Mean-square Convergence of a Symplectic Local Discontinuous Galerkin Method Applied to Stochastic Linear Schroedinger Equation

In this paper, we investigate the mean-square convergence of a novel symplectic local discontinuous Galerkin method in L^2-norm for stochastic linear Schroedinger equation with multiplicative noise. It is shown that the mean-square error is bounded not only by the temporal and spatial step-sizes, but also by their ratio. The mean-square convergence rate with respect to the computational cost is derived under appropriate assumptions for initial data and noise. Meanwhile, we show that the method preserves the discrete charge conservation law which implies an L^2-stabilityComment: 19 page

Exploiting Partially Annotated Data for Temporal Relation Extraction

Annotating temporal relations (TempRel) between events described in natural language is known to be labor intensive, partly because the total number of TempRels is quadratic in the number of events. As a result, only a small number of documents are typically annotated, limiting the coverage of various lexical/semantic phenomena. In order to improve existing approaches, one possibility is to make use of the readily available, partially annotated data (P as in partial) that cover more documents. However, missing annotations in P are known to hurt, rather than help, existing systems. This work is a case study in exploring various usages of P for TempRel extraction. Results show that despite missing annotations, P is still a useful supervision signal for this task within a constrained bootstrapping learning framework. The system described in this system is publicly available.Comment: [Final Version] short paper accepted by *SEM'1

Conservative methods for stochastic differential equations with a conserved quantity

This paper proposes a novel conservative method for numerical computation of general stochastic differential equations in the Stratonovich sense with a conserved quantity. We show that the mean-square order of the method is $1$ if noises are commutative and that the weak order is also $1$. Since the proposed method may need the computation of a deterministic integral, we analyse the effect of the use of quadrature formulas on the convergence orders. Furthermore, based on the splitting technique of stochastic vector fields, we construct conservative composition methods with similar orders as the above method. Finally, numerical experiments are presented to support our theoretical results

Approximation of Invariant Measure for Damped Stochastic Nonlinear Schr\"{o}dinger Equation via an Ergodic Numerical Scheme

In order to inherit numerically the ergodicity of the damped stochastic nonlinear Schr\"odinger equation with additive noise, we propose a fully discrete scheme, whose spatial direction is based on spectral Galerkin method and temporal direction is based on a modification of the implicit Euler scheme. We not only prove the unique ergodicity of the numerical solutions of both spatial semi-discretization and full discretization, but also present error estimations on invariant measures, which gives order $2$ in spatial direction and order ${\frac12}$ in temporal direction

Preservation of Physical Properties of Stochastic Maxwell Equations with Additive Noise via Stochastic Multi-symplectic Methods

Stochastic Maxwell equations with additive noise are a system of stochastic Hamiltonian partial differential equations intrinsically, possessing the stochastic multi-symplectic conservation law.It is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic Maxwell equations with additive noise preserves the divergence in the sense of expectation. Moreover, we propose three novel stochastic multi-symplectic methods to discretize stochastic Maxwell equations in order to investigate the preservation of these properties numerically. We made theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence. Meanwhile, we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method. Especially, the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case. Numerical experiments are performed to verify our theoretical results.Comment: This paper has been withdrawn by the author due to an error in Theorem

Road to safe autonomy with data and formal reasoning

We present an overview of recently developed data-driven tools for safety analysis of autonomous vehicles and advanced driver assist systems. The core algorithms combine model-based, hybrid system reachability analysis with sensitivity analysis of components with unknown or inaccessible models. We illustrate the applicability of this approach with a new case study of emergency braking systems in scenarios with two or three vehicles. This problem is representative of the most common type of rear-end crashes, which is relevant for safety analysis of automatic emergency braking (AEB) and forward collision avoidance systems. We show that our verification tool can effectively prove the safety of certain scenarios (specified by several parameters like braking profiles, initial velocities, uncertainties in position and reaction times), and also compute the severity of accidents for unsafe scenarios. Through hundreds of verification experiments, we quantified the safety envelope of the system across relevant parameters. These results show that the approach is promising for design, debugging and certification. We also show how the reachability analysis can be combined with statistical information about the parameters, to assess the risk level of the control system, which in turn is essential, for example, for determining Automotive Safety Integrity Levels (ASIL) for the ISO26262 standard.Comment: 7 pages, 5 figures, under submission to IEEE Design & Tes

Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order $1$ of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation

Verifying nonlinear analog and mixed-signal circuits with inputs

We present a new technique for verifying nonlinear and hybrid models with inputs. We observe that once an input signal is fixed, the sensitivity analysis of the model can be computed much more precisely. Based on this result, we propose a new simulation-driven verification algorithm and apply it to a suite of nonlinear and hybrid models of CMOS digital circuits under different input signals. The models are low-dimensional but with highly nonlinear ODEs, with nearly hundreds of logarithmic and exponential terms. Some of our experiments analyze the metastability of bistable circuits with very sensitive ODEs and rigorously establish the connection between metastability recovery time and sensitivity.Comment: 8 pages, 8 figures, a shorter version will appear on the IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 2018

The probabilistic superiority of stochastic symplectic methods via large deviations principles

It is well known that symplectic methods have been rigorously shown to be superior to non-symplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the probabilistic superiority of stochastic symplectic methods by means of the theory of large deviations. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Concerning that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the G\"artner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but non-symplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the "hitting probability" of the mean position and mean velocity of the stochastic oscillator. To the best of our knowledge, this is the first result about using large deviations principle to show the superiority of stochastic symplectic methods compared with non-symplectic ones in the existing literature

Room temperature multiplexed gas sensing using chemical-sensitive 3.5-nm-thin silicon transistors.

There is great interest in developing a low-power gas sensing technology that can sensitively and selectively quantify the chemical composition of a target atmosphere. Nanomaterials have emerged as extremely promising candidates for this technology due to their inherent low-dimensional nature and high surface-to-volume ratio. Among these, nanoscale silicon is of great interest because pristine silicon is largely inert on its own in the context of gas sensing, unless functionalized with an appropriate gas-sensitive material. We report a chemical-sensitive field-effect transistor (CS-FET) platform based on 3.5-nm-thin silicon channel transistors. Using industry-compatible processing techniques, the conventional electrically active gate stack is replaced by an ultrathin chemical-sensitive layer that is electrically nonconducting and coupled to the 3.5-nm-thin silicon channel. We demonstrate a low-power, sensitive, and selective multiplexed gas sensing technology using this platform by detecting H2S, H2, and NO2 at room temperature for environment, health, and safety in the oil and gas industry, offering significant advantages over existing technology. Moreover, the system described here can be readily integrated with mobile electronics for distributed sensor networks in environmental pollution mapping and personal air-quality monitors