Boundedness Requirement Research Articles

Nonlinear parallel-in-time simulations of multiphase flow in porous media

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution.

Parallel fully coupled methods for bound-preserving solution of subsurface flow and transport in porous media

As more powerful supercomputer systems with lots of memory and a large number of computing cores become available, the family of fully coupled algorithms is drawing more attention in scientific and engineering applications, due to its impressive robustness and scalability in extreme-scale simulations. In this work, we introduce and study some parallel domain decomposition preconditioned generalized Newton algorithms for solving the fully coupled and bound-preserving formulation of the subsurface flow and transport problem in porous media. In the approach, we present the active–set reduced–space (ASRS) method to guarantee the nonlinear consistency of the fully coupled system in a monolithic way, and meanwhile ensure the boundedness requirement of the solution. Furthermore, we focus on the application of the overlapping additive Schwarz preconditioning technique to accelerate the linear convergence of Newton iterations and be beneficial to the scalability of the inner linear solver. We present some numerical experiments to demonstrate the parallel scalability of the proposed algorithm on the Shaheen-II supercomputer with thousands of processors. In particular, the numerical results also show that the fully coupled framework has the nature of bound preservation and is efficient for the proposed reservoir flow problems.

Scalable semismooth Newton methods with multilevel domain decomposition for subsurface flow and reactive transport in porous media

Large-scale modeling and predictive simulations of the subsurface flow and reactive transport system in porous media is significantly challenging, due to the high nonlinearity of the governing equations and the strong heterogeneity of material coefficients. The design of novel mathematical models and state-of-the-art methods for the flow simulation through porous media typically needs to satisfy the so-called bound-preserving property, i.e., the computed solution should stay inside a physically meaningful range. This paper presents a robust, scalable numerical framework based on the variational inequality formulation and the semismooth Newton method in a fully implicit manner, to model and simulate highly nonlinear flows without violating the boundedness requirement of the solution. Rigorous theoretical analysis for the variational inequality formulation of the problem is provided for facilitating the design of algorithms. Specifically, our approach further enhances the numerical formulation by utilizing a family of multilevel monolithic overlapping Schwarz methods for efficiently preconditioning, and the parallel implementation of the bound-preserving solver is based on the fast and robust domain decomposition technique. Numerical experiments are presented to demonstrate the efficiency and parallel scalability of the solution strategy for both standard benchmarks as well as realistic flow problems involving strong heterogeneity and high nonlinearity. We also show that the proposed framework is more robust and efficient than the commonly used inexact Newton algorithm in terms of the bound-preserving property.

Bound-preserving inexact Newton algorithms on parallel computers for wormhole propagation in porous media

Simulating wormhole propagation during reactive dissolution of carbonates is often significantly challenging, because of the high nonlinearity of the governing equations with complex fluid physics. High resolution grids are often required to represent the complex geological heterogeneity, which demands massively parallel computers with a large number of processors. Herein, we present a parallel and scalable simulator on parallel computers for the fully implicit solution of the wormhole propagation model. Our approach is based on a family of mixed finite element methods for the spatial discretization and the implicit Backward Euler scheme for the temporal integration, to handle the combination of complicated flow physics and high resolution grids in their full complexity. Moreover, the active-set reduced-space method, as a class of bound-preserving inexact Newton algorithms, is proposed for the resultant nonlinear system of equations, to guarantee the nonlinear consistency of the fully implicit discretization in a monolithic way and to ensure the boundedness requirement of the solution. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed simulator for a set of heterogeneous medium problems. Large-scale results are provided to show the scalability for reservoir simulation with hundreds of millions of unknowns by using several thousand processors.

A 6M digital twin for modeling and simulation in subsurface reservoirs

A 6M digital twin for modeling and simulation in subsurface reservoirs

Nonlinearly preconditioned constraint-preserving algorithms for subsurface three-phase flow with capillarity

The multiphase flow model has been extensively used to describe complicated flow behaviors in subsurface formations, together with sophisticated reservoir models and well-defined fluid property. In this study, the fully implicit method, as one of most promising schemes for subsurface flow modeling, is employed to solve multiphase flow problems. In contrast to the conventional approach where mathematical models often include a pressure equation, the multiphase flow problems are modeled by up to three continuity equations so that mass conservation holds for all present phases. Another challenge that frequently shows up is the computed solution may sit outside its physically meaningful range, thereby leading to inaccurate predictions or even a failure of the simulation process. A simple remedy is to apply a cutting-off operation to the out-of-bound solution but such an action could ruin both local and global mass conservation. Instead, we replace the original model by a variational inequality formulation with box inequality constraints to protect the boundedness requirement on pressure and saturations from being violated. The variational inequality problem is then solved by a well-designed nonlinear solver consisting of the active-set reduced-space method and the nonlinear elimination preconditioning technique. A number of examples are presented to demonstrate that the proposed formulation is bound-preserving and mass-conservative for each of the present phases/components.

Parallel reservoir simulators for fully implicit complementarity formulation of multicomponent compressible flows

The numerical simulation of multicomponent compressible flow in porous media is an important research topic in reservoir modeling. Traditional semi-implicit methods for such problems are usually conditionally stable, suffer from large splitting errors, and may accompany with violations of the boundedness requirement of the numerical solution. In this study we reformulate the original multicomponent equations into a nonlinear complementarity problem and discretize it using a fully implicit finite element method. We solve the resultant nonsmooth nonlinear system of equations arising at each time step by a parallel, scalable, and nonlinearly preconditioned semismooth Newton algorithm, which is able to preserve the boundedness of the solution and meanwhile treats the possibly imbalanced nonlinearity of the system. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm on the Tianhe-2 supercomputer for both standard benchmarks as well as realistic problems in highly heterogeneous media.

A fully implicit constraint-preserving simulator for the black oil model of petroleum reservoirs

Due to the rapid advancement of supercomputing resource, there is a growing interest in developing parallel algorithms for the large-scale reservoir simulation. In this paper, we present a parallel and fully implicit simulator for the black oil model based on the variational inequality (VI) framework, which can be used to enforce important mathematical and physical properties to obtain accurate constraint-preserving solutions. In other words, this framework ensures the predicted solution to stay within the physical range. In the proposed approach, the black oil model is reformulated as a variational inequality system that naturally satisfies the basic boundedness requirement of the solution, and then a fully implicit finite volume method is applied to discretize the model equations. In addition to that, a number of nonlinear and linear fast solver technologies, including a variant of inexact Newton methods and the domain decomposition based preconditioners, are employed to guarantee the robustness and parallel scalability of the simulator. A particular emphasis of the proposed framework is placed on the parallel and algorithmic performance of the variational inequality approach across large-scale and heterogeneous problems. Several numerical results pertaining to the problems in one, two and three dimensions are presented to illustrate the efficiency, robustness, and the overall performance of the fully implicit constraint-preserving simulator.

Nonlinear evolutionary systems driven by quasi‐hemivariational inequalities

This paper is devoted to the study of the differential systems in arbitrary Banach spaces that are obtained by mixing nonlinear evolutionary equations and generalized quasi‐hemivariational inequalities (EEQHVI). We start by showing that the solution set of the quasi‐hemivariational inequality associated to problem EEQHVI is nonempty, closed, and convex. Furthermore, we establish upper semicontinuity and measurability properties for this solution set. Then, based on them, we prove the existence of solutions for problem EEQHVI and the compactness of the set of corresponding trajectories of EEQHVI. These statements extend previous results in several directions, for instance, by dropping the boundedness requirement for the set of constraints and substantially relaxing monotonicity hypotheses.

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate ...

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space $\cH$, with $dim(\cH) \geq 3$, define $\bS(\cH):= \{x\in \cH \:|\: ||x||=1\}$, and let $\nu_\cH$ be the unique regular Borel positive measure invariant under the action of the unitary operators in $\cH$, with $\nu_\cH(\bS(\cH))=1$. We prove that if a complex frame function $f : \bS(\cH)\to \bC$ satisfies $f \in \cL^2(\bS(\cH), \nu_\cH)$, then it verifies Gleason's statement: There is a unique linear operator $A: \cH \to \cH$ such that $f(u) = < u| A u>$ for every $u \in \bS(\cH)$. $A$ is Hermitean when $f$ is real. No boundedness requirement is thus assumed on $f$ {\em a priori}.

Adaptive Fault Detection and Fault-Tolerant Control of NASA Generic Transport Aircraft Model

In this paper, adaptive fault detection and fault-tolerant control schemes for aircraft flight systems are investigated and evaluated on a high-fidelity aircraft model–the nonlinear NASA generic transport model. State feedback for output tracking and output feedback for output tracking multivariable model reference adaptive control schemes are developed, where plant-model matching conditions are much less restrictive than that of the state feedback for state tracking design, to ensure stability and asymptotic output tracking for the aircraft in the presence of uncertain actuator failures and structural damage. Based on an aircraft system dynamic coupling feature and different actuator failure patterns, adaptive detectors are constructed to detect damage and actuator failures. Our adaptive detection schemes are equipped with feedback fault-tolerant control to ensure signal boundedness requirement. Desired performance of the developed schemes is demonstrated by extensive simulation studies on the nonlinear NASA generic transport model.

Convergence of Rational Interpolants to Analytic Functions with Restricted Growth

Let D be a regular domain in \(\overline{\rm C}\) with bounded boundary. For each n≥ 1, take points \(A_n=\{a_{ni}\}_{i=0}^n\) in D and points \(B_n=\{b_{ni}\}_{i=1}^n\) in \(\overline{\rm C}\), such that \tu Bn has no limit points in D. Let \gan and \gbn be the normalized point counting measures of An and Bn respectively and \(\alpha^\prime_n\) and \(\beta^\prime_n\) their swept measures onto \t6D. Denote by U\gm the logarithmic potential of the measure \gm. It has been shown by Ambroladze and Wallin, that if for every weak-star limit point of \gabn, we have \ga(D) 0 and also \(\lim_{n\to\infty}\bigg[\sup_{z\in\partial D}\bigg({n+1\over n}U_{\alpha^\prime_n}(z)-U_{\beta^\prime_n}(z)\bigg)\bigg]=0,\) then for every bounded analytic function f on D, the rational interpolants rn of degree n with poles at Bn interpolating to f at An, converge locally uniformly with geometric degree of convergence to f on D. We show that under a slightly stricter condition on {Bn}, the boundedness requirement on f can be replaced with the weaker growth condition \(\mid f(z)\mid \leq Ad(z,\partial D)^q,\) where A0, q< 0 and that this growth condition is also sharp in the sense described in this paper.

A High-Gain Scaling Technique for Adaptive Output Feedback Control of Feedforward Systems

In this note, we propose an adaptive output feedback control design technique for feedforward systems based on our recent results on dynamic high-gain scaling techniques for controller design for strict-feedback systems. The system is allowed to contain uncertain functions of all the states and the input as long as the uncertainties satisfy certain bounds. Unknown parameters are allowed in the bounds assumed on the uncertain functions. If the uncertain functions involve the input, then the output-dependent functions in the bounds on the uncertain functions need to be polynomially bounded. It is also shown that if the uncertain functions can be bounded by a function independent of the input, then the polynomial boundedness requirement can be relaxed. The designed controllers have a very simple structure being essentially a linear feedback with state-dependent dynamic gains and do not involve any saturations or recursive computations. The observer utilized to estimate the unmeasured states is similar to a Luenberger observer with dynamic observer gains. The Lyapunov functions are quadratic in the state estimates, the observer errors, and the parameter estimation error. The stability analysis is based on our recent results on uniform solvability of coupled state-dependent Lyapunov equations. The controller design provides strong robustness properties both with respect to uncertain parameters in the system model and additive disturbances. This robustness is the key to the output feedback controller design. Global asymptotic results are obtained.

Improved upper bound on step-size parameters of discrete-time recurrent neural networks for linear inequality and equation system

In this brief, an improved upper bound on the step-size parameters of a globally convergent discrete-time recurrent neural network (RNN) model proposed recently in the literature for solving the linear inequality and equation system is obtained without needing the original boundedness requirement for the solution set of the linear system while the step-size parameters being allowed different. Consequently, the rate of convergence for the discrete-time RNN model can be improved by setting the step-size parameters as large as possible no matter whether the solution set of the linear system is bounded or not. It is shown by an example that the obtained upper bound is actually tight in the sense that the RNN in the specific example is globally convergent if and only if the step-size parameters are less than the given upper bound. A numerical simulation example of a globally convergent discrete-time RNN for solving a specific linear inequality and equation system with an unbounded solution set is also provided.

Weak and strong convergence theorems for fixed points of asymptotically nonexpensive mappings

Let E be a uniformly convex Banach space, K a nonempty closed convex subset of E and T : K → K an asymptotically nonexpansive mapping with a nonempty fixed-point set. Weak and strong convergence theorems for the iterative approximation of fixed points of T are proved. Our results show that the boundedness requirement imposed on the subset K in recent results of Huang [1], Rhoades [2], and Schu [3,4] can be dropped. Furthermore, our results extend these results to more satisfactory modified Mann and Ishikawa iteration methods with errors in the sense of Xu [5].

Prevalence‐free utility‐respecting summary indices of diagnostic power do not exist

One would like to have a ranking method for diagnostic tests that justifiably makes no assumptions about the local ‘disease mix’, that is, about the pre-test probabilities of the diagnostic alternatives. Assuming the simplest possible clinical context we therefore examine the following question: can a summary index of diagnostic test performance be devised which is loyal to the expected utility paradigm and, at the same time, is incidence-free (prevalence-free)? In other words, regardless of pre-test probabilities, the index should rank rival tests in a way which agrees with how they might be ranked by expected utility calculations. From a practical point of view, the answer is negative; it is proved that such indices do exist, but they necessarily impose a very restrictive mathematical form on the utility framework and, what is worse, invariably go against what is medically appropriate by violating a boundedness requirement. Copyright © 2000 John Wiley & Sons, Ltd.

Prevalence-free utility-respecting summary indices of diagnostic power do not exist

One would like to have a ranking method for diagnostic tests that justifiably makes no assumptions about the local 'disease mix', that is, about the pre-test probabilities of the diagnostic alternatives. Assuming the simplest possible clinical context we therefore examine the following question: can a summary index of diagnostic test performance be devised which is loyal to the expected utility paradigm and, at the same time, is incidence-free (prevalence-free)? In other words, regardless of pre-test probabilities, the index should rank rival tests in a way which agrees with how they might be ranked by expected utility calculations. From a practical point of view, the answer is negative; it is proved that such indices do exist, but they necessarily impose a very restrictive mathematical form on the utility framework and, what is worse, invariably go against what is medically appropriate by violating a boundedness requirement.

Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation

The standard five-point difference approximation to the Cauchy problem for Laplace's equation satisfies stability estimates---and hence turns out to be a well-posed problem---when a certain boundedness requirement is fulfilled. The estimates are of logarithmic convexity type. Herewith, a regularization method will be proposed and associated error bounds can be derived. Moreover, the error between the given (continuous) Cauchy problem and the difference approximation obtained via a suitable minimization problem can be estimated by a discretization and a regularization term.

Aspects Of (Un)Boundedness

Abstract Resultative predicates have the aspectual effect of telicizing an atelic activity verb. The function of the postverbal constituent in accomplishments has been taken to be one of providing an end point to the activity, or of a constituent that "measures out" the event denoted by the activity verb. In either case, it delimits the event by providing it with boundaries. Looking at resultative predicates, we observe that they are subject to the requirement that they denote a bounded scale. This requirement is argued to be empirically superior to an alternative restriction stating that the resultative must be a stage-level predicate. The boundedness requirement furthermore provides direct evidence against an approach that treats the resultative as an end point, and supports the claim that it is an event measure. One piece of evidence concerns the 'make + NP+Adjective' construction, in which the adjective denotes the final stage or end point in a change of state, exactly as in the resultative construction. In contrast to the resultative, however, the adjective can be unbounded, as it is not an event measure in this case. We argue that the boundedness requirement on resultative predicates follows directly from treating it as an event measure, since a measure must be bounded as a matter of conceptual necessity.