Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation

Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise

The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the Primitive Equations and more generally. In this work we study a stochastic version of the Primitive Equations. We establish the global existence of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, $L^{p}_{t}L^{q}_{x}$ estimates on the pressure and stopping time arguments.Comment: To appear in Nonlinearit

On polyhedral projection and parametric programming

Submitted versio

Structure of shocks in Burgers turbulence with L\'evy noise initial data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by L\'evy noise, or equivalently when the initial potential is a two-sided L\'evy process $\psi_0$. When $\psi_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $\psi_0$ is abrupt we show that the shock structure is discrete. When $\psi_0$ is eroded we show that there are no rarefaction intervals.Comment: 22 page

Numerical approach to a model for quasistatic damage with spatial BV-regularization

We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems

Superquantile/CVaR Risk Measures: Second-Order Theory

Superquantiles, which refer to conditional value-at-risk (CVaR) in the same way that quantiles refer to value-at-risk (VaR), have many advantages in the modeling of risk in finance and engineering. However, some applications may benefit from a further step, from superquantiles to second- order superquantiles. Measures of risk based on second-order superquantiles have recently been explored in some settings, but key parts of the theory have been lacking: descriptions of the associated risk envelopes and risk identifiers. Those missing ingredients are supplied in this paper, and moreover not just for second-order superquantiles, but also for a much broader class of mixed superquantile measures of risk. Such dualizing expressions facilitate the development of dual methods for mixed and second-order superquantile risk minimization as well as superquantile regression, a proposed second-order version of quantile regression.U.S. Air Force Office of Scientific Research grant FA9550-11-1-0206U.S. Air Force Office of Scientific Research grant F1ATAO1194GOO1DARPA grant HR0011517798U.S. Air Force Office of Scientific Research grant FA9550-11-1-0206U.S. Air Force Office of Scientific Research grant F1ATAO1194GOO1DARPA grant HR001151779

Engineering decisions under risk-averseness

Engineering decisions are invariably made under substantial uncertainty about current and future system cost and response, including cost and response associated with low-probability, high- consequence events. A risk-neutral decision maker would rely on expected values when comparing designs, while a risk-averse decision maker might adopt nonlinear utility functions or failure probability criteria. The paper shows that these models for making decisions are related to a framework of risk measures that includes many possibilities. We describe how risk measures provide an expanded set of models for handling risk-averse decision makers. General recommendations for selecting risk measures lead to decision models for risk-averse decision making that comprehensively represent risks in engi- neering systems, avoid paradoxes, and accrue substantial benefits in subsequent risk, reliability, and cost optimization. The paper provides an overview of the framework of decision making based on risk measures.This material is based upon work supported in part by the U. S. Air Force Office of Scientific Research under grants FA9550-11-1-0206 and F1ATAO1194GOO1

Measures of Residual Risk with Connections to Regression, Risk Tracking, Surrogate Models, and Ambiguity

Measures of residual risk are developed as extension of measures of risk. They view a random variable of interest in concert with an auxiliary random vector that helps to manage, predict, and mitigate the risk in the original variable. Residual risk can be exempli ed as a quanti cation of the improved situation faced by a hedging investor compared to that of a single-asset investor, but the notion reaches further with deep connections emerging with forecasting and generalized regression. We establish the fundamental properties in this framework and show that measures of residual risk along with generalized regression can play central roles in the development of risk-tuned approximations of random variables, in tracking of statistics, and in estimation of the risk of conditional random variables. The paper ends with dual expressions for measures of residual risk, which lead to further insights and a new class of distributionally robust optimization models.U. S. Air Force Office of Scientific Research grant FA9550-11-1-0206U. S. Air Force Office of Scientific Research grant F1ATAO1194GOO1U. S. Air Force Office of Scientific Research grant F4FGA04094G003DARPA grant HR0011412251U. S. Air Force Office of Scientific Research grant FA9550-11-1-0206U. S. Air Force Office of Scientific Research grant F1ATAO1194GOO1U. S. Air Force Office of Scientific Research grant F4FGA04094G003DARPA grant HR001141225