Explicit Inversion Research Articles

A Monte Carlo-based framework for assessing the value of information and development risk in geothermal exploration

The exploration and development of geothermal energy resources carries considerable financial risk. Due to the cost of drilling, there is often large uncertainty in the prediction of resource potential as well as challenges in optimizing well placement. In this paper, we propose a comprehensive Bayesian framework that accounts for high degrees of geologic uncertainty. Although Bayesian inference methods for prediction and uncertainty quantification are well-established, limitations exist, such as incorporating model realism and reducing the computational burden of simulating a large number of forward models. Using a case study problem, we demonstrate how to turn geologic understanding into a prior probability model for a basin-scale extensional geothermal system. We then use the proposed Bayesian framework, called Bayesian Evidential Learning, to generate posterior temperature predictions constrained to a temperature well without any explicit model inversion. In this approach, the relationship between data and prediction variables is learned by Canonical Correlation Analysis of a training set of models generated by Monte Carlo simulation. Sensitivity analysis results show that temperature in a geothermal target area is most sensitive to the bulk permeability of the basement and basin rock as well as the basal heat flux.

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

Fast Stokesian dynamics

We present a new method for large scale dynamic simulation of colloidal particles with hydrodynamic interactions and Brownian forces, which we call fast Stokesian dynamics (FSD). The approach for modelling the hydrodynamic interactions between particles is based on the Stokesian dynamics (SD) algorithm (J. Fluid Mech., vol. 448, 2001, pp. 115–146), which decomposes the interactions into near-field (short-ranged, pairwise additive and diverging) and far-field (long-ranged many-body) contributions. In FSD, the standard system of linear equations for SD is reformulated using a single saddle point matrix. We show that this reformulation is generalizable to a host of particular simulation methods enabling the self-consistent inclusion of a wide range of constraints, geometries and physics in the SD simulation scheme. Importantly for fast, large scale simulations, we show that the saddle point equation is solved very efficiently by iterative methods for which novel preconditioners are derived. In contrast to existing approaches to accelerating SD algorithms, the FSD algorithm avoids explicit inversion of ill-conditioned hydrodynamic operators without adequate preconditioning, which drastically reduces computation time. Furthermore, the FSD formulation is combined with advanced sampling techniques in order to rapidly generate the stochastic forces required for Brownian motion. Specifically, we adopt the standard approach of decomposing the stochastic forces into near-field and far-field parts. The near-field Brownian force is readily computed using an iterative Krylov subspace method, for which a novel preconditioner is developed, while the far-field Brownian force is efficiently computed by linearly transforming those forces into a fluctuating velocity field, computed easily using the positively split Ewald approach (J. Chem. Phys., vol. 146, 2017, 124116). The resultant effect of this field on the particle motion is determined through solution of a system of linear equations using the same saddle point matrix used for deterministic calculations. Thus, this calculation is also very efficient. Additionally, application of the saddle point formulation to develop high-resolution hydrodynamic models from constrained collections of particles (similar to the immersed boundary method) is demonstrated and the convergence of such models is discussed in detail. Finally, an optimized graphics processing unit implementation of FSD for mono-disperse spherical particles is used to demonstrated performance and accuracy of dynamic simulations of $O(10^{5})$ particles, and an open source plugin for the HOOMD-blue suite of molecular dynamics software is included in the supplementary material.

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

High-order methods gain increased attention in computational fluid dynamics. However, due to the time step restrictions arising from the semi-implicit time stepping for the incompressible case, the potential advantage of these methods depends critically on efficient elliptic solvers. Due to the operation counts of operators scaling with the polynomial degree p times the number of degrees of freedom nDOF, the runtime of the best available multigrid solvers scales with O(p⋅nDOF). This scaling with p significantly lowers the applicability of high-order methods to high orders. While the operators for residual evaluation can be linearized when using static condensation, Schwarz-type smoothers require their inverses on fixed subdomains. No explicit inverse is known in the condensed case and matrix-matrix multiplications scale with p⋅nDOF. This paper derives a matrix-free explicit inverse for the static condensed operator in a cuboidal, Cartesian subdomain. It scales with p3 per element, i.e. nDOF globally, and allows for a linearly scaling additive Schwarz smoother, yielding a p-multigrid cycle with an operation count of O(nDOF). The resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with nDOF for polynomial degrees at least up to 48. Furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a CPU, leading to time-stepping for the incompressible Navier-Stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers.

Explicit inverse of tridiagonal matrix with applications in autoregressive modelling

AbstractWe present the explicit inverse of a class of symmetric tridiagonal matrices, which is almost Toeplitz, except that the first and last diagonal elements are different from the rest. This class of tridiagonal matrices is of special interest in complex statistical models, which uses the first-order autoregression to induce dependence in the covariance structure, for instance, in econometrics or spatial modelling. They also arise in interpolation problems using the cubic spline. We show that the inverse can be expressed as a linear combination of Chebyshev polynomials of the second kind and present results on the properties of the inverse, such as bounds on the row sums, the trace of the inverse and its square and their limits as the order of the matrix increases.

Radon transforms over lower-dimensional horospheres in real hyperbolic space

We study horospherical Radon transforms that integrate functions on the $n$-dimensional real hyperbolic space over horospheres of arbitrary fixed dimension $1\le d\le n-1$. Exact existence conditions and new explicit inversion formulas are obtained for these transforms acting on smooth functions and functions belonging to $L^p$. The case $d=n-1$ agrees with the well-known Gelfand-Graev transform.

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

Performance modeling of pulse detonation engines using the method of characteristics

A quasi-one-dimensional method of characteristics (MOC) model is developed to evaluate the single-cycle gasdynamic flow field and associated propulsive performance of a fully- and partially-filled pulse detonation engine (PDE), and PDEs equipped with diverging nozzles. A detailed description of the PDE thrust chamber flow field is provided to highlight the dominant gasdynamic wave processes encountered in a general single-cycle operation. The MOC model is developed using a simplified unit process approach with an explicit inverse time marching algorithm in order to readily construct the complex thrust chamber flow field along a predefined grid. A grid dependency study is carried out to determine the appropriate grid resolution for the purposes of minimizing computational expense and maximizing numerical accuracy. Lastly, the MOC model is validated with existing numerical and experimental performance data for PDEs operating with oxyhydrogen and hydrocarbon mixtures in both atmospheric and sub-atmospheric environments. It is shown that the current MOC model has good agreement with existing numerical and experimental data for a variety of PDE configurations and operating conditions.

The V-line transform with some generalizations and cone differentiation

The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in . The VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the fundamental theorem of calculus, which we call the cone differentiation theorem.

Configurational Entropy in Multicomponent Alloys: Matrix Formulation from Ab Initio Based Hamiltonian and Application to the FCC Cr-Fe-Mn-Ni System.

Configuration entropy is believed to stabilize disordered solid solution phases in multicomponent systems at elevated temperatures over intermetallic compounds by lowering the Gibbs free energy. Traditionally, the increment of configuration entropy with temperature was computed by time-consuming thermodynamic integration methods. In this work, a new formalism based on a hybrid combination of the Cluster Expansion (CE) Hamiltonian and Monte Carlo simulations is developed to predict the configuration entropy as a function of temperature from multi-body cluster probability in a multi-component system with arbitrary average composition. The multi-body probabilities are worked out by explicit inversion and direct product of a matrix formulation within orthonomal sets of point functions in the clusters obtained from symmetry independent correlation functions. The matrix quantities are determined from semi canonical Monte Carlo simulations with Effective Cluster Interactions (ECIs) derived from Density Functional Theory (DFT) calculations. The formalism is applied to analyze the 4-body cluster probabilities for the quaternary system Cr-Fe-Mn-Ni as a function of temperature and alloy concentration. It is shown that, for two specific compositions (CrFeMnNi and CrFeMnNi), the high value of probabilities for Cr-Fe-Fe-Fe and Mn-Mn-Ni-Ni are strongly correlated with the presence of the ordered phases L1-CrFe and L1-MnNi, respectively. These results are in an excellent agreement with predictions of these ground state structures by ab initio calculations. The general formalism is used to investigate the configuration entropy as a function of temperature and for 285 different alloy compositions. It is found that our matrix formulation of cluster probabilities provides an efficient tool to compute configuration entropy in multi-component alloys in a comparison with the result obtained by the thermodynamic integration method. At high temperatures, it is shown that many-body cluster correlations still play an important role in understanding the configuration entropy before reaching the solid solution limit of high-entroy alloys (HEAs).

The Numerical Solution of Singularly Perturbed Nonlinear Partial Differential Equations in Three Space Variables: The Adaptive Explicit Inverse Preconditioning Approach

Critical comments on the complexity of computational systems and the basic singularly perturbed (SP) concepts are given. A class of several complex SP nonlinear elliptic equations arising in various branches of science, technology, and engineering is presented. A classification of complex SP nonlinear PDEs with characteristic boundary value problems is described. A modified explicit preconditioned conjugate gradient method based on explicit inverse preconditioners is presented. The numerical solution of a characteristic 3D SP nonlinear parabolic model is analytically given and numerical results for several model problems are presented demonstrating both applicability and efficiency of the new computational methods.

Explicit determinants, inverses and eigenvalues of four band Toeplitz matrices with perturbed rows

Abstract In this paper, four-band Toeplitz matrices and four-band Hankel matrices of type I and type II with perturbed rows are introduced. Explicit determinants, inverses and eigenvalues for these matrices are derived by using a nice inverse formula of block bidiagonal Toeplitz matrices.

Inverse elastic surface scattering with far-field data

A rigorous mathematical model and an efficient computational method are proposed to solving the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. We demonstrate how an enhanced resolution can be achieved by using more easily measurable far-field data. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangular slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near (FtN) field data conversion and utilizes the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem. Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction. Results show that the proposed method is capable of stably reconstructing surfaces with resolution controlled by the slab's density.

A Weak-Form Combined Source Integral Equation With Explicit Inversion of the Combined-Source Condition

The combined source integral equation (CSIE) for the electric field on the surface of a perfect electrically conducting scatterer can be discretized very accurately with lowest-order Rao-Wilton-Glisson basis and testing functions if the combined-source (CS) condition is enforced in weak form. We introduce a technique to accelerate the iterative solution for this kind of CSIE. It is demonstrated that the iterative solution of the equation system can be performed very efficiently by explicitly inverting the weak-form CS condition in any evaluation of the forward operator. This reduces the number of unknowns and results in improved convergence behavior at negligible, linear cost. Numerical results demonstrate that the new CSIE outperforms the classical CFIE for high-accuracy simulations.

Explicit inversion formulas for properly Hessenberg and Toeplitz–Hessenberg matrices

ABSTRACT The nice inversion properties of Toeplitz–Hessenberg matrices from a Hessenbergian representation of Catalan's numbers encourage us to provide explicit inversion formulas (in terms of basic arithmetical operations involving entries from the original matrix) for the non-singular Toeplitz–Hessenberg matrices. Our approach is based on an elementary matrix inflation method, joint with the nested sums formulas. These explicit inversion formulas are then extended to apply to the entries of the inverse of every non-singular properly Hessenberg matrix.

Space and time inversions of stochastic processes and Kelvin transform

AbstractLet X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.

A note on Bernstein-Vandermonde matrices

ABSTRACTThe inverse matrix of a Vandermonde matrix with arbitrary field entries of dimension with is well-studied in the literature. Especially, an explicit inversion formula does exist. However in theory as e.g. for determining (analytically) the eigenvalues of (arbitrary) matrix polynomials, Bernstein-Vandermonde matrices must be inverted. This note derives an explicit low-cost inversion formula for this matrix class with entries over a field whose characteristic equals zero. Its derivation requires at most additions and multiplications.

The conic-paraboloid formulae for coarse woody material volume and taper and their approximation

The conic-paraboloid volume equation is receiving increased use with downed coarse woody material (CWM), but the consequences for taper have not been identified mathematically. Requiring that subdivision of a conic-paraboloid yields two smaller conic-paraboloids leads to an exact taper equation intermediate between those of cones and second-order paraboloids. This exact taper equation does not have an explicit inverse, however. An alternative, naive approach does have an explicit inverse, but subdivision does not yield two conic-paraboloids. The exact conic-paraboloid is closely approximated by Fermat’s paraboloid with exponent 7/5. The exact and naive conic-paraboloids match in volume; differences in taper are ≤2.2% of large-end cross-sectional area and ≤5.9% of large-end diameter, while differences in inverse taper are ≤3.7% of total length. Fermat’s paraboloid is always within 1.2% of total volume; differences in taper are ≤0.8% of large-end cross-sectional area and ≤2.0% of large-end diameter, while differences in inverse taper are ≤1.1% of total length. Such differences are negligible given the variety of CWM shapes and practical measurement challenges. Either the exact conic-paraboloid or the corresponding Fermat’s paraboloid provides appropriate equations for estimating the volume and taper of CWM that is intermediate between conical and ordinary paraboloid frusta.

Single scattering tomography with curved detectors

Classical computed tomography assumes that x-rays propagate without scattering. The data is modeled as the two-dimensional Radon transform of the attenuation coefficient of the scanned object. To reconstruct the image, the Radon transform has to be inverted. Recently, several new imaging methods that use singly scattered rays have been suggested. The data in these cases is modeled by integrals of a function along V-lines (two rays joined at a common vertex). In this paper we study the V-line transform arising in Single Scattering Optical Tomography in the case of a single concave detector. In this setup the direction of one of the rays constituting a V-line depends on the common vertex. We derive an explicit inversion formula which allows the recovery of the absorption coefficient. Our results are illustrated with numerical simulations.

Viscoelasticity of short fiber composites in the time domain: from three-phases micromechanics to finite element analyses

Micromechanical models can be used to calculate the mechanical properties of short glass fiber reinforced thermoplastics. In the present work, a three-step framework is used to validate a three-phases micromechanical model (RDI model) in the time domain, since the analysis of technical components by the finite element method is usually carried out in the time domain. The framework includes mechanical characterization, the implementation of the RDI model and a finite element analysis. The characterization delivers necessary information about the material phases of the composite. A dynamic mechanical analysis is performed to characterize the matrix material in order to obtain the linear viscoelastic properties. The mechanical properties of the matrix–fiber interphase are determined with an inverse calculation. In the second step, the RDI model is used to calculate the frequency depended effective stiffness of the composite. A new developed approach transforms the effective stiffness from the frequency domain into the time domain thus avoiding an explicit inverse Laplace–Carson transformation. In the third step, the RDI model is experimentally validated.