Home About us Contact
|
Home About us Contact | ||
|
|
||
Diagonal Matrix (diagonal + matrix)
Selected AbstractsA domain decomposition method for modelling Stokes flow in porous materialsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2002Guangli Liu Abstract An algorithm is presented for solving the Stokes equation in large disordered two-dimensional porous domains. In this work, it is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off-block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000-particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport. Copyright © 2002 John Wiley & Sons, Ltd. [source] A diagonal measure and a local distance matrix to display relations between objects and variables,JOURNAL OF CHEMOMETRICS, Issue 1 2010Gergely Tóth Abstract Proper permutation of data matrix rows and columns may result in plots showing striking information on the objects and variables under investigation. To control the permutation first, a diagonal matrix measureD was defined expressing the size relations of the matrix elements. D is essentially the absolute norm of a matrix where the matrix elements are weighted by their distance to the matrix diagonal. Changing the order of rows and columns increases or decreases D. Monte Carlo technique was used to achieve maximum D in the case of the object distance matrix or even minimal D in the case of the variable correlation matrix to get similar objects or variables close together. Secondly, a local distance matrix was defined, where an element reflects the distances of neighboring objects in a limited subspace of the variables. Due to the maximization of D in the local distance matrix by row and column changes of the original data matrix, the similar objects were arranged close to each other and simultaneously the variables responsible for their similarity were collected close to the diagonal part defined by these objects. This combination of the diagonal measure and the local distance matrix seems to be an efficient tool in the exploration of hidden similarities of a data matrix. Copyright © 2009 John Wiley & Sons, Ltd. [source] Partial differential equations of chemotaxis and angiogenesisMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2001B. D. Sleeman The topic of this paper is concerned with an investigation of the qualitative properties of solutions to the following problem. Let ,,Rn be a bounded domain with boundary ,,. We seek solutions P,,,Rm+1 of the system (1) subject to the ,no-flux' boundary condition (2) where n denotes the inward pointing normal to ,,. To close the system we prescribe the initial conditions (3) In this system D is a constant diffusion coefficient, P is a population density and , is a vector of nutrients or chemicals whose dynamics influences the movement of P. Notice here that the substances , do not diffuse. If they do then the second equation of (1) is modified to (4) where d is a positive semi-definite diagonal matrix. This more general system includes the so-called Keller,Segel model of Biology ([1] Keller EF, Segel LA. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology 1970; 26: 339,415). To motivate our study of system (1),(3) we begin by outlining two themes. One basic to developmental biology and the other from angiogenesis. Copyright © 2001 John Wiley & Sons, Ltd. [source] Scaling turbulent atmospheric stratification.THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 631 2008III: Space, time stratification of passive scalars from lidar data Abstract In this third and final part of the series, we concentrate on the temporal behaviour of atmospheric passive scalars. We first recall that,although the full (x, y, z, t) turbulent processes respect an anisotropic scale invariance,that due to advection,the generator will generally not be a diagonal matrix. This implies that the scaling of (1-D) temporal series will generally involve three exponents in real space: 1/3, 1/2, 3/5, for spectra ,, = 5/3, 2, 11/5, with the first and last corresponding to domination by advection (horizontal and vertical respectively), and the second to pure temporal development (no advection). We survey the literature and find that almost all the empirical ,, values are indeed in the range 5/3 to 2. We then use meteorological analyses to argue that, although pure temporal development is unlikely to be dominant for time-scales less than the eddy turnover time of the largest structures (about 2 weeks), an intermittent vertical velocity could quite easily explain the occasionally observed ,, , 2 spectra. We then use state-of-the-art vertically pointing lidar data of backscatter ratios from both aerosols and cirrus clouds yielding several (z, t) vertical space,time cross-sections with resolution of 3.75 m in the vertical, 0.5,30 s in time and spanning 3,4 orders of magnitude in temporal scale. We first test the predictions of the anisotropic, multifractal extension of the Corrsin-Obukhov law in the vertical and in time, separately finding that the cirrus and aerosol backscatters both followed the theoretical (anisotropic) scalings accurately; three of the six cases show dominance by the horizontal wind, the others by the vertical wind. In order to test the theory in arbitrary directions in this (z, t) space, and in order to get more complete information about the underlying physical scale, we develop and apply a new Anisotropic Scaling Analysis Technique (ASAT) which is based on a nonlinear space,time coordinate transformation. This transforms the original differential scaling into standard self-similar scaling; there remains only a ,trivial' anisotropy. This method is used in real space on 2-D structure functions. It is applied to both the new (z, t) data as well as the (x, z) data discussed in part II. Using ASAT, we verify the theory to within about 10% over more than three orders of magnitude of space,time scales in arbitrary directions in (x, z) and (z, t) spaces. By considering the high- (and low-) order structure functions, we verify the theory for both weak and strong structures; as predicted, their average anisotropies are apparently the same. Putting together the results for (x, z) and (z, t), and assuming that there is no overall stratification in the horizontal (x, y) plane, we find that the overall (x, y, z, t) space is found to have an effective ,elliptical dimension' characterizing the overall space,time stratification equal to Deff, st = 3.21 ± 0.05. Copyright © 2008 Royal Meteorological Society [source] The optimal density of atmospheric sounder observations in the Met Office NWP systemTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 629 2007M. L. Dando Abstract Large numbers of satellite observations are discarded from the numerical weather prediction (NWP) process because high-density observations may have a negative impact on the analysis. In current assimilation schemes, the observation error covariance matrix R is usually represented as a diagonal matrix, which assumes there are no correlations in the observation errors and that each observation is an independent piece of information. This is not the case when there are strong error correlations and this can lead to a degraded analysis. The experiments conducted in this study were designed to identify the optimal density and to determine if there were circumstances when exceeding this density might be beneficial to forecast skill. The global optimal separation distance of Advanced TIROS Operational Vertical Sounder (ATOVS) observations was identified by comparing global forecast errors produced using different densities of ATOVS. The global average of the absolute forecast error produced by each different density was found for a 3-week period from December 2004 to January 2005. The results showed that, when using the Met Office NWP system with a horizontal model resolution of ,60 km, the lowest global forecast errors were produced when using separation distances of 115,154 km. However, localized regions of the atmosphere containing large gradients such as frontal regions may benefit from thinning distances as small as 40 km and therefore the global optimal separation distance is not necessarily applicable in these circumstances. Copyright © 2007 Royal Meteorological Society [source] H, control for linear systems with state saturation nonlinearities,ASIAN JOURNAL OF CONTROL, Issue 6 2009Xiaofu Ji Abstract The problem of H, control for a class of linear systems with state saturation nonlinearities is considered in this paper. By introducing a row diagonally dominant matrix with negative diagonal elements and a diagonal matrix with positive elements, the H, control problem is reduced to a matrix inequality feasibility problem that can be solved by the proposed iterative linear matrix inequality algorithm. The effectiveness of the presented method is demonstrated by a numerical example. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source] | ||||||||||