1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

Stochastic flows and stochastic differential equations

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.

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Stochastic discrete Hamiltonian variational integrators

Crocus sativus by-products as sources of bioactive extracts: Pharmacological and toxicological focus on anthers

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

Adverse drug events (ADEs) are common and have serious consequences in older adults. ED visits are opportunities to identify and alter the course of such vulnerable patients. Current practice, however, is limited by inaccurate reporting of medication list, time-consuming medication reconciliation, and poor ADE assessment. This manuscript describes a novel approach to predict, detect, and intervene vulnerable older adults at risk of ADE using machine learning. Toxicologists’ expertise in ADE is essential to creating the machine learning algorithm. Leveraging the existing electronic health records to better capture older adults at risk of ADE in the ED may improve their care.

Machine Learning to Predict, Detect, and Intervene Older Adults Vulnerable for Adverse Drug Events in the Emergency Department

The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

Real Time Cell Analysis (RTCA) technology is used to monitor cellular changes continuously over the entire exposure period. Combining with different testing concentrations, the profiles have potential in probing the mode of action (MOA) of the testing substances. In this paper, we present machine learning approaches for MOA assessment. Computational tools based on artificial neural network (ANN) and support vector machine (SVM) are developed to analyze the time-concentration response curves (TCRCs) of human cell lines responding to tested chemicals. The techniques are capable of learning data from given TCRCs with known MOA information and then making MOA classification for the unknown toxicity. A novel data processing step based on wavelet transform is introduced to extract important features from the original TCRC data. From the dose response curves, time interval leading to higher classification success rate can be selected as input to enhance the performance of the machine learning algorithm. This is particularly helpful when handling cases with limited and imbalanced data. The validation of the proposed method is demonstrated by the supervised learning algorithm applied to the exposure data of HepG2 cell line to 63 chemicals with 11 concentrations in each test case. Classification success rate in the range of 85 to 95 % are obtained using SVM for MOA classification with two clusters to cases up to four clusters. Wavelet transform is capable of capturing important features of TCRCs for MOA classification. The proposed SVM scheme incorporated with wavelet transform has a great potential for large scale MOA classification and high-through output chemical screening.

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Machine learning algorithms for mode-of-action classification in toxicity assessment

This paper presents a study of the approximation error corresponding to a symplectic scheme of weak order one for a stochastic autonomous Hamiltonian system. A backward error analysis is done at the level of the Kolmogorov equation associated with the initial stochastic Hamiltonian system. An expansion of the weak error and expansions of the ergodic averages and of the invariant measures associated with the numerical scheme are obtained in terms of powers of the discretization step size and the solutions of the modified Kolmogorov equation.

Weak backward error analysis for stochastic Hamiltonian Systems

We present high-order symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. The approach is based on the generating function method, and we show that for the stochastic Hamiltonian systems, the coefficients of the generating function are invariant under permutations. As a consequence, the high-order symplectic schemes have a simpler form than the explicit Taylor expansion schemes with the same order. Moreover, we demonstrate numerically that the symplectic schemes are effective for long time simulations.

Cristina Anton

Papers

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

Machine learning algorithms for mode-of-action classification in toxicity assessment

Weak backward error analysis for stochastic Hamiltonian Systems

Symplectic Numerical Schemes for Stochastic Systems Preserving Hamiltonian Functions

Hopf bifurcation analysis of an aeroelastic model using stochastic normal form