ℝn) to its dual W-1, 2(ℝn). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of C with respect to the Laplacian on L 2(ℝn). In particular, we establish a complete characterization of the form boundedness of the Schrodinger operator (i ▽, + a→)2 + q with magnetic vector potential a→ ∈ Lloc2(ℝn)n and q ∈ D′(ℝn). © 2006 Wiley Periodicals, Inc.

On the Long-Time Limit of Semiclassical (Zero Dispersion Limit) Solutions of the Focusing Nonlinear Schrödinger Equation: Pure Radiation Case

Abstract: In a previous paper [13] we calculated the leading-order term q0(x,t ,e )of the solution ofq(x,t ,e ), the focusing nonlinear (cubic) Schrodinger (NLS) equation in the semiclassical limit (e → 0) for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t ≥ 0. The aim of the present paper is to calculate the long-term behavior of the semiclassical solution q(x,t ,e )in the pure radiation case. As before, our main tool is the Riemann-Hilbert problem (RHP) formulation of the inverse scattering problem and the corresponding system of “moment and integral conditions,” known also as a system of “modulation equations.” c � 2006 Wiley Periodicals, Inc.

Analytic smoothing effect for the Schrödinger equation with long-range perturbation

Abstract: We study microlocal analytic singularity of solutions to Schr\"odinger equation with analytic coefficients. Using microlocal weight estimate developped for estimating the phase space tunneling, we prove microlocal smoothing estimates that generalize results by L. Robbiano and C. Zuily. We suppose the Schr\"odinger operator is a long-range type perturbation of the Laplacian, and we employ positive commutator type estimates to prove the smoothing property.

Fine structure of the zeros of orthogonal polynomials, iii. periodic recursion coefficients

Abstract: We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them.