Many important kernel methods in the machine learning area, such as kernel principal component analysis, feature approximation, denoising, compression and prediction require the computation of the dominant set of eigenvectors of the symmetric kernel Gram matrix. Recently, an efficient incremental approach was presented for the fast calculation of the dominant kernel eigenbasis. In this manuscript we propose faster algorithms for incrementally updating and downsizing the dominant kernel eigenbasis. These methods are well-suited for large scale problems since they are both efficient in terms of complexity and data management.

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.

Here we investigate the behavior of the analytical and numerical solution of a nonlinear second kind Volterra integral equation where the linear part of the kernel has a constant sign and we provide conditions for the boundedness or decay of solutions and approximate solutions obtained by Volterra Runge-Kutta and Direct Quadrature methods.

In this paper, for the "critical case" with two delays, we establish two relations between any two solutions y(t) and y*(t) for the Volterra integral equation of non-convolution type y(t)=f(t)+\int_{t-\tau}^{t-\delta}k(t,s)g(y(s))ds and a solution z(t) of the first order differential equation \dot z(t)=\beta(t)[z(t-\delta)-z(t-\tau) , and offer a sufficient condition that limt->+?(y(t)-y*(t))=0.

A graphic processing unit (GPU) implementation of the multicomponent lattice Boltzmann equation with multirange interactions for soft-glassy materials ["glassy" lattice Boltzmann (LB)] is presented. Performance measurements for flows under shear indicate a GPU/CPU speed up in excess of 10 for 1024(2) grids. Such significant speed up permits to carry out multimillion time-steps simulations of 1024(2) grids within tens of hours of GPU time, thereby considerably expanding the scope of the glassy LB toward the investigation of long-time relaxation properties of soft-flowing glassy materials.

Phase separation of binary fluids quenched by contact with cold external walls is considered. Navier-Stokes, convection-diffusion, and energy equations are solved by lattice Boltzmann method coupled with finite-difference schemes. At high viscosity, different morphologies are observed by varying the thermal diffusivity. In the range of thermal diffusivities with domains growing parallel to the walls, temperature and phase separation fronts propagate toward the inner of the system with power-law behavior. At low viscosity hydrodynamics favors rounded shapes, and complex patterns with different length scales appear. Off-symmetrical systems behave similarly but with more ordered configurations.

Barenblatt e. a. introduced a fluid model for groundwater flow in fissurised porous media. The system consists of two diffusion equations for the groundwater levels in, respectively, the porous bulk and the system of cracks. The equations are coupled by a fluid exchange term. Numerical evidence suggests that the penetration depth of the fluid increases dramatically due to the presence of cracks and that the smallness of certain parameter values play a key role in this phenomenon. In the present paper we give precise estimates for the penetration depth in terms of the smallness of some of the parameters.

The "effective geometry" formalism is used to study the perturbations of a perfect barotropic Newtonian self-gravitating rotating and compressible fluid coupled with gravitational backreaction. The case of a uniformly rotating polytrope with index $n=1$ is investigated, due to its analytical tractability. Special attention is devoted to the geometrical properties of the underlying background acoustic metric, focusing in particular on null geodesics as well as on the analog light cone structure.

Kerr–Schild metrics have been introduced as a linear superposition of the flat spacetime metric and a squared null-vector field, say k, multiplied by some scalar function, say H. The basic assumption which led to Kerr solution was that k be both geodesic and shearfree. This condition is relaxed here and Kerr–Schild Ansatz is revised by treating Kerr–Schild metrics as exact linear perturbations of Minkowski spacetime. The scalar function H is taken as the perturbing function, so that Einstein’s field equations are solved order-by-order in powers of H. It turns out that the congruence must be geodesic and shearfree as a consequence of third- and second-order equations, leading to an alternative derivation of Kerr solution.

Spiral waves appear in many different contexts: excitable biological tissues, fungi and amoebae colonies, chemical reactions, growing crystals, fluids and gas eddies as well as in galaxies. While the existing theories explain the presence of spirals in terms of nonlinear parabolic equations, in this paper it is shown that self-sustained spiral wave regime is already present in the linear heat operator, in terms of integer Bessel functions of complex argument. Such solutions, even if commonly not discussed in the literature because diverging at spatial infinity, play a central role in the understanding of the universality of spiral process. As an example we have studied how in nonlinear reaction-diffusion models the linear part of the equations determines the wave front appearance while nonlinearities are mandatory to cancel out the blowup of solutions. The spiral wave pattern still requires however at least two cross diffusing species to be physically realized.

We consider splitting methods for the numerical integration of separable non-autonomous differential equations. In recent years, splitting methods have been extensively used as geometric numerical integrators showing excellent performances (both qualitatively and quantitatively) when applied on many problems. They are designed for autonomous separable systems, and a substantial number of methods tailored for different structures of the equations have recently appeared. Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly. We show that both procedures correspond to introducing the time as two new coordinates. We generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques. We show that the performance as well as the order of the final method can strongly depend on the particular choice. We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems. This technique is applied to different problems and its performance is illustrated for several numerical examples.

This work deals with infinite horizon optimal growth models and uses the results in the Mercenier and Michel (1994a) paper as a starting point. Mercenier and Michel (1994a) provide a one-stage Runge-Kutta discretization of the above-mentioned models which preserves the steady state of the theoretical solution. They call this feature the "steady-state invariance property". We generalize the result of their study by considering discrete models arising from the adoption of s-stage Runge-Kutta schemes. We show that the steady-state invariance property requires two different Runge-Kutta schemes for approximating the state variables and the exponential term in the objective function. This kind of discretization is well-known in literature as a partitioned symplectic Runge-Kutta scheme. Its main consequence is that it is possible to rely on the well-stated theory of order for considering more accurate methods which generalize the first order Mercenier and Michel algorithm. Numerical examples show the efficiency and accuracy of the proposed methods up to the fourth order, when applied to test models.

In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta-Kawashima ([SK]) coupling condition. This investigation is motivated by the fact that this condition is not satisfied by various physical sys- tems, and almost all the time in several space dimensions. First, we explore the role of entropy functionals around equilibrium solutions, which may not be constant, proposing a stability condition for such solutions. Then we find strictly dissipa- tive entropy functions for one dimensional 2 × 2 systems which violate the [SK] condition. Finally, we prove the existence of global smooth solutions for a class of systems such that condition [SK] does not hold, but which are linearly degenerated in the non-dissipative directions.