Implicit Discretization Scheme Research Articles

Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem

AbstractThis paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on an implicit discretization scheme, we propose a primal-dual algorithm and provide a non-ergodic convergence rate under a general setting for the inertial parameters when one objective function is continuously differentiable and convex and the other is a proper, convex and lower semicontinuous function. For this algorithm we derive a $$O\left( 1/k^2 \right) $$ O 1 / k 2 convergence rate under three classical rules proposed by Nesterov, Chambolle-Dossal and Attouch-Cabot without assuming strong convexity, which is compatible with the results of the continuous-time dynamic system. For the case when both objective functions are continuously differentiable and convex, we further present a primal-dual algorithm based on an explicit discretization. We provide a corresponding non-ergodic convergence rate for this algorithm and show that the sequence of iterates generated weakly converges to a primal-dual optimal solution. Finally, we present numerical experiments that indicate the superior numerical performance of both algorithms.

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

Discretization of prescribed-time observers in the presence of noises and perturbations

An implicit Euler discretization scheme of the prescribed-time converging observer from Holloway and Krstic, (2019) is proposed, which preserves all main properties of the continuous-time counterpart, and it is investigated how to apply recursively this discretization on any interval of time. In addition, it is demonstrated that the estimation error is robustly stable with respect to the measurement noise with a linear gain (the property that does not hold in the continuous-time setting). The efficiency of the suggested discrete-time observer is illustrated through numeric experiments.

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

Distributed Inertial Continuous and Discrete time Algorithms for Solving Resource Allocation Problem

In this paper, we investigate several distributed inertial algorithms in continuous and discrete time for solving resource allocation problem (RAP), where its objective function is convex or strongly convex. First, the original RAP is equivalently transformed into a distributed unconstrained optimization problem by introducing an auxiliary variable. Then, two distributed inertial continuous time algorithms and two discrete time algorithms are proposed and the rates of their convergence based on the gap between the objective function and their optimal function are determined. Our first distributed damped inertial continuous time algorithm is designed for RAP with a convex function, it achieves convergence rate at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O( \frac{1}{ t^{2}})$</tex-math></inline-formula> based on Lyapunov analysis method, and then we design a rate-matching distributed damped inertial discrete time algorithm by exploiting implicit and Nesterov's discretization scheme. Our second distributed fixed inertial discrete time algorithm is designed to deal with the RAP with a strongly convex objective function. Noteworthy, the transformed distributed problem is no longer strongly convex even though the original objective function is strongly convex, but it satisfies the Polyak-Ł jasiewicz ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PL</b> ) and quadratic growth ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">QG</b> ) conditions. Inspired by the Heavy-Ball method, a distributed fixed inertial continuous time algorithm is proposed, it has an explicit and accelerated exponential convergence rate. Later, a rate-matching accelerated distributed fixed inertial discrete time algorithm is also obtained by applying explicit, semi-implicit Euler discretization and sufficient decrease update schemes. Finally, the effectiveness of the proposed distributed inertial algorithms is verified by simulation.

Development of a semi‐implicit contact methodology for finite volume stress solvers

AbstractThe past decades have seen numerous efforts to apply the finite volume methodology to solid mechanics problems. However, only limited work has been done by the finite volume community toward the simulation of mechanical contact. In this article, we present a novel semi‐implicit methodology for the solution of static force‐loading contact problems with cell‐centered finite volume codes. Starting from the similarities with multi‐material problems, we derive an implicit discretization scheme for the normal contact stress with a straightforward inclusion of frictional forces and correction vectors for non‐orthogonal boundaries. With the introduction of a sigmoid blending function interpolating between contact stresses and gap pressure, the proposed approach is extended to cases with partially closed gap. The contact procedure is designed around an arbitrary mesh mapping algorithm to allow for non‐conformal meshes at the contact interface between the two bodies. Finally, we verify the contact methodology against five benchmarks cases.

Parallel multilevel domain decomposition preconditioners for monolithic solution of non-isothermal flow in reservoir simulation

In reservoir simulation, the non-isothermal flow adds a conservation of the energy equation with the involvement of a temperature variable to the complicated physics dynamics, which brings a higher nonlinearity of the corresponding PDE system and therefore exhibits significantly additional challenges on the linear preconditioner of the simulator. However, the commonly used block preconditioning techniques mostly focus on the isothermal petroleum model and do not have a specific treatment of the energy conservation equation for the non-isothermal flow. In this study, we propose and develop a family of multilevel restricted additive Schwarz (MRAS) preconditioners on parallel computers for the fully implicit solution of the non-isothermal flow in porous media. The proposed parallel reservoir simulator incorporates the restricted additive Schwarz preconditioner into a general multilevel preconditioning framework with the use of the domain decomposition technique and the multigrid method, which allows us to flexibly construct several types of multilevel Schwarz preconditioners by using the additive or multiplicative strategies. In particular, it follows the fully implicit discretization scheme for the monolithic solution, and therefore is unconditionally stable with the relaxation of the time step size by the stability condition. We numerically show that the proposed approach is highly robust and efficient for solving some non-isothermal flow problems with high nonlinearity, and good parallel scalabilities are obtained on a supercomputer with a large number of processors.

A refined two-scale model for Newtonian and non-Newtonian fluids in fractured poroelastic media

A refined model is presented which numerically resolves the fluid flow within a fracture inside a saturated poroelastic material. At the discontinuity, the mass balance of the fluid is solved using the velocity profile inside the fracture, with the velocity profile being determined numerically from the momentum balance in the integration points at the discontinuity. The resolution of the mass balance and the velocity profile in the integration points at the discontinuity is coupled to surrounding poroelastic material model in a two-scale approach. The resulting monolithic scheme allows for complex fluid behaviour to be included, which is demonstrated via the inclusion of the fluid inertia and the use of a Carreau fluid. The governing equations are discretised using T-splines, while the fracture is represented by spline-based interface elements, and an implicit time discretisation scheme is used. Mesh refinement studies are carried out for a typical case, which contains a pressurised and propagating fracture. A coarse macro-scale mesh is sufficient to obtain the correct propagation velocities, but a finer mesh is needed to prevent pressure oscillations. Time step refinement studies show the capabilities of the model to capture pressure waves within the fracture. Finally, it is shown that if inertial effects are limited and a Newtonian fluid is used, a fracture scale discretisation using a single element is sufficient. However, for a Carreau fluid or when the problem is inertia-dominated, smaller elements are needed to correctly represent the velocity profile.

Fast algorithms for high-dimensional variable-order space-time fractional diffusion equations

High-dimensional variable-order space-time fractional diffusion equations have numerous real-world applications and have attracted a lot of attention in recent years. Solving this kind of equation is difficult because of the nonlocal property of time evolution. In this paper, we consider fast algorithms for high-dimensional variable-order space-time fractional diffusion equations. We propose an implicit discretization scheme based on Grunwald formula and discuss its stability and convergence. Two preconditioning strategies based on the structure of the coefficient matrix are proposed to reduce the computational costs. Furthermore, we design two low-rank tensor minimal energy (LTME) algorithms for multidimensional problems to solve relevant huge linear systems. Numerical examples are tested to show the effectiveness of the proposed methods.

Numerical comparison of mathematical and computational models for the simulation of stochastic neutron kinetics problems

This paper concerns numerical comparisons between five mathematical models capable of modelling the stochastic behaviour of neutrons in low extraneous (extrinsic or fixed) neutron source applications. These models include analog Monte-Carlo (AMC), forward probability balance equations (FPB), generating function form of the forward probability balance equations (FGF), generating function form of the backward probability balance equations (Pál-Bell), and an Itô calculus model using both an explicit and implicit Euler-Maruyama discretization scheme. Results such as the survival probability, extinction probability, neutron population mean and standard deviation, and neutron population cumulative distribution function have all been compared. The least computationally demanding mathematical model has been found to be the use of the Pál-Bell equations which on average take four orders of magnitude less time to compute than the other methods in this study. The accuracy of the AMC and FPB models have been found to be strongly linked to the computational efficiency of the models. The computational efficiency of the models decrease significantly as the maximum allowable neutron population is approached. The Itô calculus methods, utilising explicit and implicit Euler-Maruyama discretization schemes, have been found to be unsuitable for modelling very low neutron populations. However, improved results, using the Itô calculus methods, have been achieved for systems containing a greater number of neutrons.

Feedback Linearization of Implicit Discrete Systems

Feedback linearization of a nonlinear system is an actively employed technique to control a nonlinear system. Feedback linearizability of continuous-time systems is generally not preserved under sampling and is dependent upon the choice of discretization method. This paper modifies the set of necessary and sufficient conditions for feedback linearization given by Grizzle (1986), so they can be employed for implicitly described discrete-time control system and using them establishes an equivalence between the discrete-time feedback linearizability of the Implicit and Explicit Euler discretization schemes.

Robust Learning with Implicit Residual Networks

In this effort, we propose a new deep architecture utilizing residual blocks inspired by implicit discretization schemes. As opposed to the standard feed-forward networks, the outputs of the proposed implicit residual blocks are defined as the fixed points of the appropriately chosen nonlinear transformations. We show that this choice leads to the improved stability of both forward and backward propagations, has a favorable impact on the generalization power, and allows for control the robustness of the network with only a few hyperparameters. In addition, the proposed reformulation of ResNet does not introduce new parameters and can potentially lead to a reduction in the number of required layers due to improved forward stability. Finally, we derive the memory-efficient training algorithm, propose a stochastic regularization technique, and provide numerical results in support of our findings.

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur’e dynamical systems where the set-valued feedback depends on both the time and the state. This work is a generalization of Tanwani et al. (SIAM J. Control Opti. 56(2), 751–781, 2018) where the time-dependent set-valued feedback is considered to acquire weak solutions. Obviously, strong solutions and implicit discretization scheme are nice properties, especially for numerical simulations. Conditions guarantying the exponential attractivity of solutions are provided. The obtained results can be used to study the time-varying Lur’e systems with errors in data or under the state-dependent controls applied to the moving sets. Our work is new even the set-valued feedback depends only on the time.

A Fast Block $\alpha$-Circulant Preconditoner for All-at-Once Systems From Wave Equations

In this paper, we propose a fast block $\alpha$-circulant preconditioner for solving the nonsymmetric linear system arising from an all-at-once implicit discretization scheme in time for the wave equation. As a generalization of the well-known block circulant preconditioning technique, the proposed block $\alpha$-circulant preconditioner can also be efficiently inverted in a parallel-in-time manner. The complex eigenvalues of the preconditioned matrix are fully derived in explicit expression and its diagonalizability is also shown. Furthermore, a mesh-independent convergence rate of the preconditioned GMRES method is derived under certain conditions. Building upon the proposed preconditioner, two stationary iterative methods with uniform asymptotic convergence rates were also presented. The extension of our preconditioner within a simplified Newton iteration to the nonlinear wave equation is also discussed. Both linear and nonlinear numerical examples are given to illustrate the promising performance of our proposed block $\alpha$-circulant preconditioner and also validate our theoretical results on convergence analysis.

Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

ABSTRACTIn this paper, we introduce a class of state-dependent maximal monotone differential inclusions. The existence and uniqueness of solutions are obtained by using an implicit discretization scheme and a kind of hypo-monotonicity assumption. In addition, a characterization for nonsmooth Lyapunov pairs associated with such systems is provided. Our result can be applied to study state-dependent sweeping processes and Lur'e dynamical systems. It is new even the involved maximal monotone operators depend only on the time.

Four-Point EGSOR Iteration for the Grünwald Implicit Finite Difference Solution of One-Dimensional Time-Fractional Parabolic Equations

In this paper, our main concerned is on the application of the formulation of a four-point explicit group successive over-relaxation (4EGSOR) iterative method in solving one-dimensional time-fractional parabolic equations based on the second-order Grünwald implicit approximation equation. The formulation of the 4EGSOR method is constructed by using the implicit approximation equation which is derived by the Grunwald derivative operator and the implicit finite difference discretization scheme. In order to access the performance results of the 4EGSOR iterative method, another block and point iterative methods which are the four-point EGGS (4EGGS) and the Gauss-Seidel (GS) were also presented as control methods. The results of three numerical experiments show substantial improvement in terms of the number of iterations and execution time.

Parametric analysis of the solidification of nanofluids in spherical cavities

This paper presents the results of a numerical study involving the solidification process of nanofluids in spherical cavities. The model was constructed by applying the heat conduction equation in the solid and liquid phases of the domain, including effective thermal conductivity in the liquid phase so as to accommodate the convective terms of the process. The equations of the model were solved using the finite difference method with an implicit discretization scheme and were validated by comparing results with an analytical solution from the literature. This paper seeks to analyze the effect of nanoparticles on the speed of cooling of the liquid phase and of the solidification times of the nanofluid. Four different types of nanofluids were analyzed all of which had water as the base fluid. The results showed that the nanoparticles in the base fluid accelerate the cooling process and decrease the time taken for complete solidification.

VIV simulation of riser-conductor systems including nonlinear soil-structure interactions

This paper presents a fully three-dimensional numerical approach for analyzing deepwater drilling riser-conductor system vortex-induced vibrations (VIV) including nonlinear soil-structure interactions (SSI). The drilling riser-conductor system is modeled as a tensioned beam with linearly distributed tension and is solved by a fully implicit discretization scheme. The fluid field around the riser-conductor system is obtained by Finite-Analytic Navier-Stokes (FANS) code, which numerically solves the unsteady Navier-Stokes equations. The SSI is considered by modeling the lateral soil resistance force according to nonlinear p-y curves. Overset grid method is adopted to mesh the fluid domain. A partitioned fluid-structure interaction (FSI) method is achieved by communication between the fluid solver and riser motion solver. A riser-conductor system VIV simulation without SSI is firstly presented and served as a benchmark case for the subsequent simulations. Two SSI models based on a nonlinear p-y curve are then applied to the VIV simulations. Also, the effects of two key soil properties on the VIV simulations of riser-conductor systems are studied.

A fractional PDE for first passage time of time-changed Brownian motion and its numerical solution

We show that the First-Passage-Time probability distribution of a Lévy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show that it preserves the structural properties (non-negativity, boundedness, time monotonicity) of the theoretical solution, having to be a probability distribution. Numerical experiments confirming such findings are reported. Simulations of the sample paths of the considered process are also performed and used to both provide suitable boundary conditions and to validate the numerical results.

Consistent Discretization of Finite-Time and Fixed-Time Stable Systems

Algorithms of implicit discretization for generalized homogeneous systems having discontinuity only at the origin are developed. They are based on the transformation of the original system to an equivalent one which admits an implicit or a semi-implicit discretization schemes preserving the stability properties of the continuous-time system. Namely, the discretized model remains finite-time stable (in the case of negative homogeneity degree), and practically fixed-time stable (in the case of positive homogeneity degree). The theoretical results are supported with numerical examples. 1. Introduction. Discretization issues are important for a digital implementation of estimation and control algorithms. Construction of a consistent stable discretization is complex for essentially non-linear ordinary differential equations (ODEs), which do not satisfy some classical regularity assumptions. For example, the sliding mode algorithms are known to be difficult in practical realization [1], [2], [3] due to discontinuous (set-valued) nature, which may invoke chattering caused by the discretization. The mentioned papers have discovered that the implicit discretization technique is useful for practical implementation of non-smooth and discontinuous control and estimation algorithms. In particular, chattering suppression in both input and output, as well as a good closed-loop performance has been confirmed experimentally in [1], [4], [5]. Finite-time stability is a desirable property for many control and estimation algorithms [6], [7], [8], [9], [10], [11]. It means that system trajectories reach a stable equilibrium (or a set) in a finite time, in contrast to asymptotic stability allowing this only for the time tending to infinity. If the settling (reaching) time is globally bounded for all initial conditions then the origin is fixed-time stable (see, e.g. [12]). The corresponding ODE models do not satisfy Lipschitz condition (at least at the origin). In the general case, an application of the conventional implicit or explicit discretization schemes does not guarantee that finite-time or fixed-time stability properties will be preserved (see, e.g. [13], [14], [15]). The latter means that the discrete-time model may be inconsistent with the continuous-time one. However, the discretized systems may remain globally finite-time stable in some cases (see [1], [2], [16], [17]). The aim of this paper is to study systematically the problem of consistent discretization of the so-called generalized homogeneous non-linear systems. The discretized model is consistent if it preserves the stability property (e.g. exponential, finite-time or fixed-time stability) of the original continuous-time system. Homogeneity is a certain form of symmetry studied in systems and control theory [9], [18], [19], [20],[21], [22], [23]. The standard homogeneity (introduced originally by L. Euler in 17th century) is the symmetry of a mathematical object f (e.g. function, vector field, operator, etc) with respect to the uniform dilation of the argument x → λx, namely, f (λx) = λ 1+ν f (x), λ > 0