An efficient sampling method for stochastic inverse problems

A general framework is developed to treat inverse problems with parameters that are random fields. It involves a sampling method that exploits the sensitivity derivatives of the control variable with respect to the random parameters. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the present method is a fraction of the total cost of the Monte Carlo method. The effectiveness of the method is demonstrated on an example problem governed by the Burgers equation with random viscosity. It is specifically shown that this method is two orders of magnitude more efficient compared to the conventional Monte Carlo method. In other words, for a given number of samples, the present method yields two orders of magnitude higher accuracy than its conventional counterpart.     

Quantitative measurements of micro- and macromixing in a stirred vessel using Planar Laser-Induced Fluorescence

The Planar Laser-Induced Fluorescence (PLIF) technique enables measurement of the local degree of deviation in an arbitrary plane inside the mixing vessel. This is achieved by injecting a mixture of an inert dye and a reacting fluorescent into the vessel. The inert dye serves as a tracer for the macromixing. The fluorescent characteristics of the reacting dye change while undergoing a fast chemical reaction with the vessel content and it therefore shows the micromixing indirectly. The concentration fields of the dyes are measured simultaneously. For that a laser beam is expanded into a thin light sheet which illuminates an arbitrary plane in the mixing vessel, exciting the fluorescent dye in this area. The emitted light is detected by a CCD-camera which is positioned vertical to the measurement plane. The fluorescent light passes through two optical filters which are suitable for separating the fluorescent light of the two dyes. The light is then projected on half of the camera chip each so that the same display window is detected twice, and thus the local concentration of the two dyes can be measured simultaneously. Low Reynolds number measurements are performed in a mixing vessel equipped with a Rushton turbine. The lamellar structure is clearly resolved. Areas of micromixing are detected by calculating the local degree of deviation from the concentration fields. These areas are mainly found in the boundary layer of the lamellas.     

Global error bound for convex inclusion problems

The existence of global error bound for convex inclusion problems is discussed in this paper, including pointwise global error bound and uniform global error bound. The existence of uniform global error bound has been carefully studied in Burke and Tseng (SIAM J. Optim. 6(2), 265–282, 1996) which unifies and extends many existing results. Our results on the uniform global error bound (see Theorem 3.2) generalize Theorem 9 in Burke and Tseng (1996) by weakening the constraint qualification and by widening the varying range of the parameter. As an application, the existence of global error bound for convex multifunctions is also discussed.     

Dihedral crossed products of exponent 2 are abelian

We prove that assuming enough roots of unity in the base field, a central simple algebra of exponent 2 which is split by a dihedral group, is also split by certain abelian groups.     

Study of the Structure and Sorption Properties of Some Types of Cellulose

The pulsed NMR, isopiestic and pycnometric methods are applied to investigate the physicochemical properties of softwood [GOST (State Standard) 9571-89] and hardwood [GOST 28172-89] celluloses, as well as flax cellulose [GOST 10330-76, 9394-76], both in the initial state and subjected to chemical treatment. Possible changes occurring in the supramolecular structure of a polymer are analyzed. The structural and sorption characteristics of the investigated samples are calculated. Their water absorption capacities are determined.     

An Automatic Procedure for Updating the Block Size in the Block Conjugate Gradient Method for Solving Linear Systems

The paper considers the problem of constructing an efficient automatic procedure for reducing the block size in the block conjugate gradient method insuring that the resulting rate of convergence is comparable with that of the block conjugate gradient method with constant block size. The numerical results provided show that, independently of the type of distribution of the smallest eigenvalues of the preconditioned matrix, the procedure suggested always leads to a decrease of the arithmetic costs with respect to those of the block method with constant block size. Bibliography: 8 titles.     

Formation of coherent spatial structures in the reaction-diffusion atmospheric system under the action of a planetary wave

On the example of the effect of an atmospheric wave with given amplitude on the density oscillations of minor gas constituents near the mesopause (at heights 80–90 km, we consider the influence of the quasi-resonant periodic spatial transport on a distributed system responding subharmonically to a periodic external action (diurnal sunlight variations). Numerical modeling revealed the formation of a coherent horizontal distribution of the dynamical variables (i.e., the concentrations of small constituents). Application of this effect to the mesopause conditions is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 47, No. 9, pp. 739–752, September, 2004.     

Attractors of the sine-Gordon equation in the field of a quasiperiodic external force

The well-known sine-Gordon equation, supplemented with small damping and small quasiperiodic external force, is studied under the zero Dirichlet boundary conditions at the endpoints of a finite interval. The main assumption is that all frequencies of the external force are in 1:1 resonance with certain eigenfrequencies of the unperturbed equation; i.e., the socalled fundamental multifrequency resonance is observed. It is shown that in this case, by an appropriate choice of the parameters of the external force, one can make it so that the boundary value problem has a stable invariant torus of any finite dimension that bifurcates from zero on any preassigned finite set of spatial modes. It is also shown (by numerical analysis) that in a number of cases the above-mentioned torus coexists with a chaotic attractor.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.