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Optimal stopping with signatures
No full textWe propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear functionals of the associated rough path signature, and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. The only assumption on the process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semi-martingales or Markov processes
Accelerated gradient methods with absolute and relative noise in the gradient
Get PDFIn this paper, we investigate accelerated first-order methods for smooth convex optimization problems under inexact information on the gradient of the objective. The noise in the gradient is considered to be additive with two possibilities: absolute noise bounded by a constant, and relative noise proportional to the norm of the gradient. We investigate the accumulation of the errors in the convex and strongly convex settings with the main difference with most of the previous works being that the feasible set can be unbounded. The key to the latter is to prove a bound on the trajectory of the algorithm. We also give a stopping criterion for the algorithm and consider extensions to the cases of stochastic optimization and composite nonsmooth problems
Functional SDE approximation inspired by a deep operator network architecture
Get PDFWe present a novel approach to solve Stochastic Differential Equations (SDEs) with Deep Neural Networks by a Deep Operator Network (DeepONet) architecture. The notion of Deep-ONets relies on operator learning in terms of a reduced basis. We make use of a polynomial chaos expansion (PCE) of stochastic processes and call the corresponding architecture SDEONet. The PCE has been used extensively in the area of uncertainty quantification with parametric partial differential equations. This however is not the case with SDE, where classical sampling methods dominate and functional approaches are seen rarely. A main challenge with truncated PCEs occurs due to the drastic growth of the number of components with respect to the maximum polynomial degree and the number of basis elements. The proposed SDEONet architecture aims to alleviate the issue of exponential complexity by learning a sparse truncation of the Wiener chaos expansion. A complete convergence analysis is presented, making use of recent Neural Network approximation results. Numerical experiments illustrate the promising performance of the suggested approach in 1D and higher dimensions
Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints
Get PDFIn this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In several rural areas in Africa with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: forecasting errors of solar power and load; frequency and outages duration from the main-grid. These uncertainties pose new challenges to the classical power system's operation tasks. Three alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC), Expected-Value Model (EVM) and a so called regular model that ignores outages and forecasting uncertainties. The JCC model has the capability to guarantee a high probability of meeting the local demand throughout an outage event by keeping appropriate reserves for Diesel generation and battery discharge. In contrast, the easier to handle ICC model guarantees such probability only individually for different time steps, resulting in a much less robust dispatch. The even simpler EVM focuses solely on average values of random variables. We illustrate the four models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. The results show the dispatch modifications for battery and Diesel reserve planning, with the JCC model providing the most robust results, albeit with a small increase in costs
Primal and dual optimal stopping with signatures
Get PDFWe propose two signature-based methods to solve the optimal stopping problem - that is, to price American options - in non-Markovian frameworks. Both methods rely on a global approximation result for Lp-functionals on rough path-spaces, using linear functionals of robust, rough path signatures. In the primal formulation, we present a non-Markovian generalization of the fa- mous Longstaff--Schwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrize the space of square-integrable martingales using linear functionals of the signature, and apply a sample average approximation. We prove convergence for both methods and present first numerical examples in non-Markovian and non-semimartingale regimes
Stochastic homogenization on perforated domains I: Extension operators
No full textThis preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W1,p to W1,r with the strict inequality rr-norm of the extended gradient in terms of the Lp-norm of the original gradient. Similar relations hold for the symmetric gradients (for ℝd-valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local (δ,M)-regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process, for which we can explicitly estimate the connectivity terms
Optical mode calculation in large-area photonic crystal surface-emitting lasers
Get PDFWe discuss algorithms and numerical challenges in constructing and resolving spectral prob- lems for photonic crystal surface-emitting lasers (PCSELs) with photonic crystal layers and large (up to several tens of mm2) emission areas. We show that finite difference schemes created using coarse numerical meshes provide sufficient accuracy for several major (lowest-threshold) modes of particular device designs. Our technique is applied to the example of large-area all- semiconductor PCSELs, showing how it can be used to optimize device performance
Optimal control of conveyor-mode spin-qubit shuttling in a Si/SiGe quantum bus in the presence of charged defects
Get PDFSpin-qubit shuttles are novel functional elements in modular architectures of semiconductor quantum processors, that have the capability of solving the scalability problem. Such coherent quantum links serve to interconnect different processor units and enable the transfer of quantum information over longer distances across the chip by physical transport of electrons. The shuttling fidelity is limited by hardly avoidable material defects and fabrication imperfections, which can cause spin dephasing. In this paper, we present a numerical simulation framework for conveyormode spin-qubit shuttling in Si/SiGe and investigate the impact of charged defects in the channel on the orbital state dynamics of the transported electron. Quantum optimal control theory is employed to engineer control pulses that enable nearly deterministic passage of the electron through the channel by minimizing the accumulated energy uncertainty. The resulting control pulses facilitate quasi-adiabatic driving of the electron by circumventing critical regions in the channel without reducing the shuttling speed. Moreover, we demonstrate that trailing electrons subject to the same control pulse at a defect-free segment of the channel are not disturbed by the control. The theoretical results serve as a guideline for fine-tuning the controls in spin-qubit shuttling experiments
CoRDI 2023
No full textResearch data form the basis for knowledge and innovation throughout all scientific disciplines. They play a fundamental role in the progress of our society. The key to using these data treasures is an effective infrastructure. With the first edition of the Conference on Research Data Infrastructure from 12 to 14 September 2023, the Association German National Research Data Infrastructure (NFDI) is initiating a conference that will focus on establishing interdisciplinary research data management (RDM). Under the theme Connecting Communities, national and international stakeholders from all research fields as well as from the infrastructure sector are invited to present their contributions to an excellent RDM of the future and to exchange information about the latest developments. NFDI is organizing the conference in cooperation with the Karlsruhe Institute of Technology (KIT). NFDI contributors as well as all other RDM interested stakeholders will have the opportunity to meet at the KIT South Campus. Over the course of three days, topics related to RDM and the joint development of an effective research data infrastructure for Germany and beyond will be examined from a wide variety of perspectives. Scientific presentations, a panel discussion, exciting keynotes, a poster session and networking activities are planned. The Conference on Research Data Infrastructure stands for more comprehensive knowledge through better use of research data, for innovations and the resulting social benefits