Search Results (74)

In this work, we proposed a dynamic inverse solution with spatio-temporal constraints of the nonlinear heat diffusion problem in 1D and 2D based on a regularized Gauss–Newton and Krylov subspace with a preconditioner. The preconditioner is computed by approximating the Jacobian of the nonlinear system at each Gauss–Newton iteration. The proposed approach is used for estimation of the initial value from measurements of the last value by considering spatial and spatio-temporal constraints. The system is compared to a dynamic Tikhonov inverse solution and generalized minimal residual method (GMRES) with and without a preconditioner. The system is evaluated under noise conditions in order to verify the robustness of the proposed approach. It can be seen that the proposed spatio-temporal regularized Gauss–Newton method with GMRES and a preconditioner shows better estimation results than the other methods for both spatial and spatio-temporal constraints. Full article

In this work, we propose a nonlinear dynamic inverse solution to the diffusion problem based on Krylov Subspace Methods with spatiotemporal constraints. The proposed approach is applied by considering, as a forward problem, a 1D diffusion problem with a nonlinear diffusion model. The dynamic inverse problem solution is obtained by considering a cost function with spatiotemporal constraints, where the Krylov subspace method named the Generalized Minimal Residual method is applied by considering a linearized diffusion model and spatiotemporal constraints. In addition, a Jacobian-based preconditioner is used to improve the convergence of the inverse solution. The proposed approach is evaluated under noise conditions by considering the reconstruction error and the relative residual error. It can be seen that the performance of the proposed approach is better when used with the preconditioner for the nonlinear diffusion model under noise conditions in comparison with the system without the preconditioner. Full article

Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank–Nicolson method and the tempered weighted and shifted Grünwald formula are used to discretize the tempered fractional diffusion equations. The discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite (SPD) Toeplitz matrix. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the $\tau$ matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. Full article

We study overlapping additive Schwarz (OAS) preconditioners for the solution of elliptic boundary value problems discretized using isogeometric collocation methods based on generalized B-splines (GB-splines). Through a series of numerical experiments, we demonstrate the scalability of the proposed preconditioning strategy with respect to the number of subdomains, as well as its robustness with respect to the parameters of the isogeometric discretization. Full article

This study analyzes and evaluates the performance of various solvers and preconditioners for reservoir simulations of CO₂ injection and long-term storage using the model 11B of SPE CSP (Society of Petroleum Engineers, 11th Comparative Solution Project) and the MATLAB Reservoir Simulation Toolbox (MRST). The SPE CSP 11 model serves as a benchmark for testing numerical methods for solving partial differential equations (PDEs) in reservoir simulations. The research focuses on the Biconjugate Gradient Stabilized (BiCGSTAB) and Loose Generalized Minimum Residual (LGMRES) solver methods, as well as multiple preconditioning techniques designed to improve convergence rates and reduce computational effort for CO₂ storage. Extensive simulations were performed to compare the performance of different solver-preconditioner combinations under varying reservoir conditions, leveraging MRST’s flexible simulation capabilities. Key performance metrics, including iteration counts and computational time, were analyzed for the project. The results reveal trade-offs between computational efficiency and solution accuracy, providing valuable insights into the effectiveness of each approach. This study offers practical guidance for reservoir engineers and researchers seeking to analyze and optimize simulation workflows within MRST by identifying the strengths and limitations of specific solver-preconditioner combinations for complex linear systems. Full article

For solving the continuous Sylvester equation, a class of Hermitian and skew-Hermitian based multiplicative splitting iteration methods is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations, and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite, we can choose Hermitian and skew-Hermitian (HS) splittings of matrices A and B in the first equation, and the splitting of the Jacobi iterations for matrices A and B in the second equation in the multiplicative splitting iteration method. Convergence conditions of this method are studied in depth, and numerical experiments show the efficiency of this method. Moreover, by numerical computation, we show that multiplicative splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the continuous Sylvester equation. Full article

In mineral, environmental, and engineering explorations, we frequently encounter geological bodies with varied sizes, depths, and conductivity contrasts with surround rocks and try to interpret them with single survey data. The conventional three-dimensional (3-D) inversions significantly rely on the size of the grids, which should be smaller than the smallest geological target to achieve a good recovery to anomalous electric conductivity. However, this will create a large amount of unknowns to be solved and cost significant time and memory. In this paper, we present a multi-scale (MS) stochastic inversion scheme based on shearlet transform for airborne electromagnetic (AEM) data. The shearlet possesses the features of multi-direction and multi-scale, allowing it to effectively characterize the underground conductivity distribution in the transformed domain. To address the practical implementation of the method, we use a compressed sensing method in the forward modeling and sensitivity calculation, and employ a preconditioner that accounts for both the sampling rate and gradient noise to achieve a fast stochastic 3-D inversion. By gradually updating the coefficients from the coarse to fine scales, we obtain the multi-scale information on the underground electric conductivity. The synthetic data inversion shows that the proposed MS method can better recover multiple geological bodies with different sizes and depths with less time consumption. Finally, we conduct 3-D inversions of a field dataset acquired from Byneset, Norway. The results show very good agreement with the geological information. Full article

The purpose of this work is to study the efficient numerical solvers for time-dependent conservative space fractional diffusion equations. Specifically, for the discretized Toeplitz-like linear system, we aim to study efficient preconditioning based on a matrix-splitting iteration method. We propose a scaled tridiagonal and Toeplitz-like splitting iteration method. Its asymptotic convergence property is first established. Further, based on the induced preconditioner, a fast circulant-like preconditioner is developed to accelerate the convergence of the Krylov Subspace iteration methods. Theoretical results suggest that the fast preconditioner can inherit the effectiveness of the original induced preconditioner. Numerical results also demonstrate its efficiency. Full article

In recent years, CO₂ flooding has become an important technical measure for oil and gas field enterprises to further improve oil and gas recovery and achieve the goal of “dual carbon”. It is also one of the concrete application forms of CCUS. Numerical simulation based on CO₂-EOR plays an indispensable role in the study of the mechanism of CO₂ flooding and buried percolation, allowing for technical indicators to be selected and EOR/EGR prediction to be improved for reservoir engineers. This paper discusses the numerical simulation techniques related to CO₂ flooding and storage, including mathematical models and solving algorithms. A multiphase and multicomponent mathematical model is developed to describe the flow mechanism of hydrocarbon components–CO₂–water underground and to simulate the phase diagram of the components. The two-phase P-T flash calculation with SSI (+DEM) and the Newton method is adopted to obtain the gas–liquid phase equilibrium parameters. The extreme value judgment of the TPD function is used to form the phase stability test and miscibility identification model. A tailor-made multistage preconditioner is built to solve the linear equation of the strong-coupled, multiphase, multicomponent reservoir simulation, which includes the variables of pressure, saturation, and composition. The multistage preconditioner improves the computational efficiency significantly. A numerical simulation of CO₂ injection in a carbonate reservoir in the Middle East shows that it is effective for researching the recovery factor and storage quantity of CO₂ flooding based on the above numerical simulation techniques. Full article

The high-resolution 3D groundwater flow and transport simulation problem requires massive discrete linear systems to be solved, leading to significant computational time and memory requirements. The domain decomposition method is a promising technique that facilitates the parallelization of problems with minimal communication overhead by dividing the computation domain into multiple subdomains. However, directly utilizing a domain decomposition scheme to solve massive linear systems becomes impractical due to the bottleneck in algebraic operations required to coordinate the results of subdomains. In this paper, we propose a two-level domain decomposition method, named dual-domain decomposition, to efficiently solve the massive discrete linear systems in high-resolution 3D groundwater simulations. The first level of domain decomposition partitions the linear system problem into independent linear sub-problems across multiple subdomains, enabling parallel solutions with significantly reduced complexity. The second level introduces a domain decomposition preconditioner to solve the linear system, known as the Schur system, used to coordinate results from subdomains across their boundaries. This additional level of decomposition parallelizes the preconditioning of the Schur system, addressing the bottleneck of the Schur system solution while improving its convergence rates. The dual-domain decomposition method facilitates the partition and distribution of the computation to be solved into independent finely grained computational subdomains, substantially reducing both computational and memory complexity. We demonstrate the scalability of our proposed method through its application to a high-resolution 3D simulation of chromium contaminant transport in groundwater. Our results indicate that our method outperforms both the vanilla domain decomposition method and the algebraic multigrid preconditioned method in terms of runtime, achieving up to 8.617× and 5.515× speedups, respectively, in solving massive problems with approximately 108 million degrees of freedom. Therefore, we recommend its effectiveness and reliability for high-resolution 3D simulations of groundwater flow and transport. Full article

The extrusion processing of food powder relies heavily on its moisture content to aid in flow and proper cooking, shaping, and/or puffing. This study focused on the impact of the moisture content on the dynamic flow and shear properties of coarse food powders (corn meal, wheat farina, and granulated sugar). The dynamic flow properties explored were the specific basic flowability energy (SBFE), specific energy, stability index, and flow rate index. The shear properties were the angle of internal friction, unconfined yield strength, major principal stress, wall friction angle, flow factor (FF), and compressibility. Corn meal exhibited an increase in SBFE as the moisture content increased (6.70 mJ/g at 13.13% to 9.14 mJ/g at 19.61%) but no change in FF (4.94 to 5.11); wheat farina also showed an increase in energy requirement as the moisture increased (5.81 mJ/g at 13.73% to 9.47 mJ/g 19.57%) but a marked decrease in FF ratings (18.47 to 6.1); granulated sugar showed a decrease in energy requirements as the moisture increased (51.73 mJ/g at 0.06% moisture content to 13.58 mJ/g at 0.78% moisture content) and a decrease in FF ratings (8.53 to 3.47). Overall, upon the addition of moisture, corn meal became cohesive yet free-flowing; wheat farina became less compressible and more cohesive; and granulated sugar became more cohesive and compressible and less free-flowing. Full article

The recently deployed Surface Water and Ocean Topography (SWOT) mission for the first time has observed the ocean surface at a spatial resolution of 1 km, thus giving an opportunity to directly monitor submesoscale sea surface height (SSH) variations that have a typical magnitude of a few centimeters. This progress comes at the expense of the necessity to take into account numerous uncertainties in calibration of the quality-controlled altimeter data. Of particular importance is the proper filtering of spatially correlated errors caused by the uncertainties in geometry and orientation of the on-board interferometer. These “systematic” errors dominate the SWOT error budget and are likely to have a notable signature in the SSH products available to the oceanographic community. In this study, we explore the utility of the block-circulant (BC) approximation of the SWOT precision matrix developed by the Jet Propulsion Laboratory for assessment of a mission’s accuracy, including the possible impact of the systematic errors on the assimilation of the wide-swath altimeter data into numerical models. It is found that BC approximation of the precision matrix has sufficient (90–99%) accuracy for a wide range of significant wave heights of the ocean surface, and, therefore, could potentially serve as an efficient preconditioner for data assimilation problems involving altimetry observations by space-borne interferometers. An extensive set of variational data assimilation (DA) experiments demonstrates that BC approximation provides more accurate SSH retrievals compared to approximations, assuming a spatially uncorrelated observation error field as is currently adopted in operational DA systems. Full article

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy-$\alpha$-stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method. Full article

Jacobian-free Newton Krylov (JFNK) is an attractive method to solve nonlinear equations in the nuclear engineering community, and has been successfully applied to steady-state neutron diffusion k-eigenvalue problems and multi-physics coupling problems. Preconditioning technique plays an important role in the JFNK algorithm, significantly affecting its computational efficiency. The key point is how to automatically construct a high-quality preconditioning matrix that can improve the convergence rate and perform the preconditioning matrix factorization efficiently and robustly. A reordering-based ILU(k) preconditioner is proposed to achieve the above objectives. In detail, the finite difference technique combined with the coloring algorithm is utilized to automatically construct a preconditioning matrix with low computational cost. Furthermore, the reordering algorithm is employed for the ILU(k) to reduce the additional non-zero elements and pursue robust computational performance. A 2D LRA neutron steady-state benchmark problem is used to evaluate the performance of the proposed preconditioning technique, and a steady-state neutron diffusion k-eigenvalue problem with thermal-hydraulic feedback is also utilized as a supplement. The results show that coloring algorithms can automatically and efficiently construct the preconditioning matrix. The computational efficiency of the FDP with coloring could be about 60 times higher than that of the preconditioner without the coloring algorithm. The reordering-based ILU(k) preconditioner shows excellent robustness, avoiding the effect of the fill-in level k choice in incomplete LU factorization. Moreover, its performances under different fill-in levels are comparable to the optimal computational cost with natural ordering. Full article

Driven by the emergence of Graphics Processing Units (GPUs), the solution of increasingly large and intricate numerical problems has become feasible. Yet, the integration of GPUs into Computational Fluid Dynamics (CFD) codes still presents a significant challenge. This study undertakes an evaluation of the computational performance of GPUs for CFD applications. Two Compute Unified Device Architecture (CUDA)-based implementations within the Open Field Operation and Manipulation (OpenFOAM) environment were employed for the numerical solution of a 3D Kaplan turbine draft tube workbench. A series of tests were conducted to assess the fixed-size grid problem speedup in accordance with Amdahl’s Law. Additionally, tests were performed to identify the optimal configuration utilizing various linear solvers, preconditioners, and smoothers, along with an analysis of memory usage. Full article