Efficient computer manipulation of tensor products with applications to multidimensional approximation

A new approach to the application of direct method in petroleum reservoir simulation is introduced, This new approach is based on a modified form of the alternating diagonal ordering scheme that will rearrange each type of elements along different secondary diagonals in the coefficient matrix. In a typical 2-dimensional 3-phase petroleum reservoir simulation problem of 345 grid system, the new method is substantially faster than the band algorithm applied to the matrix resulting from the standard ordering. Introduction In petroleum reservoir simulation, a linear system of equations is obtained. The coefficient matrix of such a linear system takes different forms depending on the ordering of the grid system. The resulting system of equations can be solved either by direct methods or by iterative methods. Direct solutions are more reliable, but they normally require larger storage space and more computer time. Therefore, a development of a better direct method approach is needed. The ordering technique can be improved by the use of processes which are geared specifically to the direct solution. The method is the generate-and-solve algorithm used by Woo, et al. The drawback of this technique is the fact it requires large computer storage and overhead work to store and generate the nonlooping code. The yet known systematic optimal ordering scheme in petroleum reservoir simulation is the alternating diagonal scheme developed by Price and Coats. However, there is a major drawback to this ordering scheme. In the alternating diagonal scheme, the elements in the upper right and lower left corners of the coefficient matrix do not form perfect secondary diagonals. Thus, the solution of the resulting matrix requires band algorithm which treats the zeros within the band as nonzeros. The treatment of the zeros as nonzeros makes the band matrix technique require larger storage space and more computational labor. In this paper a modification to the alternating diagonal ordering is proposed. This modification will rearrange each type of elements along different secondary diagonals in the coefficient matrix. By this rearrangement no new element will be introduced in the process of factorization except in the lower right process of factorization except in the lower right corner of the composite matrix. Also the location of the original and new elements are known beforehand. This enables us to develop a general procedure to solve the system of equations in reservoir simulation. This general procedure eliminates the overhead calculations and extra storage needed for the generate-and-solve algorithm. Also this general procedure would eliminate the extra calculations required procedure would eliminate the extra calculations required on the zero elements in the band matrix algorithm. A NEW DIRECT SOLUTION APPROACH The new direct solution approach will be outlined to solve the following system of equations (1) The new approach begins with the definition of a new ordering scheme. Then the LU factorization will be applied to this new ordering. Finally the implementation of the new ordering scheme in typical reservoir simulators will be outlined. RESTRICTED ALTERNATING DIAGONAL ORDERING Here a modification to the alternating diagonal ordering that produces perfect secondary diagonals in the original A matrix is proposed. This ordering will be called restricted alternating diagonal (RAD). Figure 1 depicts the RAD ordering. Figure 2 shows the resulting coefficient matrix corresponding to RAD ordering scheme. The numbering of this scheme must go along the shortest diagonal to reduce the band width of the new nonzero elements that will be generated later in the process of factorization. process of factorization.

The approximation of functions and linear functionals: best vs. good approximation

An efficient algorithm based on group theory and the Woodbury formula for the dynamic responses of periodic structures

Residual gradient (RG) was proposed as an alternative to TD(0) for policy evaluation when function approximation is used, but there exists little formal analysis comparing them except in very limited cases. This paper employs techniques from online learning of linear functions and provides a worst-case (non-probabilistic) analysis to compare these two types of algorithms when linear function approximation is used. No statistical assumptions are made on the sequence of observations, so the analysis applies to non-Markovian and even adversarial domains as well. In particular, our results suggest that RG may result in smaller temporal differences, while TD(0) is more likely to yield smaller prediction errors. These phenomena can be observed even in two simple Markov chain examples that are non-adversarial.

Sometimes it is desirable to visualize complex-valued functions in polar co-ordinates that always have difficulties. But actually it is sufficient to visualize separately the real and imaginary parts of a complex-valued function. For a fast visualization process, local spline interpolation of functions from two variables is the most convenient in applications and gives approximations with the required order of accuracy. This paper deals with local complex-valued splines, constructed using tensor product spline interpolation in a disc with a centre of zero and a radius of 1. For constructing the tensor product we use local basis splines of a radial variable and local complex-valued basis splines of an angular variable. For the construction of the grid we consider a number of circles in the disc of radius 1 with a center of zero, and get a number of points on the boundary of this disc, arranged from the centre to the edge. The points at which those lines cross each circle form the grid nodes. The approximation is constructed separately in each elementary segment, formed by two arcs and two line segments. For the approximation of a complex-valued function we use the values of this function in several nodes near this elementary segment and the tensor product of basis splines. The order of the approximation depends on the splines’ properties which we use in the tensor product. In this paper we use local exponential and polynomial splines of second and third order approximation.

Sparse p-version BEM for first kind boundary integral equations with random loading

The task of simultaneous localization and mapping (SLAM) is to build environmental map and locate the position of mobile robot at the same time. FastSLAM 2.0 is one of powerful techniques to solve the SLAM problem. However, there are two obvious limitations in FastSLAM 2.0, one is the linear approximations of nonlinear functions which would cause the filter inconsistent and the other is the “particle depletion” phenomenon. A kind of PSO & H∞-based FastSLAM 2.0 algorithm is proposed. For maintaining the estimation accuracy, H∞ filter is used instead of EKF for overcoming the inaccuracy caused by the linear approximations of nonlinear functions. The unreasonable proposal distribution of particle greatly influences the pose state estimation of robot. A new sampling strategy based on PSO (particle swarm optimization) is presented to solve the “particle depletion” phenomenon and improve the accuracy of pose state estimation. The proposed approach overcomes the obvious drawbacks of standard FastSLAM 2.0 algorithm and enhances the robustness and efficiency in the parts of consistency of filter and accuracy of state estimation in SLAM. Simulation results demonstrate the superiority of the proposed approach.

The goal of this paper is to investigate linear approximations of random functions and permutations. Our motivation is twofold. First, before the distinguishability of a practical cipher from an ideal one can be analysed, the cryptanalyst must have an accurate understanding of the statistical behaviour of the ideal cipher. Secondly, this issue has been neglected both in old and in more recent studies, particularly when multiple linear approximations are being used simultaneously. Traditional models have been based on the average behaviour and simplified using other assumptions such as independence of the linear approximations. Multidimensional cryptanalysis was introduced to avoid making artificial assumptions about statistical independence of linear approximations. On the other hand, it has the drawback of including many trivial approximations that do not contribute to the attack but just cause a waste of time and memory. We show for the first time in this paper that the trivial approximations reduce the degree of freedom of the related χ2 distribution. Previously, the affine linear cryptanalysis was proposed to allow removing trivial approximations and, at the same time, admitting a solid statistical model. In this paper, we identify another type of multidimensional linear approximation, called Davies-Meyer approximation, which has similar advantages, and present full statistical models for both the affine and the Davies-Meyer type of multidimensional linear approximations. The new models given in this paper are realistic, accurate and easy to use. They are backed up by standard statistical tools such as Pearson’s χ2 test and finite population correction and demonstrated to work accurately using practical examples.

The optimal minimax solution to the N segment piecewise linear approximation of arbitrary convex differentiable functions over a finite range is described. The optimal solution is uniquely described by the derivatives at N distinct points. The optimality of the solution is proven and a recursive descent algorithm is proposed. The efficacy of the algorithm and optimality of the solution are demonstrated in example solutions for commonly used functions.

The tensor product is a fundamental mathematical concept with applications spanning linear algebra, graph theory, quantum computing, and representation theory. In graph theory, the tensor product provides a framework for analyzing structural relationships, particularly through the study of complete graphs, which yield complex networks from simple structures. Closely related is the Kronecker product of matrices, an essential tool for investigating tensor products via adjacency matrices. The Kronecker product preserves key algebraic properties, including linearity, distributivity, and associativity, and has played a central role in matrix analysis, systems theory, and signal processing. This work presents the definitions and core properties of the tensor product, supported by illustrative examples with complete graphs, and explores the Kronecker product along with its fundamental properties and theorems. By combining theoretical foundations with applications, the study offers both conceptual insights and practical perspectives on these algebraic constructions.

This article will explore and discuss the nature of the solution to the problems related to the Linear Programming Problem(LPP) and system of Linear equations if the coefficient matrix is any of the matrix Symmetric, Skew symmetric, Idempotent, Involutory and Orthogonal matrices

The regularization network (RN) is a network for learning input‐output mapping proposed by Poggio and Girosi from the viewpoint of learning = function approximation. the RN, which is derived from the standard regularization theory, is extended to an approximation of multivalued mappings from examples to make the learning of inverse models of nonlinear systems possible. This extension is called multivalued regularization network (MVRN). the MVRN is derived based on the multivalued standard regularization theory (MVSRT), which in turn is a based on direct representation of the mulitivalued functions by using Kronecker's tensor product. In the MVRN, the learning of the weight parameters from the traning data results in simultaneous linear equations that are similar to those of conventional regularization networks. In the present theory, the clustering operations for separating training data into those correponding to single‐valued element functions are unnecessary. As with the conventional methods, the radial basis function (RBF), generalized radial basis function (GRBF), spline approximations, and Hyper‐BF networks can be extended to those of multivalued functions by specializaing MVRN or by marking the number of basis functions smaller than the number of training samples. Direct representation of the multivalued function used in the present theory has broad application as a basic equation for more general classes of networks to learn multivalued mappings.

Positivity for non-Metzler systems and its applications to stability of time-varying delay systems

This paper presents an approach to combinatorial optimization problems using linear function approximation (LFA) to solve the Travelling Salesman Problem (TSP). We create a reinforcement learning model in which we parameterize our policy using linear function approximation instead of the more commonly used neural networks. We then evaluated our models based on two factors: training time and optimality. When we compared our results with a state-of-the-art neural network solver, we found that our model was able to solve the TSP accurately while using drastically less computational resources and time to train than the neural network algorithm (Kool et al., 2019).

An algorithm for low-rank approximation of bivariate functions using splines