Corrector theory for MsFEM and HMM in random media

We analyze the random fluctuations of several multi-scale algorithms such as the multi-scale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multi-scale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multi-scale algorithms in the presence of random environments with short-range and long-range correlations. What we find is that the computationally more expensive method MsFEM captures the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.Comment: 41 page

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension $n=2$, it is known that the number of critical points (where $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $\sigma$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $\sigma(x)$ such that $\nabla u$ vanishes at least at one point in $X$. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $\sigma(x)$. These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $\sigma(x)$ for which the stability of the reconstructions will inevitably degrade.Comment: 26 pages, 4 figure

Inverse Transport Theory of Photoacoustics

We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are then used to construct the deposited thermal energy map. The latter depends on the constitutive optical parameters in a nontrivial manner. In this paper, we develop and use an inverse transport theory with internal measurements to extract information on the optical coefficients from knowledge of the deposited thermal energy map. We consider the multi-measurement setting in which many electromagnetic radiation patterns are used to probe the domain of interest. By developing an expansion of the measurement operator into singular components, we show that the spatial variations of the intrinsic attenuation and the scattering coefficients may be reconstructed. We also reconstruct coefficients describing anisotropic scattering of photons, such as the anisotropy coefficient $g(x)$ in a Henyey-Greenstein phase function model. Finally, we derive stability estimates for the reconstructions

Inverse Diffusion Theory of Photoacoustics

This paper analyzes the reconstruction of diffusion and absorption parameters in an elliptic equation from knowledge of internal data. In the application of photo-acoustics, the internal data are the amount of thermal energy deposited by high frequency radiation propagating inside a domain of interest. These data are obtained by solving an inverse wave equation, which is well-studied in the literature. We show that knowledge of two internal data based on well-chosen boundary conditions uniquely determines two constitutive parameters in diffusion and Schroedinger equations. Stability of the reconstruction is guaranteed under additional geometric constraints of strict convexity. No geometric constraints are necessary when $2n$ internal data for well-chosen boundary conditions are available, where $n$ is spatial dimension. The set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics (CGO) solutions.Comment: 24 page

Insular Carcinoma of Thyroid Presenting as a Giant Skull Lesion: A Dilemma in Treatment.

Thyroid surgeons are becoming increasingly more aware of a histologically distinct subset of thyroid carcinoma whose classification falls between well-differentiated and anaplastic carcinomas with respect to both cell differentiation and clinical behavior. This subtype of tumors has been categorized as poorly differentiated or insular carcinoma, based on its characteristic cell groupings. Although the differentiation of insular carcinoma from other thyroid carcinomas has important prognostic and therapeutic significance, relatively little about insular carcinoma has been published in the otolaryngology literature. In this article, we discuss a case of insular carcinoma of thyroid presenting with concurrent distant metastasis to skull, lung, ribs, and inguinal region with review of the literature. We conclude that insular thyroid carcinoma warrants aggressive management with total thyroidectomy and excision of accessible giant lesion followed by radioactive iodine ablation of any remaining thyroid tissue

Role of Paclobutrazol and Ethephon in Reproductive Growth of 'Allahabad Safeda' Guava (Psidium guajava L.) Plants at Different Spacing

A study on 4-year 'Allahabad Safeda' guava plants was made to assess the influence of Paclobutrazol (PP 333), [(2RS, 3RS)-1-(4-chlorophenyl)-4,4-dimethyl-2-(1,2,4 triazol-1-yl) pentan-3-ol], a gibberellin-inhibitor, and Ethephon [(2-chloroethyl) phosphonic acid], a vegetative growth inhibitor and a ripening promoter, on reproductive growth of plants. Treatments in the form of foliar application at 500 and 1000 ppm were applied consecutively during March 2007 and 2008 on plants at 6m x 2m, 6m x 3m, 6m x 4m and 6m x 5m spacing. Maximum flowering and fruit set was recorded in paclobutrazol treated plants in both rainy and winter crops. Ethephon reduced flower bud density (FBD) and fruit set during both the cropping seasons. However, Ethephon treated plants exhibited slightly higher fruit retention. Ethephon advances fruit maturity by upto a week during rainy season and two weeks during winter season. Paclobutrazol treated plants exhibited significantly higher fruit number, fruit yield, yield efficiency, fruiting density compared to Ethephon treated and control plants. Reproductive growth of plants at wider spacing of 6m x 5m and 6m x 4m significantly improved compared to closer spacings of 6m x 2m and 6m x 3m during both cropping seasons. Plants at wider spacing responded better to Paclobutrazol applications with respect to flowering and fruiting

Effect of Growth Regulator and Nutrients Spray on Control of Fruit Drop, Fruit Size and Quality of Ber under Sub-Montane Zone of Punjab

The effect of foliar spray of plant growth regulators and nutrients on fruit drop, fruit size and quality was studied in ber cv Sanuar-2 at Krishi Vigyan Kendra, Hoshiarpur, Punjab Agricultural University, Ludhiana, during 2005-2006. Plants were sprayed with NAA (20, 30 or 40 ppm), KNO3 (0.5, 1.0 or 1.5%) and ZnSO43333</sub (1.5%)

Survey of the sea fisheries of India

A preliminary account is given of the design and technique of the sampling method employed for estimating the landings of sea fish at some selected centres along the East and West coasts of India. Statistics of marine fishing villages, fishing populations, boats and nets are presented along with app roximate percentages of the occurrence of important fishes and also the estimated monthly landings of fish during 1949 at some representauve places