A Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing

Our goal is to identify the volatility function in Dupire's equation from given option prices. Following an optimal control approach in a Lagrangian framework, we propose a globalized sequential quadratic programming (SQP) algorithm with a modified Hessian - to ensure that every SQP step is a descent direction - and implement a line search strategy. In each level of the SQP method a linear-quadratic optimal control problem with box constraints is solved by a primal-dual active set strategy. This guarantees L1 constraints for the volatility, in particular assuring its positivity. The proposed algorithm is founded on a thorough first- and second-order optimality analysis. We prove the existence of local optimal solutions and of a Lagrange multiplier associated with the inequality constraints. Furthermore, we prove a sufficient second-order optimality condition and present some numerical results underlining the good properties of the numerical scheme.Dupire equation, parameter identification, optimal control, optimality conditions, SQP method, primal-dual active set strategy

Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming

AbstractThe design of an optimal (output feedback) reduced order control (ROC) law for a dynamic control system is an important example of a difficult and in general non-convex (nonlinear) optimal control problem. In this paper we present a novel numerical strategy to the solution of the ROC design problem if the control system is described by partial differential equations (PDE). The discretization of the ROC problem with PDE constraints leads to a large scale (non-convex) nonlinear semidefinite program (NSDP). For reducing the size of the high dimensional control system, first, we apply a proper orthogonal decomposition (POD) method to the discretized PDE. The POD approach leads to a low dimensional model of the control system. Thereafter, we solve the corresponding small-sized NSDP by a fully iterative interior point constraint trust region (IPCTR) algorithm. IPCTR is designed to take advantage of the special structure of the NSDP. Finally, the solution is a ROC for the low dimensional approximation of the control system. In our numerical examples we demonstrate that the reduced order controller computed from the small scaled problem can be used to control the large scale approximation of the PDE system

Model order reduction approaches for infinite horizon optimal control problems via the HJB equation

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods

The ACS Exams Institute Undergraduate Chemistry Anchoring Concepts Content Map I: General Chemistry

To provide tools for programmatic assessment related to the use of ACS Exams in undergraduate chemistry courses, the ACS Exams Institute has built a content map that applies to the entire undergraduate curriculum. At the top two levels, the grain size of the content classification is large and spans the entire undergraduate curriculum. At the bottom two levels, the grain size of the content is more fine and tuned to specific course levels of the curriculum. This paper presents all four levels of the map as identified for first-year general chemistry

Order reduction approaches for the algebraic Riccati equation and the LQR problem

We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices

Life cycle assessment of Polychlorinated Biphenyl contaminated soil remediation processes

Goal and scope. A life-cycle assessment (LCA) was performed to evaluate the environmental impacts of the remediation of industrial soils contaminated by polychlorobiphenyl (PCB). Two new bioremediation treatment options were compared with the usual incineration process. In this attributional LCA, only secondary impacts were considered. The contaminated soil used for the experiments contained 200 mg of PCB per kg. Methods. Three off-site treatments scenarios were studied: 1) bioremediation with mechanical aeration, 2) bioremediation with electric aeration and 3) incineration with natural gas. Bioremediation processes were designed from lab-scale, scale-up and pilot experiments. The incineration technique was inspired by a French plant. A semi-quantitative uncertainty analysis was performed on the data. Environmental impacts were evaluated with the CML 2001 method using the Simapro software program. Results and discussion. In most compared categories, the bioremediation processes are favorable. Of the bioremediation options, the lowest environmental footprint was observed for electric aeration. The uncertainty analysis supported the results that compared incineration and bioremediation but decreased the difference between the options of aeration. The distance of transportation was one of the most sensitive parameters, especially for bioremediation. At equal distances between the polluted sites and the treatment plant, bioremediation had fewer impacts than incineration in eight out of thirteen categories. Conclusions. The use of natural gas for the incineration process generated the most impacts. Irrespective of the aeration option, bioremediation was better than incineration. Recommendations. The time of treatment should be taken into account. More precise and detailed data are required for the incineration scenario. More parameters of biological treatments should be measured. LCA results should be completed using ecological and health risk assessment and an acceptability evaluation

Energy Dissipating Devices in Falling Rock Protection Barriers

Rockfall is a phenomenon which, when uncontrolled, may cause extensive material damage and personal injury. One of the structures used to avoid accidents caused by debris flows or rockfalls is flexible barriers. The energy dissipating devices which absorb the energy generated by rock impact and reduce the mechanical stresses in the rest of the elements of the structure are an essential part of these kinds of structures. This document proposes an overview of the performance of energy dissipating devices, as well as of the role that they fulfil in the barrier. Furthermore, a compilation and a description of the dissipating elements found in the literature are proposed. Additionally, an analysis has been performed of the aspects taken into account in the design, such as experimental (quasi-static and dynamic) tests observing the variation of the behaviour curve depending on the test speed and numerical simulations by means of several finite element software packages