Albert Fannjiang

Convection enhanced diffusion for periodic flow

This paper studies the influence of convection by periodic or cellular flows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by two-dimensional, steady, divergence-free, periodic velocity fields.

Coherence Pattern-Guided Compressive Sensing with Unresolved Grids

Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such coherent sensing matrices. These techniques are embedded in the existing compressed sensing algorithms, such as Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), Iterative Hard Thresholding (IHT), Basis Pursuit (BP), and Lasso, and result in the modified algorithms BLOOMP, BLOSP, BLOIHT, BP-BLOT, and Lasso-BLOT, respectively. Under appropriate conditions, it is proved that BLOOMP can reconstruct sparse, widely separated objects up to one Rayleigh length in the Bottleneck distance independent of the grid spacing. One of the most distinguishing attributes of BLOOMP is its capability of dealing with large dynamic ranges. The BLO-based algorithms are systematically tested with respect to four performance metrics: dynamic range, noise stability, sparsity, and resolution. With respect to dynamic range and noise stability, BLOOMP is the best performer. With respect to sparsity, BLOOMP is the best performer for high dynamic range, while for dynamic range near unity BP-BLOT and Lasso-BLOT with the optimized regularization parameter have the best performance. In the noiseless case, BP-BLOT has the highest resolving power up to certain dynamic range. The algorithms BLOSP and BLOIHT are good alternatives to BLOOMP and BP/Lasso-BLOT: they are faster than both BLOOMP and BP/Lasso-BLOT and share, to a lesser degree, BLOOMPs amazing attribute with respect to dynamic range. Detailed comparisons with the algorithms Spectral Iterative Hard Thresholding (SIHT) and the frame-adapted BP demonstrate the superiority of the BLO-based algorithms for the problem of sparse approximation in terms of highly coherent, redundant dictionaries.

Compressed Remote Sensing of Sparse Objects

The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a logsquare factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The predictions of theorems are confirmed by numerical simulations.

Integral equations with hypersingular kernels-theory and applications to fracture mechanics

Abstract Hypersingular integrals of the type I α (T n ,m,r)= −1 1 T n (s)(1−s 2 ) m−1/2 (s−r) α d s, |r| and I α (U n ,m,r)= −1 1 U n (s)(1−s 2 ) m−1/2 (s−r) α d s, |r| are investigated for general integers α (positive) and m (nonnegative), where Tn(s) and Un(s) are the Chebyshev polynomials of the first and second kinds, respectively. Exact formulas are derived for the cases α=1,2,3,4 and m=0,1,2,3; most of them corresponding to new solutions derived in this paper. Moreover, a systematic approach for evaluating these integrals when α>4 and m>3 is provided. The integrals are also evaluated as |r|>1 in order to calculate the stress intensity factors. Examples involving crack problems are given and discussed with emphasis on the linkage between mathematics and mechanics of fracture. The examples include classical linear elastic fracture mechanics, functionally graded materials, and gradient elasticity theory. An appendix, with an alternative derivation of the formulae, supplements the paper.

The crack problem for nonhomogeneous materials under antiplane shear loading — a displacement based formulation

Abstract This paper presents a displacement based integral equation formulation for the mode III crack problem in a nonhomogeneous medium with a continuously differentiable shear modulus, which is assumed to be an exponential function, i.e., G(x)=G0exp(βx). This formulation leads naturally to a hypersingular integral equation. The problem is solved for a finite crack and results are given for crack profiles and stress intensity factors. The results are affected by the parameter β describing the material nonhomogeneity. This study is motivated by crack problems in strain-gradient elasticity theories where higher order singular integral equations naturally arise even in the slope-based formulation.

Diffusion in turbulence

SummaryWe prove long time diffusive behavior (homogenization) for convection-diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix. Simple shear flow examples show that this result is sharp for flows that have stationary stream matrices.

Absolute uniqueness of phase retrieval with random illumination

Random illumination is proposed to enforce absolute uniqueness and resolve all types of ambiguity, trivial or nontrivial, in phase retrieval. Almost sure irreducibility is proved for any complex-valued object whose support set has rank 2. While the new irreducibility result can be viewed as a probabilistic version of the classical result by Bruck, Sodin and Hayes, it provides a novel perspective and an effective method for phase retrieval. In particular, almost sure uniqueness, up to a global phase, is proved for complex-valued objects under general two-point conditions. Under a tight sector constraint absolute uniqueness is proved to hold with probability exponentially close to unity as the object sparsity increases. Under a magnitude constraint with random amplitude illumination, uniqueness modulo global phase is proved to hold with probability exponentially close to unity as object sparsity increases. For general complex-valued objects without any constraint, almost sure uniqueness up to global phase is established with two sets of Fourier magnitude data under two independent illuminations. Numerical experiments suggest that random illumination essentially alleviates most, if not all, numerical problems commonly associated with the standard phasing algorithms. (Some figures may appear in colour only in the online journal)

Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials—Part II: Crack Parallel to the Material Gradation

ing partial differential equations PDEs are derived, and the Fourier transform method is introduced and applied to convert the governing PDE into an ordinary differential equation ODE .A fterward, the crack boundary value problem is described, and a specific complete set of boundary conditions is given. The governing hypersingular integrodifferential equation is derived and discretized using the collocation method. Next, various relevant aspects of the numerical discretization are described in detail. Subsequently, numerical results are given, conclusions are inferred, and potential extensions of this work are discussed. One appendix, providing the hierarchy of the PDEs and the corresponding integral equations, supplements the paper.

Convection-enhanced diffusion for random flows

We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of Isichenko and Kalda that the effective diffusivity behaves likeɛ3/13 when the molecular diffusivityɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows.

Compressive inverse scattering: I. High-frequency SIMO/MISO and MIMO measurements

Inverse scattering from discrete targets with the single-input–multiple-output (SIMO), multiple-input–single-output (MISO) or multiple-input–multiple-output (MIMO) measurements is analyzed by compressed sensing theory with and without the Born approximation. High-frequency analysis of (probabilistic) recoverability by the L1-based minimization/regularization principles is presented. In the absence of noise, it is shown that the L1-based solution can recover exactly the target of sparsity up to the dimension of the data either with the MIMO measurement for the Born scattering or with the SIMO/MISO measurement for the exact scattering. The stability with respect to noisy data is proved for weak or widely separated scatterers. Reciprocity between the SIMO and MISO measurements is analyzed. Finally a coherence bound (and the resulting recoverability) is proved for diffraction tomography with high-frequency, few-view and limited-angle SIMO/MISO measurements.