Inverse problems in dynamic cognitive modeling

Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov-Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.

Finding the smallest eigenvalue by the inverse Monte Carlo method with refinement

Finding the smallest eigenvalue of a given square matrix A of order n is computationally very intensive problem. The most popular method for this problem is the Inverse Power Method which uses LU-decomposition and forward and backward solving of the factored system at every iteration step. An alternative to this method is the Resolvent Monte Carlo method which uses representation of the resolvent matrix [I -qA](-m) as a series and then performs Monte Carlo iterations (random walks) on the elements of the matrix. This leads to great savings in computations, but the method has many restrictions and a very slow convergence. In this paper we propose a method that includes fast Monte Carlo procedure for finding the inverse matrix, refinement procedure to improve approximation of the inverse if necessary, and Monte Carlo power iterations to compute the smallest eigenvalue. We provide not only theoretical estimations about accuracy and convergence but also results from numerical tests performed on a number of test matrices.

Investigation into low power of a 2D Inverse Discrete CosineTransform (IDCT) in FPGAs

Design for low power in FPGA is rather limited since technology factors affecting power are either fixed or limited for FPGA families. This paper investigates opportunities for power savings of a pipelined 2D IDCT design at the architecture and logic level. We report power consumption savings of over 25% achieved in FPGA circuits obtained from clock gating implementation of optimizations made at the algorithmic level(1).

Parallel Monte Carlo approach for integration of the rendering equation

This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images. In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method. This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.

A new solution of the rendering equation with stratified Monte Carlo approach

This paper is turned to the advanced Monte Carlo methods for realistic image creation. It offers a new stratified approach for solving the rendering equation. We consider the numerical solution of the rendering equation by separation of integration domain. The hemispherical integration domain is symmetrically separated into 16 parts. First 9 sub-domains are equal size of orthogonal spherical triangles. They are symmetric each to other and grouped with a common vertex around the normal vector to the surface. The hemispherical integration domain is completed with more 8 sub-domains of equal size spherical quadrangles, also symmetric each to other. All sub-domains have fixed vertices and computable parameters. The bijections of unit square into an orthogonal spherical triangle and into a spherical quadrangle are derived and used to generate sampling points. Then, the symmetric sampling scheme is applied to generate the sampling points distributed over the hemispherical integration domain. The necessary transformations are made and the stratified Monte Carlo estimator is presented. The rate of convergence is obtained and one can see that the algorithm is of super-convergent type.

Quasi Monte Carlo approach with uniform separation for solving of the rendering equation, ICNAAM 2007

This paper is directed to the advanced parallel Quasi Monte Carlo (QMC) methods for realistic image synthesis. We propose and consider a new QMC approach for solving the rendering equation with uniform separation. First, we apply the symmetry property for uniform separation of the hemispherical integration domain into 24 equal sub-domains of solid angles, subtended by orthogonal spherical triangles with fixed vertices and computable parameters. Uniform separation allows to apply parallel sampling scheme for numerical integration. Finally, we apply the stratified QMC integration method for solving the rendering equation. The superiority our QMC approach is proved.

Kinetics and mechanism of the interconversion of inverse bicontinuous cubic mesophases

This paper describes time-resolved x-ray diffraction data monitoring the transformation of one inverse bicontinuous cubic mesophase into another, in a hydrated lipid system. The first section of the paper describes a mechanism for the transformation that conserves the topology of the bilayer, based on the work of Charvolin and Sadoc, Fogden and Hyde, and Benedicto and O'Brien in this area. We show a pictorial representation of this mechanism, in terms of both the water channels and the lipid bilayer. The second section describes the experimental results obtained. The system under investigation was 2:1 lauric acid: dilauroylphosphatidylcholine at a hydration of 50% water by weight. A pressure-jump was used to induce a phase transition from the gyroid (Q(II)(G)) to the diamond (Q(II)(D)) bicontinuous cubic mesophase, which was monitored by time-resolved x-ray diffraction. The lattice parameter of both mesophases was found to decrease slightly throughout the transformation, but at the stage where the Q(II)(D) phase first appeared, the ratio of lattice parameters of the two phases was found to be approximately constant for all pressure-jump experiments. The value is consistent with a topology-preserving mechanism. However, the polydomain nature of our sample prevents us from confirming that the specific pathway is that described in the first section of the paper. Our data also reveal signals from two different intermediate structures, one of which we have identified as the inverse hexagonal (H-II) mesophase. We suggest that it plays a role in the transfer of water during the transformation. The rate of the phase transition was found to increase with both temperature and pressure-jump amplitude, and its time scale varied from the order of seconds to minutes, depending on the conditions employed.

Inverse model control using recurrent networks

This paper illustrates how internal model control of nonlinear processes can be achieved by recurrent neural networks, e.g. fully connected Hopfield networks. It is shown that using results developed by Kambhampati et al. (1995), that once a recurrent network model of a nonlinear system has been produced, a controller can be produced which consists of the network comprising the inverse of the model and a filter. Thus, the network providing control for the nonlinear system does not require any training after it has been trained to model the nonlinear system. Stability and other issues of importance for nonlinear control systems are also discussed.

Modelling and control of Hammerstein system using B-spline approximation and the inverse of De Boor algorithm

In this article a simple and effective controller design is introduced for the Hammerstein systems that are identified based on observational input/output data. The nonlinear static function in the Hammerstein system is modelled using a B-spline neural network. The controller is composed by computing the inverse of the B-spline approximated nonlinear static function, and a linear pole assignment controller. The contribution of this article is the inverse of De Boor algorithm that computes the inverse efficiently. Mathematical analysis is provided to prove the convergence of the proposed algorithm. Numerical examples are utilised to demonstrate the efficacy of the proposed approach.

Issues in the assimilation of the sources of a trace gas by inverse modelling

The problem of reconstructing the (otherwise unknown) source and sink field of a tracer in a fluid is studied by developing and testing a simple tracer transport model of a single-level global atmosphere and a dynamic data assimilation system. The source/sink field (taken to be constant over a 10-day assimilation window) and initial tracer field are analysed together by assimilating imperfect tracer observations over the window. Experiments show that useful information about the source/sink field may be determined from relatively few observations when the initial tracer field is known very accurately a-priori, even when a-priori source/sink information is biased (the source/sink a-priori is set to zero). In this case each observation provides information about the source/sink field at positions upstream and the assimilation of many observations together can reasonably determine the location and strength of a test source.

B-spline neural network based digital baseband predistorter solution using the inverse of De Boor algorithm

In this paper a new nonlinear digital baseband predistorter design is introduced based on direct learning, together with a new Wiener system modeling approach for the high power amplifiers (HPA) based on the B-spline neural network. The contribution is twofold. Firstly, by assuming that the nonlinearity in the HPA is mainly dependent on the input signal amplitude the complex valued nonlinear static function is represented by two real valued B-spline neural networks, one for the amplitude distortion and another for the phase shift. The Gauss-Newton algorithm is applied for the parameter estimation, in which the De Boor recursion is employed to calculate both the B-spline curve and the first order derivatives. Secondly, we derive the predistorter algorithm calculating the inverse of the complex valued nonlinear static function according to B-spline neural network based Wiener models. The inverse of the amplitude and phase shift distortion are then computed and compensated using the identified phase shift model. Numerical examples have been employed to demonstrate the efficacy of the proposed approaches.

Efficient non-linear data assimilation in geophysical fluid dynamics

New ways of combining observations with numerical models are discussed in which the size of the state space can be very large, and the model can be highly nonlinear. Also the observations of the system can be related to the model variables in highly nonlinear ways, making this data-assimilation (or inverse) problem highly nonlinear. First we discuss the connection between data assimilation and inverse problems, including regularization. We explore the choice of proposal density in a Particle Filter and show how the ’curse of dimensionality’ might be beaten. In the standard Particle Filter ensembles of model runs are propagated forward in time until observations are encountered, rendering it a pure Monte-Carlo method. In large-dimensional systems this is very inefficient and very large numbers of model runs are needed to solve the data-assimilation problem realistically. In our approach we steer all model runs towards the observations resulting in a much more efficient method. By further ’ensuring almost equal weight’ we avoid performing model runs that are useless in the end. Results are shown for the 40 and 1000 dimensional Lorenz 1995 model.