Semismooth Newton methods for variational problems with inequality constraints

Hager, Corinna; Wohlmuth, Barbara

doi:10.1002/gamm.201010002

Cited by 17 publications

(12 citation statements)

References 45 publications

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“…The gas production process aws simulated until t end = 360 days. The maximum time step P r e p r i n t 20 S. Gupta et al We identify a domain of interest, Ω I , for the gas production process as: 0m ≤ r ≤ 250m, −150m ≥ z ≥ −550m. Outside this domain, pressure, temperature, and saturation profiles do not change much.…”

Section: Numerical Simulation and Resultsmentioning

confidence: 99%

“…As a general outline, we cast the inequality constraints arising from the vapour-liquid-equilibrium (VLE) assumption (e.g., [24]) for the CH 4 − H 2 O system into a set of complementarity conditions which lead to the mathematical structure of a variational inequality (e.g., [12,44]). We reformulate the complementarity conditions as a set of non-differentiable but semi-smooth functions which are solved together with the governing PDEs of the methane hydrate model fully implicitly using a semi-smooth Newton method (See, e.g., [20] and the references therein). We implement our semi-smooth Newton method using an active-set strategy (e.g., [25,27]), where the Jacobian is uniquely determined based on the local phase states which are partitioned into active/inactive sets using the semi-smooth NCP functions.…”

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confidence: 99%

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An All-At-Once Newton Strategy for Marine Methane Hydrate Reservoir Models

2020

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Marine gas hydrate systems are characterized by highly dynamic transport-reaction processes in an essentially water-saturated porous medium that are coupled to thermodynamic phase transitions between solid gas hydrates, free gas and dissolved methane in the aqueous phase. These phase transitions are highly nonlinear and strongly coupled, and cause the mathematical model to rapidly switch the phase states and pose serious convergence issues for the classical Newton's method. One of the common methods of dealing with such phase transitions is the primary variable switching (PVS) method where the choice of the primary variables is adapted locally to the phase state 'outside' the Newton loop. In order to ensure that the phase states are determined accurately, the PVS strategy requires an additional iterative loop, which can get quite expensive for highly nonlinear problems. For methane hydrate reservoir models, the PVS method shows poor convergence behaviour and often leads to extremely small time step sizes. In order to overcome this issue, we have developed a nonlinear complementary constraints method (NCP) for handling phase transitions, and implemented it within a non-smooth Newtons linearization scheme using an active-set strategy. Here, we present our numerical scheme and show its robustness through field scale applications based on the highly dynamic geological setting of the Black Sea.

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Section: Numerical Simulation and Resultsmentioning

confidence: 99%

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confidence: 99%

An All-At-Once Newton Strategy for Marine Methane Hydrate Reservoir Models

2020

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“…In a more generalized solution method, the nonlinear equilibrium equation as well as the inequality constraints can be treated within a single Newton iteration, which can be implemented as a primal-dual active set strategy (e.g. [24,25]). We have implemented our numerical scheme in C++ based on the DUNE PDELab framework [2,17].…”

Section: 5mentioning

confidence: 99%

Efficient parameter estimation for a methane hydrate model with active subspaces

et al. 2018

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Methane gas hydrates have increasingly become a topic of interest because of their potential as a future energy resource. There are significant economical and environmental risks associated with extraction from hydrate reservoirs, so a variety of multiphysics models have been developed to analyze prospective risks and benefits. These models generally have a large number of empirical parameters which are not known a priori. Traditional optimization-based parameter estimation frameworks may be ill-posed or computationally prohibitive. Bayesian inference methods have increasingly been found effective for estimating parameters in complex geophysical systems. These methods often are not viable in cases of computationally expensive models and high-dimensional parameter spaces. Recently, methods have been developed to effectively reduce the dimension of Bayesian inverse problems by identifying low-dimensional structures that are most informed by data. Active subspaces is one of the most generally applicable methods of performing this dimension reduction. In this paper, Bayesian inference of the parameters of a state-of-the-art mathematical model for methane hydrates based on experimental data from a triaxial compression test with gas hydrate-bearing sand is performed in an efficient way by utilizing active subspaces. Active subspaces are used to identify low-dimensional structure in the parameter space which is exploited by generating a cheap regression-based surrogate model and implementing a modified Markov chain Monte Carlo algorithm. Posterior densities having means that match the experimental data are approximated in a computationally efficient way.

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“…This makes the use of a primal-dual active set strategy for the high-dimensional problem associated with the snapshots computationally attractive cf. [18,20,26].…”

Section: Solution Of the Detailed And Reduced Problemmentioning

confidence: 99%

A Reduced Basis Method for Parametrized Variational Inequalities

Haasdonk¹,

Salomon²,

Wohlmuth³

2012

SIAM J. Numer. Anal.

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International audienceReduced basis methods are an eﬃcient tool for signiﬁcantly reducing the computational complexity of solving parametrized partial diﬀerential equations. Originally introduced for elliptic equations, they have been generalized during the last decade to various types of elliptic, parabolic and hyperbolic systems. In this article, we extend the reduction technique to parametrized variational inequalities. Firstly, we propose a reduced basis variational inequality scheme in a saddle-point form and prove existence and uniqueness of the solution. We state some elementary analytical properties of the scheme such as reproduction of solutions, a-priori stability with respect to the data and Lipschitz- continuity with respect to the parameters. Secondly, we provide rigorous a-posteriori error bounds. An oﬄine/online decomposition guarantees an eﬃcient assembling of the reduced scheme, which can be solved by constrained quadratic programming. The reduction scheme is applied to a one-dimensional obstacle problem with a two-dimensional parameter space. The numerical results conﬁrm the theoret- ical ones and demonstrate the eﬃciency of the reduction technique

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Semismooth Newton methods for variational problems with inequality constraints

Cited by 17 publications

References 45 publications

An All-At-Once Newton Strategy for Marine Methane Hydrate Reservoir Models

An All-At-Once Newton Strategy for Marine Methane Hydrate Reservoir Models

Efficient parameter estimation for a methane hydrate model with active subspaces

A Reduced Basis Method for Parametrized Variational Inequalities

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