Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations

This paper introduces tensorial calculus techniques in the framework of Proper Orthogonal Decomposition (POD) to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree $p$. Such nonlinear terms have an on-line complexity of $\mathcal{O}(k^{p+1})$, where $k$ is the dimension of POD basis, and therefore is independent of full space dimension. However it is efficient only for quadratic nonlinear terms since for higher nonlinearities standard POD proves to be less time consuming once the POD basis dimension $k$ is increased. Numerical experiments are carried out with a two dimensional shallow water equation (SWE) test problem to compare the performance of tensorial POD, standard POD, and POD/Discrete Empirical Interpolation Method (DEIM). Numerical results show that tensorial POD decreases by $76\times$ times the computational cost of the on-line stage of standard POD for configurations using more than $300,000$ model variables. The tensorial POD SWE model was only $2-8\times$ slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM shallow water equation model to compute its off-line stage faster than the standard and tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table

A Randomized Proper Orthogonal Decomposition Method for Reducing Large Linear Systems

The proper orthogonal decomposition (POD) method is a powerful tool for reducing large data systems which can quickly overwhelm modern computing tools. In this thesis we provide a link between randomized projections and statistical methods by introducing the randomized POD method. We also apply the POD method to a heat transfer finite element model and image compression. In doing so we demonstrate the practical use and quantify the error introduced by the POD method.Aerospace Engineerin

Characterizing bean pod rot in Arkansas and Missouri

Green beans are an important crop grown for processing in both Arkansas and Missouri. Green beans are harvested mechanically using non-selective picking fingers. Harvested beans are then transported in bulk to processing plants that are located at various locations throughout the midSouth. Thus, the crop is managed for high quality, avoiding pod blemishes caused by insects and diseases. One of the consistent quality problems that affect Arkansas and Missouri green bean crops is pod rot. Two of the causal agents of pod rot that have been reported by researchers and vegetable companies alike are Pythium aphanidermatum and an unidentified Phytophthora sp. In this study, 15 growers’ fields were selected and soil samples (at planting), pod samples (at harvest), and environmental data were taken from each field. Disease incidence for field sites ranged from 0 to 7.3%. Pathogens associated with pod rot were Sclerotinia sclerotiorum, Rhizoctonia solani, a Phytophthora sp., and Pythium spp. The two suspected causal agents for pod rot, Pythium and Phytophthora spp., were found in all but one of the 12 field sites assessed for pod rot. Pythium inoculum potential, as determined by a baiting technique, was not a good indicator of pod rot incidence. In addition, soil temperature and water were not associated with pod rot. Pods collected at harvest having symptoms of pod rot were either in direct contact with the soil, senescing leaf tissue, or other diseased pods

Selective Pod Abortion by \u3ci\u3eBaptista Leucantha\u3c/i\u3e (Fabaceae) as Affected by a Curculionid Seed Predator, \u3ci\u3eApion Rostrum\u3c/i\u3e (Coleoptera)

The effect of a seed predator, Apion rostrum (Coleoptera: Curculionidae), on selective pod abortion from Baptisia leucantha (Fabaceae) was investigated in a restored tallgrass prairie plot. Weevil densities in and undamaged seed contents of attached and detached pods were compared over four occasions during the summer of 1993. Detached pods had similar to lower counts of weevils/pod and fewer seeds/pod than attached pods. Weevil density in pods appears only important in promoting pod abortIon through affects on seed number/pod as pods having fewer seeds are selectively aborted

Local Improvements to Reduced-Order Approximations of Optimal Control Problems Governed by Diffusion-Convection-Reaction Equation

We consider the optimal control problem governed by diffusion convection reaction equation without control constraints. The proper orthogonal decomposition(POD) method is used to reduce the dimension of the problem. The POD method may be lack of accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. We are interested in the perturbation of diffusion term. To increase the accuracy and robustness of the basis, we compute three bases additional to the baseline POD. The first two of them use the sensitivity information to extrapolate and expand the POD basis. The other one is based on the subspace angle interpolation method. We compare these different bases in terms of accuracy and complexity and investigate the advantages and main drawbacks of them.Comment: 19 pages, 5 figures, 2 table

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet