skip to main content

Title: Approximate analytical solutions for a class of nonlinear stochastic differential equations
An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
European Journal of Applied Mathematics
Page Range / eLocation ID:
1 to 17
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Predicting the process of porosity-based ductile damage in polycrystalline metallic materials is an essential practical topic. Ductile damage and its precursors are represented by extreme values in stress and material state quantities, the spatial probability density function (PDF) of which are highly non-Gaussian with strong fat tails. Traditional deterministic forecasts utilizing sophisticated continuum-based physical models generally lack in representing the statistics of structural evolution during material deformation. Computational tools which do represent complex structural evolution are typically expensive. The inevitable model error and the lack of uncertainty quantification may also induce significant forecast biases, especially in predicting the extreme events associated with ductile damage. In this paper, a data-driven statistical reduced-order modeling framework is developed to provide a probabilistic forecast of the deformation process of a polycrystal aggregate leading to porosity-based ductile damage with uncertainty quantification. The framework starts with computing the time evolution of the leading few moments of specific state variables from the spatiotemporal solution of full- field polycrystal simulations. Then a sparse model identification algorithm based on causation entropy, including essential physical constraints, is utilized to discover the governing equations of these moments. An approximate solution of the time evolution of the PDF is obtained from the predicted moments exploiting the maximum entropy principle. Numerical experiments based on polycrystal realizations of a representative body-centered cubic (BCC) tantalum illustrate a skillful reduced-order model in characterizing the time evolution of the non-Gaussian PDF of the von Mises stress and quantifying the probability of extreme events. The learning process also reveals that the mean stress is not simply an additive forcing to drive the higher-order moments and extreme events. Instead, it interacts with the latter in a strongly nonlinear and multiplicative fashion. In addition, the calibrated moment equations provide a reasonably accurate forecast when applied to the realizations outside the training data set, indicating the robustness of the model and the skill for extrapolation. Finally, an information-based measurement is employed to quantitatively justify that the leading four moments are sufficient to characterize the crucial highly non-Gaussian features throughout the entire deformation history considered. 
    more » « less
  2. Sun, Weichao ; Yao, Bin (Ed.)
    This paper introduces an analytically tractable and computationally efficient model for legged robot dynamics during locomotion on a dynamic rigid surface (DRS), along with an approximate analytical solution and a real-time walking pattern generator synthesized based on the model and solution. By relaxing the static-surface assumption, we extend the classical, time-invariant linear inverted pendulum (LIP) model for legged locomotion on a static surface to dynamic-surface locomotion, resulting in a time-varying LIP model termed as “DRS-LIP”. Sufficient and necessary stability conditions of the time-varying DRS-LIP model are obtained based on the Floquet theory. This model is also transformed into Mathieu’s equation to derive an approximate analytical solution that provides reasonable accuracy with a relatively low computational cost. Using the extended model and its solution, a walking pattern generator is developed to efficiently plan physically feasible trajectories for quadrupedal walking on a vertically oscillating surface. Finally, simulations and hardware experiments from a Laikago quadrupedal robot walking on a pitching treadmill (with a maximum vertical acceleration of 1 m/s ) confirm the accuracy and efficiency of the proposed analytical solution, as well as the efficiency, feasibility, and robustness of the pattern generator, under various surface motions and gait parameters. 
    more » « less
  3. Abstract

    We study the evolution of a nonrelativistically expanding thin shell in radio-emitting tidal disruption events (TDEs) based on a one-dimensional spherically symmetric model considering the effect of both a time-dependent mass-loss rate of the disk wind and the ambient mass distribution. The analytical solutions are derived in two extreme limits; one is the approximate solution near the origin in the form of the Taylor series, and the other is the asymptotic solution in which the ambient matter is dominant far away from the origin. Our numerical solutions are confirmed to agree with the respective analytical solutions. We find that no simple power law of the time solution exists in early to middle times because the mass-loss rate varies over time, affecting the shell dynamics. We also discuss the application of our model to the observed radio-emitting TDE, AT 2019dsg.

    more » « less
  4. These files are supplementary data for this publication:

    Uhl JH & Leyk S (2022). "Assessing the relationship between morphology and mapping accuracy of built-up areas derived from global human settlement data (

    Each geopackage (GPKG) file contains a set of point locations (in EPSG:3857) attributed with focal accuracy metrics of the GHS-BUILT-R2018A epochs 1975 and 2014, calculated within different levels of spatial support (i.e., focal window size) and for different analytical units (i.e., 30m grid cells, and 3x3 grid cell blocks). Moreover, each location is attributed with focal landscape metrics of built-up areas calculated in the same focal windows using the software Fragstats. These landscape metrics are calculated based on both, GHS built-up areas and reference built-up areas. Reference built-up areas were derived from the Multi-temporal building footprint database for 33 U.S. counties (MTBF-33). These datasets can be used for spatially explicit predictive modeling of the GHS-BUILT R2018A data accuracy using landscape metrics as predictor variables.

    File nomenclature:

    lsm_ref_accuracy_sample_2014_1000.gpkg : landscape metrics calculated from the reference built-up areas, for the epoch 2014, using a quadratic focal window of 1,000m x 1,000m.

    lsm_ghs_accuracy_sample_1975_10000.gpkg : landscape metrics calculated from the ghs built-up areas, for the epoch 1975, using a quadratic focal window of 10,000m x 10,000m.

    Data processing: Johannes H. Uhl, University of Colorado Boulder (USA), 2020-2022.

    more » « less
  5. Fugacity is a fundamental thermodynamical property of gas and gas mixtures to determine their behavior and dynamics in complex systems. Fugacity can be deduced experimentally from the measurements of volume as a function of pressure at constant temperature or calculated iteratively using analytical equations of states (EOS). Experimental measurement is time-consuming, and analytical models based on EOS are computationally demanding, especially when an approximate but quick estimation is desired. In this work, machine learning (ML) is employed as a viable alternative to analytical EOSs for quick and accurate approximation of CO2 fugacity coefficients. Five different ML algorithms are used to estimate the fugacity coefficients of pure CO2 as a function of pressure (≤ 2000 bar) and temperature (≤ 1000 °C). A combination of experimental and pseudo-experimental (obtained from an analytical EOS) data of CO2 fugacity coefficients is used to train, validate, and test the models. The best results were found using the Extreme Gradient Boosting algorithm, which showed a mean square error of only 0.0002 in the validation data and an average deviation of only 1.3% in the test data (pure prediction). To quantify the effectiveness of the machine learning techniques, results from the best-performing model are compared with two state-of-the-art analytical models. The ML model with significantly less computational complexity showed similar accuracy to the analytical models. The estimated fugacity data are then used to compute the CO2 solubility in aqueous NaCl solution of different concentrations, and a maximum deviation of only 3.2% from the experimental data is observed. 
    more » « less