RSVD-CUR Decomposition for Matrix Triplets

We propose a restricted SVD based CUR (RSVD-CUR) decomposition for matrix triplets $(A, B, G)$. Given matrices $A$, $B$, and $G$ of compatible dimensions, such a decomposition provides a coordinated low-rank approximation of the three matrices using a subset of their rows and columns. We pick the subset of rows and columns of the original matrices by applying either the discrete empirical interpolation method (DEIM) or the L-DEIM scheme on the orthogonal and nonsingular matrices from the restricted singular value decomposition of the matrix triplet. We investigate the connections between a DEIM type RSVD-CUR approximation and a DEIM type CUR factorization, and a DEIM type generalized CUR decomposition. We provide an error analysis that shows that the accuracy of the proposed RSVD-CUR decomposition is within a factor of the approximation error of the restricted singular value decomposition of given matrices. An RSVD-CUR factorization may be suitable for applications where we are interested in approximating one data matrix relative to two other given matrices. Two applications that we discuss include multi-view/label dimension reduction, and data perturbation problems of the form $A_E=A + BFG$, where $BFG$ is a nonwhite noise matrix. In numerical experiments, we show the advantages of the new method over the standard CUR approximation for these applications

Block Discrete Empirical Interpolation Methods

We present two block variants of the discrete empirical interpolation method (DEIM); as a particular application, we will consider a CUR factorization. The block DEIM algorithms are based on the rank-revealing QR factorization and the concept of the maximum volume of submatrices. We also present a version of the block DEIM procedures, which allows for adaptive choice of block size. Experiments demonstrate that the block DEIM algorithms may provide a better low-rank approximation, and may also be computationally more efficient than the standard DEIM procedure

Tillage Slows Fecal Bacteria Infiltration through Soil

Bacterial pathogens can degrade ground water quality by infiltrating and eroding from land treated with poultry wastes. The potential for ground water contamination (as well as associated health risks and cost of water treatment) greatly depends on the depth of soil to the water table or bedrock and soil structure. Pathogens must move through the soil profile to contaminate ground water (although sinkholes can provide a direct channel from the soil surface to the water table in karst areas). Deep soils have less potential for contamination than shallow soils. Structureless soils retain fecal bacteria better than well structured soils. Research at UK indicates that surface-applied fecal bacteria, and other contaminants, travel rapidly toward ground water through soil pores in well structured, intact soil. Tillage disrupts pores and channels in the tilled layer, and increases water and bacteria contact with soil. To improve our understanding of bacterial movement, and of the potential for ground water contamination, we decided to examine whether tillage affected fecal coliform transport through intact soil amended with poultry wastes. We used poultry wastes because their disposal is an increasingly important waste management issue in western Kentucky

Fecal Coliform Transport through Intact Soil Blocks Amended with Poultry Manure

Poultry production in Kentucky increased almost 200% between 1991 and 1995. Their waste is typically land applied, and fecal pathogen runoff and infiltration may cause nonpoint source groundwater pollution. We looked at the preferential flow of fecal coliforms through undisturbed soil blocks since fecal bacteria typically infiltrate the soil profile to contaminate groundwater. Poultry manure was uniformly distributed on top of sod-covered or tilled (upper 12.5 cm) soil blocks and the blocks were irrigated. Drainage was collected in 100 uniformly spaced cells beneath each block and analyzed for fecal coliform content and drainage volume. The spatial distribution of drainage and fecal coliforms through the soil blocks was not uniform. Fecal coliforms appeared where most drainage flowed. Drainage water from each soil block consistently exceeded 200 000 fecal coliforms per 100 mL and was as great as 30 million fecal coliforms per 100 mL of leachate collected. Fecal coliforms leached as a pulse, but the breakthrough of fecal coliforms through tilled blocks was delayed with respect to the breakthrough of fecal coliforms through sod-covered blocks. Rainfall on a well-structured soil will cause the preferential movement of fecal bacteria, even with unsaturated flow conditions, and could contribute to fecal coliform concentrations in shallow groundwater that exceed standards for domestic discharge and primary contact water in Kentucky (200 fecal coliforms/100 mL)

Fecal Coliform Transport through Intact Soil Blocks Amended with Poultry Manure

Poultry production in Kentucky increased almost 200% between 1991 and 1995. Their waste is typically land applied, and fecal pathogen runoff and infiltration may cause nonpoint source groundwater pollution. We looked at the preferential flow of fecal coliforms through undisturbed soil blocks since fecal bacteria typically infiltrate the soil profile to contaminate groundwater. Poultry manure was uniformly distributed on top of sod-covered or tilled (upper 12.5 cm) soil blocks and the blocks were irrigated. Drainage was collected in 100 uniformly spaced cells beneath each block and analyzed for fecal coliform content and drainage volume. The spatial distribution of drainage and fecal coliforms through the soil blocks was not uniform. Fecal coliforms appeared where most drainage flowed. Drainage water from each soil block consistently exceeded 200 000 fecal coliforms per 100 mL and was as great as 30 million fecal coliforms per 100 mL of leachate collected. Fecal coliforms leached as a pulse, but the breakthrough of fecal coliforms through tilled blocks was delayed with respect to the breakthrough of fecal coliforms through sod-covered blocks. Rainfall on a well-structured soil will cause the preferential movement of fecal bacteria, even with unsaturated flow conditions, and could contribute to fecal coliform concentrations in shallow groundwater that exceed standards for domestic discharge and primary contact water in Kentucky (200 fecal coliforms/100 mL)

Solute Transport as Related to Soil Structure in Unsaturated Intact Soil Blocks

Concern about soil and groundwater pollution has resulted in numerous studies focused on solute transport. The objectives of our study were to investigate the effect of soil type and land-use management on solute movement. Transport of water and Cl− were measured through intact blocks of Maury (fine, mixed, semiactive, mesic Typic Paleudalf) and Cecil (fine, kaolinitic, thermic Typic Kanhapludult) soils, under steady-state, unsaturated flow conditions. Three replicate blocks for the Maury soil and two replicate blocks for the Cecil soil were studied per land-use treatment. The land-use treatments were conventional-till corn (Zea mays L.) production and long-term grass pasture. Individual blocks were instrumented with time domain reflectometry (TDR) probes at the 5-, 15-, and 25-cm depths. The effluent Cl− and TDR breakthrough curves were fitted using the convection dispersion equation (CDE); the estimated parameters were pore water velocity (v), dispersion coefficient (D), and, for the TDR breakthrough curves, maximum bulk electrical conductivity (BECmax). The CDE fitted the data very well, with model R 2 values ranging from 0.971 to 0.999. Volumetric water content (θ), total porosity, the soil water retention curve, and saturated hydraulic conductivity were determined on the same blocks. Volumetric water content increased (R2 = 0.25) as the slope of the water retention curve decreased. Increasing θ resulted in decreasing v (R2 =0.20) and thus, because of the linear relationship between D and v(R2 = 0.26), decreasing D Structural controls on solute dispersion in this study were mainly indirect, and related to variations in water content produced by differences in pore-size distribution

Phase Transition in Liquid Drop Fragmentation

A liquid droplet is fragmented by a sudden pressurized-gas blow, and the resulting droplets, adhered to the window of a flatbed scanner, are counted and sized by computerized means. The use of a scanner plus image recognition software enables us to automatically count and size up to tens of thousands of tiny droplets with a smallest detectable volume of approximately 0.02 nl. Upon varying the gas pressure, a critical value is found where the size-distribution becomes a pure power-law, a fact that is indicative of a phase transition. Away from this transition, the resulting size distributions are well described by Fisher's model at coexistence. It is found that the sign of the surface correction term changes sign, and the apparent power-law exponent tau has a steep minimum, at criticality, as previously reported in Nuclear Multifragmentation studies [1,2]. We argue that the observed transition is not percolative, and introduce the concept of dominance in order to characterize it. The dominance probability is found to go to zero sharply at the transition. Simple arguments suggest that the correlation length exponent is nu=1/2. The sizes of the largest and average fragments, on the other hand, do not go to zero but behave in a way that appears to be consistent with recent predictions of Ashurst and Holian [3,4].Comment: 10 pages, 11 figures. LaTeX (revtex4) with psfig/epsfi

Multiscale Soil Investigations: Physical Concepts And Mathematical Techniques

Soil variability has often been considered to be composed of “functional” (explained) variations plus random fl uctuations or noise. However, the distinction between these two components is scale dependent because increasing the scale of observation almost always reveals structure in the noise (Burrough, 1983). Soils can be seen as the result of spatial variation operating over several scales, indicating that factors infl uencing spatial variability differ with scale. Th is observation points to variability as a key soil attribute that should be studied