An Implicit-Function Theorem for B-Differentiable Functions

A function from one normed linear space to another is said to be Bouligand differentiable (B-differentiable) at a point if it is directionally differentiable there in every direction, and if the directional derivative has a certain uniformity property. This is a weakening of the classical idea of Frechet (F-) differentiability, and it is useful in dealing with optimization problems and in other situations in which F-differentiability may be too strong. In this paper we introduce a concept of strong B-derivative, and we employ this idea to prove an implicit-function theorem for B-differentiable functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with B-differentiability in place of F-differentiability. Therefore it is applicable to a considerably wider class of functions than is the classical theorem

Trade-Offs Model Of Multi-Objective Reservoir Operation With Uncertainties

As the increase of water resources management and exploitation goals, it is gaining increasing weights for reservoir operation to seek optimal options for the balance between multiple and contradictory water resources use objectives. This study develops a trade-offs model to quantify the economic benefits of reservoir operation rules on the downstream water supply yield. Uncertainties of different water use benefits are considered by using a Monte Carlo method in the trade-offs model. The case study is analyzed to evaluate its performance in terms of water use benefits of agriculture, hydropower, flood control and environmental water requirements in the Yellow River, China. Trade-offs results are got among water resources needs of social development and environmental protections under the reservoir operation. The results indicate that there are magnificent trade-offs between ecological benefit and social economic development under different management policies and scenarios. This study could provide a simple but robust framework for quantifying the consequences of management options with reservoir operations under control. The results could be used for reference of compromised solutions to the ecological and human negotiations for water. Acknowledgement: This project has been funded with support from the European Commission. This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein

Global Convergence of Damped Newton's Method for Nonsmooth Equations, via the Path Search

A natural damping of Newton's method for nonsmooth equations is presented. This damping, via the path search instead of the traditional line search, enlarges the domain of convergence of Newton's method and therefore is said to be globally convergent. Convergence behavior is like that of line search damped Newton's method for smooth equations, including Q-quadratic convergence rates under appropriate conditions. Applications of the path search include damping Robinson-Newton's method for nonsmooth normal equations corresponding to nonlinear complementarity problems and variational inequalities, hence damping both Wilson's method (sequential quadratic programming) for nonlinear programming and Josephy-Newton's method for generalized equations. Computational examples from nonlinear programming are given

Averaging Schemes for Solving Fived Point and Variational Inequality Problems

We develop and study averaging schemes for solving fixed point and variational inequality problems. Typically, researchers have established convergence results for solution methods for these problems by establishing contractive estimates for their algorithmic maps. In this paper, we establish global convergence results using nonexpansive estimates. After first establishing convergence for a general iterative scheme for computing fixed points, we consider applications to projection and relaxation algorithms for solving variational inequality problems and to a generalized steepest descent method for solving systems of equations. As part of our development, we also establish a new interpretation of a norm condition typically used for establishing convergence of linearization schemes, by associating it with a strong-f-monotonicity condition. We conclude by applying our results to transportation networks

On a semismooth* Newton method for solving generalized equations

In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness\ast, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models. © 2021 Society for Industrial and Applied Mathematic

Multiphase Flow and Transport in Porous Media with Phase Transition at Multiple Scales: Modeling, Numerical Analysis, and Simulation

In this dissertation we consider two application specific flow and transport models in porous media at multiple scales: 1) methane gas transport models for hydrate formation and dissociation in the subsurface under two-phase conditions, and 2) coupled flow and biomass-nutrient model for biofilm growth in complex geometries with biofilm, and its impact via upscaling from pore scale to Darcy scale on Darcy scale permeability. Both projects are motivated by the challenges from real-life applications in the subsurface. First we consider the simplified methane gas transport models at the Darcy scale under equilibrium and non-equilibrium conditions. The equilibrium model (EQ) is a conservation law with a nonsmooth space-dependent flux function, similar to those that are known in other applications including the two-phase flow in a heterogeneous porous medium, traffic flow on roads, and nonlinear elasticity in mixed materials. There are two unknowns in (EQ) models which are bound together by a relationship called nonlinear complementarity constraint and represented by a multivalued graph. Our main result is the weak stability of an upwind-implicit scheme for a regularized (EQ). To our best knowledge, this is the first such result for the transport model. We also consider kinetic models which approximate (EQ) and are useful when we simulate the hydrate phase change at shorter time scales, e.g., after a seismic event. After a rigorous analysis of three kinetic models, we focus on the analysis of a particular model robust across the unsaturated and saturated conditions. We also prove the weak stability of this model and confirm the rate of convergence $O(\sqrt{h})$ for both equilibrium and kinetic models. We choose various equilibrium and non-equilibrium scenarios relevant to the applications, and we provide 1d simulation results which illustrate the theory. Next we study the coupled biomass-nutrient-flow dynamics in a complicated pore scale geometry. Our goal is to describe a new monolithic coupled flow and biomass-nutrient model and to show its robustness through various numerical experiments. The biomass-nutrient model is of variational inequality type blended with nonsingular diffusivity to ensure the volume constraint while enhancing the biofilm growth mechanism. For the flow, we consider the Brinkman flow with spatially varying permeability which accounts for the flow in (somewhat) permeable domains as well as around these. We apply the flow and biofilm growth model to the entire domain so that the model and the coupling are monolithic. Our overall scheme follows operator splitting and time lagging: we solve advection explicitly by the upwind method and diffusion-reaction together using CCFD with time-lagged diffusion coefficients. For flow, we use our version of the Marker-And-Cell method adapted to the heterogeneous Brinkman model on a time-staggered grid. We also present simulation results to show the robustness of our model. To handle the sensitivity of the biomass-nutrient model to its initial data, we introduce a new modeling construction which "promotes" the adhesion of biofilm to the surface. Then we perform the Monte Carlo simulations and construct the probability distributions of upscaled permeability which represent the randomness of complex geometry with biofilm

Inexact Newton Methods For Solving Nonsmooth Equations

This paper investigates inexact Newton methods for solving systems of nonsmooth equations. We define two inexact Newton methods for locally Lipschitz functions and we prove local (linear and superlinear) convergence results under the assumptions of semismoothness and BD-regularity at the solution. We introduce a globally convergent inexact iteration function based method. We discuss implementations and we give some numerical examples. © 1995.601-2127145Brown, Saad, Hybrid Krylov Methods for Nonlinear Systems of Equations (1990) SIAM Journal on Scientific and Statistical Computing, 11, pp. 450-481Broyden, A class of methods for solving nonlinear simultaneous equations (1965) Mathematics of Computation, 19, pp. 577-593Broyden, Dennis, More, On the local and superlinear convergence of quasi-Newton methods (1973) J. Inst. Math. 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