Definition of Complex One‐Parameter Generalized Moore–Penrose Inverses Using Differential Transformations

Let denote the set of matrices with the same ray pattern as a complex matrix . If the Moore-Penrose (M-P) inverses of and have the same ray pattern for every , then is said to have ray M-P inverse. In this paper, we give complete characterizations of matrices with ray M-P inverse, which extends some results in [Shao JY, Shan HY, Matrices with signed generalized inverses. Linear Algebra Appl. 2001;322:105–127].

Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints

We consider matrices arising out of the one-way dissection method for solving large sparse systems of linear equations. The systems that we consider are those that may have singular diagonal blocks. Such systems arise in certain fluid flow problems. Gunzberger and Nicholaides proposed a method for resolving the singularity in the diagonal blocks. This method uses the Moore-Penrose pseudoinverse. We propose two improvements to the Gunzberger-Nicholaides procedure: the substitution of a weighted pseudoinverse for the Moore-Penrose pseudoinverse; a more elegant implementation of the back substitution procedure. A stability analysis of both our procedure and the Gunzberger-Nicholaides procedure is given. Both our analysis and empirical tests show that our method has better numerical stability properties than the Gunzberger-Nicholaides procedure. We also implement our procedure on Intel iPSC/1 Hypercube. Our improvement to the back substitution method makes the natural parallelism in the problem easier to exploit. 13 refs., 2 tabs.

In this paper we present a new method to model heterogeneity and flow channeling in petroleum reservoirs—specially reservoirs containing interconnected microfractures. The method is applicable both to conventional and unconventional reservoirs where the interconnected microfractures form the major flow path. The flow equations, which could include flow contributions from matrix blocks of various size, permeability and porosities, are solved by the Laplace transform analytical solutions and finite-difference numerical solutions. The accuracy of flow from and into nano-Darcy matrix blocks is of great interest to those dealing with unconventional reservoirs; thus, matrix flow equations are solved using both pseudo-steady- state (PSS) and un-steady-state (USS) formulations and the results are compared. The matrix blocks can be of different sizes and properties within the representative elementary volume (REV) in the analytical solutions, and within each control volume (CV) in the numerical solutions. While the analytical solutions were developed for slightly compressible rock-fluid linear systems, the numerical solutions are general and can be used for non- linear multi-phase, multi-component flow problems. The mathematical solutions were used to analyze the long-term performance of a gas well and an oil well in two separate unconventional reservoirs. Finally, the formulations were used to assess enhanced oil recovery potential from a typical nano- Darcy matrix block. It is concluded that matrix contribution to flow is very slow in a typical low-permeability unconventional reservoir and much of the enhanced production is from the fluids contained in the microfractures than in the matrix. In addition to field applications, the mathematical formulations and solution methods are presented in a transparent fashion to allow easy utilization of the techniques for reservoir and engineering applications.

This paper is a comprehensive report on test matrices for the generalized inversion of matrices. Two principles are described how to construct singular square or arbitrary rectangular test matrices and their Moore-Penrose inverses. By prescribing the singular values of the matrices or by suitably choosing the free parameters test matrices with condition numbers of any size can be obtained. We also deal with test matrices which are equal to their Moore-Penrose inverse. In addition to many advices how to construct test matrices the paper presents many test matrices explicitly, in particular singular square matrices of ordern, sets of 7×6 and 7×5 matrices of different rank, a set of 5×5 matrices which are equal to their Moore-Penrose inverse and some special test matrices known from literature. For the set of 7×6 parameter matrices also the singular values corresponding to six values of the parameter are listed. For three simple parameter matrices of order 5×4 and 6×5 even test results obtained by eight different algorithms are quoted.

Double-porosity/naturally-fractured reservoir (NFRs) models have traditionally been used to represent the flow and pressure behavior for highly fractured carbonate reservoirs. Given that unconventional reservoirs such as shale oil/gas reservoirs may or may not be considered to be multi-porosity media, the use of the traditional/classical "double-porosity" models may not be adequate (or appropriate). The recent development of anomalous diffusion models has opened the possibility of adapting double-porosity models to estimate reservoir (and related) parameters for unconventional reservoirs. The primary objective of this work is to develop and demonstrate analytical reservoir models that provide (possible) physical explanations for the anomalous diffusion phenomenon. The models considering anomalous diffusion in reservoirs with Euclidean shape are developed using a convolved (i.e., time-dependent) version of Darcy's law. The use of these models can yield a power-law (straight-line) behavior for the pressure and/or rate performance — similar to the fractal reservoir models. The main advantage of using anomalous diffusion models compared to models considering fractal geometry is the reduction from two parameters (i.e., the Fractal Dimension and the Conductivity Index) to only one parameter (i.e., the anomalous diffusion exponent). However, the anomalous diffusion exponent does not provide information about the geometry or spatial distribution of the reservoir properties. To provide an alternative explanation for the anomalous diffusion phenomenon in petroleum reservoirs, we have developed double-porosity models considering matrix blocks with fractal geometry and fracture networks with either radial or fractal fracture networks. The flows inside the matrix blocks and the fractal fracture network assume that Darcy's Law is valid in its space-dependent (fractal) form, whereas the classical version of Darcy's Law is assumed for the radial fracture network case. The transient interporosity transfer is modeled using the classical convolution schemes given in the literature. For the system composed by a radial fracture network and fractal matrix blocks, we have investigated three cases by changing the producing conditions for the well and the flowing conditions for the matrix blocks: These three cases are: A well producing at a constant-rate and closed matrix blocks,A well producing at a constant-rate and "infinite-acting" matrix blocks, andA well producing at a variable-rate (time-dependent inner boundary condition) and "infinite-acting" matrix blocks. We have defined the matrix blocks to be "infinite-acting" in order to represent the nano/micro permeability of shale reservoirs. For the system defined by a fractal fracture network and infinite-acting fractal matrix blocks, we have investigated the influence of the fractal parameters (both matrix and fracture network) in the pressure- and rate-transient performance behaviors. We have defined the periods of flow that can be observed in these sorts of systems and we have developed analytical solutions for pressure-transient analysis. We demonstrate that the use of the convolved version of Darcy's Law results in a model very similar to the diffusivity equation for double-porosity systems (which incorporates transient interporosity flow). In performing this work, we establish the following obervations/conclusions using our new solutions: We have found that the assumption of a well producing at variable-rate (time-dependent inner boundary condition) has a more significant impact on the pressure (and derivative) functions and hinders the effects of the properties of the reservoir.We have demonstrated that the anomalous diffusion phenomena in unconventional reservoirs can be related to their multi-porosity nature.The pressure and pressure derivative responses may be used in the diagnosis of flow periods and the evaluation of reservoir parameters in unconventional reservoirs.

Approximation and continuity of Moore–Penrose inverses of orthogonal row block matrices

On level-2 condition number for the weighted Moore–Penrose inverse

The previous matrix-fracture transfer models overlooked the effect of the fracture surface conditions on oil recovery by water imbibition. This paper presents an improved matrix-fracture-transfer model and approximate analytical solutions for waterflooding of oil reservoirs by hindered-capillary imbibition into matrix blocks surrounded by natural fractures. The rectangular and spherical type simpler shapes are considered to represent the matrix blocks formed by intersecting fractures. The immiscible displacement equations are simplified by means of the unit-end-point-mobility-ratio approximation and solved analytically. The restricted and hindered capillary imbibition phenomenon is considered in order to account for the finite-skin effect of the matrix block surfaces by various processes, including the formation of a stationary fluid film or an actual skin because of formation damage by various processes, such as deposition of minerals and other debris, and adverse rock-fluid interactions. Then, full scale, and short and long-term analytical solutions are presented and validated by experimental data. By means of these analytical models, various laboratory experimental data are analyzed, and it is demonstrated that the condition of the matrix block surface is an important factor in hindering the rate of water imbibition into matrix blocks and the matrix-to-fracture oil expulsion, and therefore reducing the oil recovery factor. Comparison with previous simplified models shows that the present model is superior and can accurately predict the matrix-to-fracture oil transfer over the full range of the oil recovery period, while the previous models can only represent either the early- or the late-time behavior with limited accuracy. In addition, the effect of the reservoir rock wettability on the rate of oil recovery by hindered-capillary imbibition is investigated by means of the capillary dispersion coefficient, determined by correlating the experimental data using the new models. This study reveals that there is a strong relationship between the oil recovery factor and the capillary dispersion coefficient. The analytical models developed here can be utilized for rapid prediction of oil recovery from naturally fractured reservoirs undergoing a water flooding process.

We consider the one-parameter conjugated analogs of Sylvester type matrix equations. On the basis of simple transformations we obtainthe equivalent matrix equation containing only the unknown matrix which should be determined. Next, using the apparatus of Kroneckerproducts of matrices, the analytical solution of the problem was obtained, which is limited in practical applications, but serves as a basis for the development of numerical-analytical methods for solving the original problem. Serial and parallel numerical and analytical solution methods are proposed based on the differential transformations G. E. Pukhov. With sequential numerical-analytical method we operate with numeric recursive procedures in the first stage of the calculation, and analytical relations – in the second stage. In parallel numerical-analytical method we operate with linear hypersystem of numerical equations in the first stage of the calculation, and analytical relations – in the second stage. For all the methods the relevant conditions for the unique solvability of the problem were obtained. A model example was considered for which using numerical and analytical methods an exact solution of the Taylor was obtained. The proposed numerical-analytical techniques can be efficiently implemented by means of modern information technology.

On the Moore-Penrose Pseudo Inverse of the Incidence Matrix for Weighted Undirected Graph

Software implementations of analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore-Penrose mat-rices are presented. Analytical methods are based on the first two Moore-Penrose con-ditions: in both cases the same solution to the problem is obtained, as it should be. Numerical-analytical methods are based on the obtained analytical relations and differential Pukhov transformations. Computational methods are implemented using modern infor-mation technologies, in particular Python programming language, NumPy and SymPy libraries. Software implementations consist of programs for entering the di-mensions and elements of a given matrix, implementation of an analytical method, de-termination of matrix elements derivatives and their values, calculation of matrix discretes and restoration of original matrices, as well as determining pseudo-inverse matrix. In addi-tion, the software implementation of the numerical-analytical method also contains a cus-tom implemented functions for determining the given matrix elements derivatives and for calculating the matrix discretes at the approximation center with a given scaling coeffi-cient. Algorithm flowcharts and Python source codes for all listed programs are presented. The software implementation is tested on a model example with a square matrix, for which an analytical solution is obtained using two approaches - the determination of the usual inverse one-parameter matrix and the decomposition method for determining the same matrix. This matrix is defined in exactly the same way as numerical-analytical decomposition methods.

On a new generalized inverse for matrices of an arbitrary index

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore–Penrose inverse that is not a classical inverse of V. This also answers a recent open problem on infinite matrices.

Linear regression (LR) is a core model in supervised machine learning performing a regression task. One can fit this model using either an analytic/closed-form formula or an iterative algorithm. Fitting it via the analytic formula becomes a problem when the number of predictors is greater than the number of samples because the closed-form solution contains a matrix inverse that is not defined when having more predictors than samples. The standard approach to solve this issue is using the Moore–Penrose inverse or the L2 regularization. We propose another solution starting from a machine learning model that, this time, is used in unsupervised learning performing a dimensionality reduction task or just a density estimation one—factor analysis (FA)—with one-dimensional latent space. The density estimation task represents our focus since, in this case, it can fit a Gaussian distribution even if the dimensionality of the data is greater than the number of samples; hence, we obtain this advantage when creating the supervised counterpart of factor analysis, which is linked to linear regression. We also create its semisupervised counterpart and then extend it to be usable with missing data. We prove an equivalence to linear regression and create experiments for each extension of the factor analysis model. The resulting algorithms are either a closed-form solution or an expectation–maximization (EM) algorithm. The latter is linked to information theory by optimizing a function containing a Kullback–Leibler (KL) divergence or the entropy of a random variable.