Fast Direct Methods Research Articles

Preliminary study on direct measurements and diagnostics for chemical reaction dynamics of NOx by using laser wavelength modulation spectroscopy

Studies on the kinetics of gas-phase chemical reactions currently rely on calculations or simulations and lack simple, fast, and accurate direct measurement methods. We developed a tunable laser molecular absorption spectroscopy measurement system to achieve direct measurements of such reactions by using wavelength modulated spectroscopy and performed online measurements and diagnostics of molecular concentration, reaction temperature, and pressure change during the redox reaction of ozone with nitrogen oxides (NOx) with 0.1 s temporal resolution. This study provides a promising diagnostic tool for studying gas-phase chemical reaction kinetics.

Sensitive SARS-CoV-2 detection, air travel Covid-19 testing, variant determination and fast direct PCR detection, using ddPCR and RT-qPCR methods.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) monitoring in air traffic is important in the prevention of the virus spreading from abroad. The gold standard for SARS-CoV-2 detection is RT-qPCR; however, for early and low viral load detection, a much more sensitive method, such as droplet digital PCR (ddPCR), is required. Our first step was to developed both, ddPCR and RT-qPCR methods, for sensitive SARS-CoV-2 detection. Analysis of ten swab/saliva samples of five Covid-19 patients in different stages of disease showed positivity in 6/10 samples with RT-qPCR and 9/10 with ddPCR. We also used our RT-qPCR method for SARS-CoV-2 detection without the need of RNA extraction, obtaining results in 90-120 minutes. We analyzed 116 self-collected saliva samples from passengers and airport staff arriving from abroad. All samples were negative by RT-qPCR, while 1 was positive, using ddPCR. Lastly, we developed ddPCR assays for SARS-CoV-2 variants identification (alpha, beta, gamma, delta/kappa) that are more economically advantageous when compared to NGS. Our findings demonstrated that saliva samples can be stored at ambient temperature, as we did not observe any significant difference between a fresh sample and the same sample after 24 hours (p = 0.23), hence, saliva collection is the optimal route for sampling airplane passengers. Our results also showed that droplet digital PCR is a more suitable method for detecting virus from saliva, compared to RT-qPCR. Keywords: COVID-19; RT-PCR; ddPCR; SARS-CoV-2; nasopharyngeal swab; saliva.

A Review on Fast Direct Methods of Surface Integral Equations for Analysis of Electromagnetic Scattering from 3-D PEC Objects

This paper reviews a series of fast direct solution methods for electromagnetic scattering analysis, aiming to significantly alleviate the problems of slow or even non-convergence of iterative solvers and to provide a fast and robust numerical solution for integral equations. Then the advantages and applications of fast direct solution methods and the research trends are introduced in detail. Three different main methods are discussed, namely hierarchically off-diagonal low-rank matrices (HODLR) and skeletonization direct methods based on weak and strong admissibility condition. Numerical examples of computational complexity and electromagnetic scattering analysis of jet models are presented to demonstrate the efficiency and accuracy of each approach. Finally, a brief discussion is given on the main challenges and possible strategies of fast direct solution methods which still exist.

Reconstructing Urban Heritage Character

The old town area gives a significant character on urban heritage, in particular regarding Kota Pusaka programme in Indonesia. While enhancing the character, in this case, part of Siak Sri Indrapura City has burned out, and raise serious issues to the local government due to lack of written data on how to reconstruct the area. In this case, extracting field data by conventional measurement from remaining buildings found ineffective. Not only consume time, but these methods have also created social and psychological problems for local people. Therefore various indirect field methods such as desktop research, digitising, and documentation are needed to be completed by more fast and interactive direct field methods. This paper confirms that field data which compiled by various field methods can more contribute to reconstructing the character of the old town.

TWO-DIMENSIONAL SPLITTING SCHEMES FOR HYPERBOLIC EQUATIONS

The article considers splitting schemes in geometric directions that approximate the initial-boundary value problem for p-dimensional hyperbolic equation by chain of two-dimensional-one-dimensional problems. Two ways of constructing splitting schemes are considered with an operator factorized on the upper layer, algebraically equivalent to the alternating direction scheme, and additive schemes of total approximation. For the first scheme, the restrictions on the shape of the region G at p=3 can be weakened in comparison with schemes of alternating directions, which are a chain of three-point problems on the upper time layer, the region can be a connected union of cylindrical regions with generators parallel to the axis OX3. In the second case, for a three-dimensional equation of hyperbolic type, an additive scheme is constructed, which is a chain «two-dimensional problem – one-dimensional problem» and approximates the original problem in a summary sense (at integer time steps). For the numerical implementation of the constructed schemes – the numerical solution of two-dimensional elliptic problems – one can use fast direct methods based on the Fourier algorithm, cyclic reduction methods for three-point vector equations, combinations of these methods, and other methods. The proposed two-dimensional splitting schemes in a number of cases turn out to be more economical in terms of total time expenditures, including the time for performing computations and exchanges of information between processors, compared to traditional splitting schemes based on the use of three-point difference problems for multiprocessor computing systems, with different structures of connections between processors type «ruler», «matrix», «cube», with universal switching.

Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations

Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fréchet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.

Fast Direct Methods for Gaussian Processes.

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the n-dimensional setting, however, it requires the inversion of an n ×n covariance matrix, C, as well as the evaluation of its determinant, det(C). In many cases, such as regression using Gaussian processes, the covariance matrix is of the form C = σ(2) I + K, where K is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix C is typically dense, causing standard direct methods for inversion and determinant evaluation to require O(n(3)) work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix C can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an O (n log(2) n) algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant det(C), permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining K. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.

Matrix decomposition algorithms for arbitrary order [formula omitted] tensor product finite element systems

Matrix decomposition algorithms (MDAs) are fast direct methods for the solution of systems of linear algebraic equations which arise in the approximation of Poisson’s equation on the unit square using various techniques such as finite difference, spline collocation and spectral methods. The attraction of MDAs is that they employ fast Fourier transforms and require O(N2logN) operations on an N×N uniform partition of the unit square. In this paper, MDAs are formulated for the solution of the finite element Galerkin equations arising when spaces of C0 piecewise polynomials of degree k≥3 are employed. Results of numerical experiments exhibit the expected optimal global convergence rates and superconvergence phenomena.

Shifted L-BFGS systems

We investigate fast direct methods for solving systems of the form (B+G)x=y, where B is a limited-memory Broyden-Fletcher-Goldfarb-Shanno matrix and G is a symmetric positive-definite matrix. These systems, which we refer to as shifted L-BFGS systems, arise in several settings, including trust-region methods and preconditioning techniques for interior-point methods. We show that under mild assumptions, the system (B+G)x=y can be solved in an efficient manner that mitigates instability via a recursion that requires only vector inner products. We consider various shift matrices G and demonstrate the effectiveness and accuracy of the recursion method in numerical experiments.

Accelerated Direct Solution of the Method-of-Moments Linear System

This paper addresses the direct (noniterative) solution of the method-of-moments (MoM) linear system, accelerated through block-wise compression of the MoM impedance matrix. Efficient matrix block compression is achieved using the adaptive cross-approximation (ACA) algorithm and the truncated singular value decomposition (SVD) postcompression. Subsequently, a matrix decomposition is applied that preserves the compression and allows for fast solution by backsubstitution. Although not as fast as some iterative methods for very large problems, accelerated direct solution has several desirable features, including: few problem-dependent parameters; fixed time solution avoiding convergence problems; and high efficiency for multiple excitation problems [e.g., monostatic radar cross section (RCS)]. Emphasis in this paper is on the multiscale compressed block decomposition (MS-CBD) algorithm, introduced by Heldring , which is numerically compared to alternative fast direct methods. A new concise proof is given for the N2 computational complexity of the MS-CBD. Some numerical results are presented, in particular, a monostatic RCS computation involving 1 043 577 unknowns and 1000 incident field directions, and an application of the MS-CBD to the volume integral equation (VIE) for inhomogeneous dielectrics.

Multi-objective optimization procedure for the wing design at cruise and low-speed conditions

The procedure for wing aerodynamic design based on the algorithm of simultaneous multiregime optimization of cruise and low-speed performance is considered. The method is based on the combination of fast direct methods for subsonic and transonic wing analysis, geometry variation module and the optimization procedure. The description of all the components and examples of practical application of the developed technique for design of the conventional medium-haul airplane wing and the ‘Flying Wing’ configuration are presented.

Acousto-capacitive tomography of liquid multiphase systems

This paper discusses a tomographic approach focusing on ultrasonic measurements to monitor liquid multiphase mixtures. Separately a capacitive tomography low-cost setup is regarded. Both sensor arrays aim for the localization of variable phase boundaries and the physical characterisation of spatially distributed phases. Focusing on a real time processing, a reduced number of transducers in combination with a fast linear modelling and direct image reconstruction methods are used. Experimental results of a layered 3-phase-system validate the potential and limits of physical resolution of both approaches. Finally, the prospectively intended data fusion of both approaches is discussed.

Matrix decomposition algorithms for the C 0-quadratic finite element Galerkin method

Explicit expressions for the eigensystems of one-dimensional finite element Galerkin (FEG) matrices based on C0 piecewise quadratic polynomials are determined. These eigensystems are then used in the formulation of fast direct methods, matrix decomposition algorithms (MDAs), for the solution of the FEG equations arising from the discretization of Poisson’s equation on the unit square subject to several standard boundary conditions. The MDAs employ fast Fourier transforms and require O(N2log N) operations on an N×N uniform partition. Numerical results are presented to demonstrate the efficacy of these algorithms.

Optimal pulse observation of one type of systems with delay

Consideration is given to a linear problem of optimal pulse observation of a non-stationary dynamic system with delay in an equation of its mathematical model. To compute estimates of an unknown vector parameter of the initial state of the system, fast direct and dual methods are proposed. The main role belongs to quasi-reduction of the fundamental matrix of solutions to systems with delay. As is shown, to perform iterations of the method, integration of auxiliary systems of ordinary differential equations on small time intervals is sufficient. An algorithm of the operation of an optimal estimator--device for computing estimates of current states--is described. The results are illustrated by the problem of optimal observation of the fourth-order system with delay.

Characterization of Surface-Breaking Cracks Using One Tangential Component of Magnetic Leakage Field Measurements

We propose a procedure for full characterization of rectangular surface-breaking cracks based on measurements of only one tangential component of the magnetic field with the magnetic flux leakage (MFL) technique. The parameters of interest include orientation, length, and depth of the cracks. We assume that the length and the depth of the investigated cracks are much larger than the crack width, so that the variation of the MFL response with respect to the width is negligible. Our procedure employs fast direct methods that provide reliable estimation of the crack parameters in three separate consecutive steps. We propose denoising and correction techniques as well. We confirmed the accuracy of the methods by simulations based on the finite-element method (FEM) as well as by experimental MFL observations. A procedure is proposed for full characterization of rectangular surface breaking cracks based on measurements of only one tangential component of the magnetic field with the magnetic flux leakage (MFL) technique. The parameters of interest include orientation, length and depth of the cracks. We assume that the length and the depth of the investigated cracks are much larger than the crack width such that the variation of the MFL response with respect to the width is negligible. The proposed procedure employs fast direct methods which provide reliable estimation of the crack parameters in three separate consecutive steps. De-noising and correction techniques are proposed as well. The accuracy of the proposed estimation methods is examined via simulations based on the finite element method (FEM) as well as experimental MFL data.

A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

Iterative numerical schemes for solving the electrostatic partial differential equation with variable conductivity, using fast and high-order accurate direct methods for preconditioning, are compared. Two integral method schemes of this type, based on previously suggested splittings of the equation, are proposed, analyzed, and implemented. The schemes are tested for large problems on a square. Particular emphasis is paid to convergence as a function of geometric complexity in the conductivity. Differences in performance of the schemes are predicted and observed in a striking manner.

Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box.

An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Bénard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single nonstaggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.

Matrix Decomposition Algorithms for Modified Spline Collocation for Helmholtz Problems

We consider the solution of various boundary value problems for the Helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal (fourth-) order approximations. For the solution of the collocation equations, we formulate matrix decomposition algorithms, fast direct methods which employ fast Fourier transforms and require O(N2 log N) operations on an N × N uniform partition of the unit square. A computational study confirms the published analysis for the Dirichlet problem and indicates that similar results hold for Neumann, mixed, and periodic boundary conditions. The numerical results also exhibit superconvergence phenomena not reported in earlier studies.

Fast direct solution methods for symmetric banded Toeplitz systems, based on the sine transform

We present new fast direct methods for solving a large symmetric banded Toeplitz system of order n with bandwidth p. We make use of structured matrices which can be diagonalized by the discrete sine transform matrix, sometimes called τ-matrices. A first method writes the Toeplitz matrix as the sum of a τ-matrix and a low rank matrix. A second method embeds the Toeplitz matrix in a larger τ-matrix of order m. The methods are similar to Jain [IEEE Trans. Acoust. Speech Signal Process. 26 (1978) 121] and Linzer [Linear Algebra Appl. 170 (1992) 1], who worked with circulant matrices. Both algorithms consist in solving two τ-systems and two smaller systems. A τ-system of order n can be solved in O( nlog n) by using a discrete sine transform if n+1 has small prime factors. Therefore, the second algorithm is preferable, since we can choose m such that m+1 has small prime factors. On the other hand, in the second method the smaller systems can become large when m differs too much from n, while in the first method the order is always p−1. In both methods, the small systems have low displacement rank, so we can use fast methods to solve them.

Waveform Relaxation with Fast Direct Methods as Preconditioner

For a restricted class of parabolic PDEs one can devise a practical numerical solver with a parallel complexity that is theoretically optimal. The method uses a multidimensional FFT to decouple the unknowns in the spatial domain into independent scalar ODEs. These are discretized to give recurrence relations in the time dimension solved by parallel cyclic reduction. This is the FFT/CR algorithm. We discuss the use of FFT/CR as a preconditioner to iteratively solve more general parabolic PDEs. This approach naturally leads to a waveform relaxation scheme. Waveform relaxation was developed as an iterative method for solving large systems of ODEs. It is the continuous-in-time analogue of stationary iterative methods for linear algebraic equations. Using the FFT/CR solver as a preconditioner preserves most of the potential for concurrency that accounts for the attractiveness of waveform relaxation with simple preconditioners like Jacobi or red-black Gauss--Seidel, while showing an important advantage: the convergence rate of the resulting iteration is independent of the mesh size used in the spatial discretization. The method can be accelerated by applying an appropriate scaling of the system before preconditioning.