Modified Cholesky Factorization Research Articles

A Line-Search-Based Algorithm for Multiphase Flash Calculations with CO2-Hydrocarbon System.

Multiphase flash calculations are pivotal in compositional simulation, necessitating a robust and efficient computational algorithm. In this work, we have developed a line-search-based algorithm framework for stability analysis and multiphase flash calculations. This algorithm is rooted in the modified Newton step and line-search method. The modified Newton step, derived from modified Cholesky factorization, ensures a descent direction, while the line search determines the degree of decrease. This combination facilitates convergence even in challenging regions for phase stability analysis and multiphase flash calculations, exhibiting superlinear convergence speed. Unlike traditional approaches that rely on successive substitution iteration and may resort to Newton iteration only if the Hessian matrix is positive definite, our algorithm incorporates modification via modified Cholesky factorization upon encountering a nonpositive definite Hessian matrix. We tested our algorithm with several classical fluids, demonstrating its efficiency and robustness. Furthermore, we assessed the algorithm's performance by computing the pressure-composition diagram for the CO2-hydrocarbon system, where all calculations achieved rapid convergence without failure. This newly developed algorithm for phase stability analysis and multiphase flash calculations represents a significant advancement for compositional or chemical process simulations.

A time-varying GARCH mixed-effects model for isolating high- and low- frequency volatility and co-volatility

This article studies long-term, short-term volatility and co-volatility in stock markets by introducing modelling strategies to the multivariate data analysis that deal with serially correlated innovations and cross-section dependence. In particular, it presents an innovative mixed-effects model through a GARCH process, allowing for heterogeneity effects and time-series dynamics. We propose a non-parametric regression model of the penalized low-rank smoothing spline to present time trends into the variance and covariance equations. The strategy provides flexible modelling of the low-frequency volatility and co-volatility in equity markets. The decomposed low-frequency matrix was modelled using the modified Cholesky factorization. The Hamiltonian Monte Carlo technique is implemented as a Bayesian computing process for estimating parameters and latent factors. The advantage of our modelling strategy in empirical studies is highlighted by examining the effect of latent financial factors on a panel across 10 equities over 110 weekly series. The model can differentiate non-parametrically dynamic patterns of high and low frequencies of variance–covariance structural equations and incorporate economic features to predict variabilities in stock markets regarding time-series evidence.

Asymmetric Density Fitting with Modified Cholesky Decomposition Applied to Second-Order Electron Propagator.

Computation of molecular orbital electron repulsion integrals (MO-ERIs) as a transformation from atomic orbital ERIs (AO-ERIs) is the bottleneck of second-order electron propagator calculations when a single orbital is studied. In this contribution, asymmetric density fitting is combined with modified Cholesky decomposition to generate efficiently the required MO-ERIs. The key point of the presented algorithms is to keep track of integrals through partial contractions performed on three-center AO-ERIs; these contractions are stored in RAM instead of the AO-ERIs. Two implementations are provided, an in-core, which reduces the arithmetic and memory scaling factors as compared to the four-center AO-ERIs contraction method, and a semidirect, which overcomes memory limitations by evaluating antisymmetrized MO-ERIs in batches. On the numerical side, the proposed approach is fast and stable. The bad effects due to ill conditioning, namely, several negative and close to zero eigenvalues due to machine round off errors of the matrix associated with the density fitting process, are effectively controlled by means of a modified Cholesky factorization that avoids the matrix inversion needed to perform the asymmetrical density fitting implementation. The numerical experience presented shows that the in-core implementation is highly competitive to perform calculations on medium and large basis sets, while the semidirect implementation has small variations in time by changes in the available memory. The general applicability is illustrated on a set of selected relatively large-size molecules.

Volume-based phase stability analysis including capillary pressure

The effect of capillary pressure on phase equilibrium is very important in tight formations, such as shale oil and gas reservoirs. In confined fluids, the capillary pressure induced by highly curved interfaces causes bubble points suppression and an inflated dew point locus, with a shift of cricondentherm points towards higher temperatures, as compared to the bulk fluid. In the conventional phase stability testing including capillary pressure, the problem is solved using the classical tangent plane distance (TPD) function (related to the Gibbs free energy surface) and mole numbers as primary variables. In this work, a volume-based approach (in which the equation of state needs not to be solved for volume) is used to solve this problem, as a bound and linear inequality constrained minimization of a TPD function with respect to the component molar densities. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure (to ensure that iterates remain in the feasible domain and the objective function is decreased) are used in Newton iterations; a proper change of variables strengthens the robustness. The Weinaug-Katz equation (widely used in chemical and petroleum industry), which is a function of molar densities only, is used for interfacial tensions; the additional partial derivatives in the gradient vector and Hessian matrix have very simple expressions, unlike in conventional formulations. The proposed method is tested for two hydrocarbon mixtures (an oil and a gas condensate) in a wide range of pressures, temperatures and curvature radii, with a special attention paid to the convergence behavior near the singularities, that is, the stability test limit locus (STLL) and the spinodal. The method proves to be highly robust and exhibits fast convergence, showing a similar convergence behavior and almost the same computational effort as in the case of a bulk fluid. The results are only slightly different from conventional methods (except at high curvatures near the cricondentherm and at low pressures); differences are due to the fact that the dependence of capillary pressure on composition is taken into account in deriving the stationarity conditions, unlike in the conventional approach. The proposed method is not dependent on the thermodynamic model and can be used with any capillary pressure representation in which the interfacial tension model is explicit in molar densities.

Designing stellarator coils by a modified Newton method using FOCUS

To find the optimal coils for stellarators, nonlinear optimization algorithms are applied in existing coil design codes. However, none of these codes have used the information from the second-order derivatives. In this paper, we present a modified Newton method in the recently developed code FOCUS. The Hessian matrix is calculated with analytically derived equations. Its inverse is approximated by a modified Cholesky factorization and applied in the iterative scheme of a classical Newton method. Using this method, FOCUS is able to recover the W7-X modular coils starting from a simple initial guess. Results demonstrate significant advantages.

Estimating MA Parameters through Factorization of the Autocovariance Matrix and an MA‐Sieve Bootstrap

A new method to estimate the moving‐average (MA) coefficients of a stationary time series is proposed. The new approach is based on the modified Cholesky factorization of a consistent estimator of the autocovariance matrix. Convergence rates are established, and the new estimates are used to implement an MA‐type sieve bootstrap. Finite‐sample simulations corroborate the good performance of the proposed methodology.

A volume-based approach to phase equilibrium calculations at pressure and temperature specifications

Phase equilibrium calculations at pressure and temperature specifications, consisting in the minimization of the Gibbs free energy with respect to mole numbers, are the most commonly used and are well documented in the literature. A very attractive alternative is given by the volume-based calculations, in which the volume and mole numbers, which are the natural variables for pressure explicit equations of state (EoS) are treated as independent variables; in this case pressure equality is an additional equation for each phase and there is no need to solve the EoS for volume. The problem is a bound- and linear inequality-constrained optimization problem; the objective function is a formal Gibbs free energy. A major disadvantage is that, unlike in conventional calculations, the successive substitution method cannot be used, thus robust modified Newton iterations are required. Using the block structure of the Hessian matrix, a link is established between conventional and volume-based Newton iterations and it is shown that the latter are inherently slower than the former due to a lower implicitness level in the Hessian. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure (ensuring that iterates remain in the feasible domain and the objective function is decreasing between two iterations) are used in Newton iterations. Numerical experiments carried out on several test mixtures show that two-phase volume-based flashes initialized from the ideal equilibrium constants pose systematic convergence problems near phase boundaries, while the algorithm is rapid and robust when initial guesses from stability testing are used. The proposed algorithm is not dependent on the thermodynamic model and any pressure explicit EoS can be used, provided the required partial derivatives are available.

Volume-based phase stability testing at pressure and temperature specifications

Conventional phase equilibrium calculations at temperature and pressure specifications require the resolution of the equation of state (EoS). In volume-based calculations, the EoS must not be solved for volume, which is a primary variable. The tangent plane distance (TPD) function can be expressed in terms of mole numbers and volume or of component molar densities. The stationary points of the TPD function, as well as the location of the stability test limit locus (STLL) are different for the two formulations. A modified TPD function in terms of mole numbers and volume is proposed here. Using the block structure of the Hessian matrix, it is shown that Newton iterations in volume-based stability are inherently slower than those in the conventional PT stability, since the Hessian matrix is evaluated at a lower implicitness level. The minimization of several TPD functions (in volume-based and conventional stability testing) is analyzed using various sets of independent variables and scaling procedures. A modified Cholesky factorization and a two-stage line search procedure ensure a sequence of decreasing TPD functions in all cases. The proposed methods are tested for several mixtures, with emphasis on the vicinities of singularities (STLL and spinodal). With proper scaling, the modified Newton iterations are robust and converge very fast for most conditions and reasonably fast for very difficult conditions. The proposed algorithm is not model-dependent; any pressure explicit EoS can be used, provided the required partial derivatives are available.

Modified Cholesky Riemann Manifold Hamiltonian Monte Carlo: exploiting sparsity for fast sampling of high-dimensional targets

Riemann manifold Hamiltonian Monte Carlo (RMHMC) has the potential to produce high-quality Markov chain Monte Carlo output even for very challenging target distributions. To this end, a symmetric positive definite scaling matrix for RMHMC is proposed. The scaling matrix is obtained by applying a modified Cholesky factorization to the potentially indefinite negative Hessian of the target log-density. The methodology is able to exploit the sparsity of the Hessian, stemming from conditional independence modeling assumptions, and thus admit fast implementation of RMHMC even for high-dimensional target distributions. Moreover, the methodology can exploit log-concave conditional target densities, often encountered in Bayesian hierarchical models, for faster sampling and more straightforward tuning. The proposed methodology is compared to alternatives for some challenging targets and is illustrated by applying a state-space model to real data.

Fast and robust phase stability testing at isothermal-isochoric conditions

In this paper, a robust and efficient phase stability testing procedure at isotherm-isochoric conditions is proposed. The tangent plane distance function is expressed in terms of component molar densities. Convergence behavior using various sets of independent variables is analyzed and it is shown that the choice of independent variables is very important for both robustness and computational speed. A particular attention is paid for conditions in which the Hessian matrix is notoriously ill-conditioned, that is, in the vicinity of the stability test limit locus (STLL) and of the spinodal, where the convergence is slow and problematic. It is found that poor conditioning occurs also at certain conditions in the two-phase region at low temperatures and towards high densities. A modified Cholesky factorization (to ensure a descent direction) and a line search procedure are used in Newton iterations. The equation of state (EoS) must not be solved for volume and any pressure explicit EoS can be used; only expressions of pressure, fugacity and their partial derivatives with respect to mole numbers are required. The proposed method is tested on several examples from the literature, using a refined grid in the temperature-molar density plane. The variables obtained by scaling the ideal part of the Hessian matrix lead by far to the most robust formulation, with no failure observed for the investigated mixtures, and much faster than previous methods, with a low average number of iterations required for convergence for all mixtures.

Fast implementation of kernel simplex volume analysis based on modified Cholesky factorization for endmember extraction

Endmember extraction is a key step in the hyperspectral image analysis process. The kernel new simplex growing algorithm (KNSGA), recently developed as a nonlinear alternative to the simplex growing algorithm (SGA), has proven a promising endmember extraction technique. However, KNSGA still suffers from two issues limiting its application. First, its random initialization leads to inconsistency in final results; second, excessive computation is caused by the iterations of a simplex volume calculation. To solve the first issue, the spatial pixel purity index (SPPI) method is used in this study to extract the first endmember, eliminating the initialization dependence. A novel approach tackles the second issue by initially using a modified Cholesky factorization to decompose the volume matrix into triangular matrices, in order to avoid directly computing the determinant tautologically in the simplex volume formula. Theoretical analysis and experiments on both simulated and real spectral data demonstrate that the proposed algorithm significantly reduces computational complexity, and runs faster than the original algorithm.

Bounds for the Distance to the Nearest Correlation Matrix

In a wide range of practical problems correlation matrices are formed in such a way that, while symmetry and a unit diagonal are assured, they may lack semidefiniteness. We derive a variety of new upper bounds for the distance from an arbitrary symmetric matrix to the nearest correlation matrix. The bounds are of two main classes: those based on the eigensystem and those based on a modified Cholesky factorization. Bounds from both classes have a computational cost of $O(n^3)$ flops for a matrix of order $n$ but are much less expensive to evaluate than the nearest correlation matrix itself. For unit diagonal $A$ with $|a_{ij}|\le 1$ for all $i\ne j$ the eigensystem bounds are shown to overestimate the distance by a factor at most $1+n\sqrt{n}$. We show that for a collection of matrices from the literature and from practical applications the eigensystem-based bounds are often good order of magnitude estimates of the actual distance; indeed the best upper bound is never more than a factor $5$ larger than a related lower bound. The modified Cholesky bounds are less sharp but also less expensive, and they provide an efficient way to test for definiteness of the putative correlation matrix. Both classes of bounds enable a user to identify an invalid correlation matrix relatively cheaply and to decide whether to revisit its construction or to compute a replacement, such as the nearest correlation matrix.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Constraint aggregation for rigorous global optimization

In rigorous constrained global optimization, upper bounds on the objective function help to reduce the search space. Obtaining a rigorous upper bound on the objective requires finding a narrow box around an approximately feasible solution, which then must be verified to contain a feasible point. Approximations are easily found by local optimization, but the verification often fails. In this paper we show that even when the verification of an approximate feasible point fails, the information extracted from the results of the local optimization can still be used in many cases to reduce the search space. This is done by a rigorous filtering technique called constraint aggregation. It forms an aggregated redundant constraint, based on approximate Lagrange multipliers or on a vector valued measure of constraint violation. Using the optimality conditions, two-sided linear relaxations, the Gauss---Jordan algorithm and a directed modified Cholesky factorization, the information in the redundant constraint is turned into powerful bounds on the feasible set. Constraint aggregation is especially useful since it also works in a tiny neighborhood of the global optimizer, thereby reducing the cluster effect. A simple introductory example demonstrates how our new method works. Extensive tests show the performance on a large benchmark.

Fast and robust algorithm for calculation of two-phase equilibria at given volume, temperature, and moles

We develop a new algorithm for the calculation of phase splitting at constant volume, temperature, and moles. The method is based on the direct minimization of the total Helmholtz free energy of the mixture with respect to the mole- and volume-balance constraints. The algorithm uses the Newton–Raphson minimization method with line-search and modified Cholesky factorization of the Hessian to produce a sequence of states with decreasing values of the total Helmholtz free energy. The algorithm is initialized using an initial guess that is constructed using the results of the constant volume stability testing. The speed and robustness of the algorithm are demonstrated by a number of examples of two-phase equilibrium calculations.

Solving the Simulations Equations Arising in the Rigid-Plastic Finite Element Analysis by a Modified Cholesky Factorization Method

Solving the Simulations Equations Arising in the Rigid-Plastic Finite Element Analysis by a Modified Cholesky Factorization Method

Solving the Simulations Equations Arising in the Rigid-Plastic Finite Element Analysis by a Modified Cholesky Factorization Method

Solving the Simulations Equations Arising in the Rigid-Plastic Finite Element Analysis by a Modified Cholesky Factorization Method

A regularized Newton method for degenerate unconstrained optimization problems

A regularized Newton method is presented in this paper to solve unconstrained nonconvex minimization problems without the nonsingularity assumption of solutions. The modified Cholesky factorization method proposed by Cheng and Higham is employed to calculate the regularized Newton step. Under suitable conditions, the global convergence of the regularized Newton method is established, and the fast local convergence result is achieved under the local error bound conditions. Some preliminary numerical results are also reported.

A new method for three-carrier GNSS ambiguity resolution

A new method for resolving the carrier-phase integer ambiguity in Global Navigation Satellite Systems (GNSS) is presented: the MOdified Cholesky factorization for Ambiguity (MOCA) resolution. The characteristics and features of this method are described and results obtained using a software simulator and an emulator are presented to validate its efficiency. The results are then compared to those obtained using another existing method and good performance of the MOCA method in new GNSS systems is shown. Furthermore, the proposed method yields accurate results even when short time spans are used or when there are poor estimations of measurement error, making it immune to non-ideal conditions and ultimately a practical solution for real applications.

The cholesky factorization in interior point methods

The paper concerns the Cholesky factorization of symmetric positive definite matrices arising in interior point methods. Our investigation is based on a property of the Cholesky factorization which interprets “small” diagonal values during factorization as degeneracy in the scaled optimization problem. A practical, scaling independent technique, based on the above property, is developed for the modified Cholesky factorization of interior point methods. This technique increases the robustness of Cholesky factorizations performed during interior point iterations when the optimization problem is degenerate. Our investigations show also the limitations of interior point methods with the recent implementation technology and floating point arithmetic standard. We present numerical results on degenerate linear programming problems of NETLIB.