Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.

Application of complex extreme learning machine to multiclass classification problems with high dimensionality: a THz spectra classification problem

We extend extreme learning machine (ELM) classifiers to complex Reproducing Kernel Hilbert Spaces (RKHS) where the input/output variables as well as the optimization variables are complex-valued. A new family of classifiers, called complex-valued ELM (CELM) suitable for complex-valued multiple-input–multiple-output processing is introduced. In the proposed method, the associated Lagrangian is computed using induced RKHS kernels, adopting a Wirtinger calculus approach formulated as a constrained optimization problem similarly to the conventional ELM classifier formulation. When training the CELM, the Karush–Khun–Tuker (KKT) theorem is used to solve the dual optimization problem that consists of satisfying simultaneously smallest training error as well as smallest norm of output weights criteria. The proposed formulation also addresses aspects of quaternary classification within a Clifford algebra context. For 2D complex-valued inputs, user-defined complex-coupled hyper-planes divide the classifier input space into four partitions. For 3D complex-valued inputs, the formulation generates three pairs of complex-coupled hyper-planes through orthogonal projections. The six hyper-planes then divide the 3D space into eight partitions. It is shown that the CELM problem formulation is equivalent to solving six real-valued ELM tasks, which are induced by projecting the chosen complex kernel across the different user-defined coordinate planes. A classification example of powdered samples on the basis of their terahertz spectral signatures is used to demonstrate the advantages of the CELM classifiers compared to their SVM counterparts. The proposed classifiers retain the advantages of their ELM counterparts, in that they can perform multiclass classification with lower computational complexity than SVM classifiers. Furthermore, because of their ability to perform classification tasks fast, the proposed formulations are of interest to real-time applications.

Does repeated burial of skeletal muscle tissue (Ovis aries) in soil affect subsequent decomposition?

The repeated introduction of an organic resource to soil can result in its enhanced degradation. This phenomenon is of primary importance in agroecosystems, where the dynamics of repeated nutrient, pesticide, and herbicide amendment must be understood to achieve optimal yield. Although not yet investigated, the repeated introduction of cadaveric material is an important area of research in forensic science and cemetery planning. It is not currently understood what effects the repeated burial of cadaveric material has on cadaver decomposition or soil processes such as carbon mineralization. To address this gap in knowledge, we conducted a laboratory experiment using ovine (Ovis aries) skeletal muscle tissue (striated muscle used for locomotion) and three contrasting soils (brown earth, rendzina, podsol) from Great Britain. This experiment comprised two stages. In Stage I skeletal muscle tissue (150 g as 1.5 g cubes) was buried in sieved (4.6 mm) soil (10 kg dry weight) calibrated to 60% water holding capacity and allowed to decompose in the dark for 70 days at 22 °C. Control samples comprised soil without skeletal muscle tissue. In Stage II, soils were weighed (100 g dry weight at 60% WHC) into 1285 ml incubation microcosms. Half of the soils were designated for a second tissue amendment, which comprised the burial (2.5 cm) of 1.5 g cube of skeletal muscle tissue. The remaining half of the samples did not receive tissue. Thus, four treatments were used in each soil, reflecting all possible combinations of tissue burial (+) and control (−). Subsequent measures of tissue mass loss, carbon dioxide-carbon evolution, soil microbial biomass carbon, metabolic quotient and soil pH show that repeated burial of skeletal muscle tissue was associated with a significantly greater rate of decomposition in all soils. However, soil microbial biomass following repeated burial was either not significantly different (brown earth, podsol) or significantly less (rendzina) than new gravesoil. Based on these results, we conclude that enhanced decomposition of skeletal muscle tissue was most likely due to the proliferation of zymogenous soil microbes able to better use cadaveric material re-introduced to the soil.

Data Assimilation and Inverse Methods in Terms of a Probabilistic Formulation

The weak-constraint inverse for nonlinear dynamical models is discussed and derived in terms of a probabilistic formulation. The well-known result that for Gaussian error statistics the minimum of the weak-constraint inverse is equal to the maximum-likelihood estimate is rederived. Then several methods based on ensemble statistics that can be used to find the smoother (as opposed to the filter) solution are introduced and compared to traditional methods. A strong point of the new methods is that they avoid the integration of adjoint equations, which is a complex task for real oceanographic or atmospheric applications. they also avoid iterative searches in a Hilbert space, and error estimates can be obtained without much additional computational effort. the feasibility of the new methods is illustrated in a two-layer quasigeostrophic model.