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Exploiting expansion basis sparsity for efficient stochastic response determination of nonlinear systems via the Wiener path integral technique


The computational efficiency of the Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear dynamical systems is enhanced herein by relying on advanced compressive sampling concepts and tools. Specifically, exploiting the sparsity of appropriately selected expansions for the joint response probability density function (PDF), and leveraging the localization capabilities of the WPI technique for direct evaluation of specific PDF points, yield an underdetermined linear system of equations to be solved for the PDF expansion coefficients. This is done by resorting to \( L_p \)-norm \( (0<p<1) \) minimization formulations and algorithms, which exhibit an enhanced sparsity-promoting behavior compared to standard \( L_1 \)-norm minimization approaches. This translates into a significant reduction of the associated computational cost. In fact, for approximately the same accuracy degree, it is shown that the herein developed technique based on \( L_p \)-norm \( (0<p<1) \) minimization requires, in some cases, even up to 40% fewer boundary value problems to be solved as part of the solution scheme than a standard \( L_1 \)-norm minimization approach. The reliability of the technique is demonstrated by comparing WPI-based response PDF estimates with pertinent Monte Carlo simulation (MCS) data (10,000 realizations). In this regard, realizations compatible with the excitation stochastic process are generated, and response time-histories are obtained by integrating numerically the nonlinear system equations of motion. Next, MCS-based PDF estimates are computed based on statistical analysis of the response time-histories. Several numerical examples are considered pertaining to various stochastically excited oscillators exhibiting diverse nonlinear behaviors. These include a Duffing oscillator, an oscillator with asymmetric nonlinearities, and a nonlinear vibro-impact oscillator.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.


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I. A. Kougioumtzoglou gratefully acknowledges the support through his CAREER award by the CMMI Division of the National Science Foundation, USA (Award no. 1748537). Y. Zhang acknowledges the support by “the Fundamental Research Funds for the Central Universities (WUT:2021IVA091)”. F. Kong acknowledges the support by the National Natural Science Foundation of China (Grant no. 52078399).

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In this Appendix, more details are provided on the derivation of the Euler–Lagrange Eq. (5) to be solved in conjunction with the boundary conditions of Eq. (6) for determining the WPI most probable path (see also Refs. [11, 12]). Specifically, considering the Lagrangian functional of Eq. (3), the stochastic action \( S[x,{\dot{x}},\ddot{x}] \) is defined as

$$\begin{aligned} S[x,{\dot{x}},\ddot{x}]=\int _{t_{i}}^{t_{f}} L[x,{\dot{x}},\ddot{x}] dt \end{aligned}$$

Next, by utilizing a functional Taylor-type series expansion of the stochastic action \( S[x,{\dot{x}},\ddot{x}] \) and by expressing x(t) as

$$\begin{aligned} x(t)=x_{c}(t)+X(t) \end{aligned}$$

the stochastic action S is written as:

$$\begin{aligned} \begin{aligned} S[x]=S[x_{c}+X] =&S[x_{c}]+ \delta S[x_{c},X] \\&+\frac{1}{2!} \delta ^2 S[x_{c},X] + ... \end{aligned} \end{aligned}$$

In Eq. (A.3), \( x_{c}(t) \) is the path associated with the maximum probability of occurrence and X(t) , with \( X(t_{i})=X(t_{f})= {\dot{X}}(t_{i}) ={\dot{X}}(t_{f}) =0 \), represents the fluctuations around \( x_{c}(t) \) (e.g., [8]). In Eq. (A.3), \( S[x,{\dot{x}},\ddot{x}] \) is denoted as S[x] for simplicity, and \( \delta S[x_{c},X] \) represents the functional differential (or variation) of S evaluated on \( x_{c} \). Moreover, according to calculus of variations [17] and considering Eqs. (A.1) and (A.3), the differential \( \delta S[x_{c},X] \) takes the form

$$\begin{aligned} \begin{aligned} \delta S[x_{c},X]=&\int _{t_i}^{t_f} \bigg ( \dfrac{\partial L}{\partial x}\bigg |_{x=x_c} X + \dfrac{\partial L}{\partial {\dot{x}}}\bigg |_{x=x_c} {\dot{X}} \\&+ \dfrac{\partial L}{\partial \ddot{x}}\bigg |_{x=x_c} {\ddot{X}} \bigg ) dt \end{aligned} \end{aligned}$$

Further, considering Eqs. (2) and (A.1), it is seen that the maximum probability of occurrence corresponds to minimum \( S[x,{\dot{x}},\ddot{x}] \). Thus, \( x_{c}(t) \) is associated with an extremum of the functional \( S[x,{\dot{x}},\ddot{x}] \) . In this context, calculus of variations dictates [17] that the first variation of \( S[x,{\dot{x}},\ddot{x}] \) vanishes for \( x(t)=x_{c}(t) \), i.e.,

$$\begin{aligned} \begin{aligned} \delta S[x_{c},X]= 0 \end{aligned} \end{aligned}$$

Therefore, Eq. (A.3) becomes

$$\begin{aligned} \begin{aligned} S[x]=S[x_{c}]+ \frac{1}{2!} \delta ^2 S[x_{c},X] + ... \end{aligned} \end{aligned}$$

Furthermore, combining Eq. (A.4) and the extremality condition of Eq. (A.5) yields

$$\begin{aligned} \begin{aligned} \int _{t_i}^{t_f} \bigg ( \dfrac{\partial L}{\partial x}\bigg |_{x=x_c} X&+ \dfrac{\partial L}{\partial {\dot{x}}}\bigg |_{x=x_c} \dfrac{d}{dt} X \\&+ \dfrac{\partial L}{\partial \ddot{x}}\bigg |_{x=x_c} \dfrac{d^2}{dt^2} X \bigg ) dt =0 \end{aligned} \end{aligned}$$

Next, integrating Eq. (A.7) by parts leads to

$$\begin{aligned} \begin{aligned}&\underbrace{ \bigg [ \bigg ( \dfrac{\partial L}{\partial {\dot{x}}}\bigg |_{x=x_c} - \dfrac{d}{dt} \dfrac{\partial L}{\partial \ddot{x}}\bigg |_{x=x_c} \bigg ) X(t) \bigg ]_{t_i}^{t_f} }_{{\mathscr {A}}_0} + \underbrace{ \bigg [ \dfrac{\partial L}{\partial \ddot{x}}\bigg |_{x=x_c} {\dot{X}}(t) \bigg ]_{t_i}^{t_f} }_{{\mathscr {A}}_1} \\&\qquad + \underbrace{ \int _{t_i}^{t_f} \bigg ( \dfrac{\partial L}{\partial x}\bigg |_{x=x_c} - \dfrac{d}{dt} \dfrac{\partial L}{\partial {\dot{x}}}\bigg |_{x=x_c} + \dfrac{d^2}{dt^2} \dfrac{\partial L}{\partial \ddot{x}}\bigg |_{x=x_c} \bigg ) X(t) dt }_{{\mathscr {B}}} \\&=0 \end{aligned} \end{aligned}$$

where, since \( X(t_i) = X(t_f ) = {\dot{X}} (t_i) = {\dot{X}} (t_f) = 0 \), the terms \( {\mathscr {A}}_0 \) and \( {\mathscr {A}}_1 \) vanish. Thus, Eq. (A.8) yields the Euler–Lagrange Eq. (5) to be solved in conjunction with the boundary conditions of Eq. (6) for determining the WPI most probable path \( x_c \).

Note that \( x_c \) can be used, at least in principle, for evaluating the higher-order terms in the expansion of Eq. (A.6). However, in the majority of practical implementations of the WPI technique, only the first term \( S[x_c ] \) is retained in the expansion of Eq. (A.6), since the evaluation of higher-order terms exhibits considerable analytical and computational challenges. In fact, in the most probable path approximation shown in Eq. (7) the remaining terms in the expansion of Eq. (A.6) are treated collectively as a constant C. The interested reader is also directed to [12], where a quadratic approximation was developed for the WPI that accounts explicitly for the contribution also of the second variation term in Eq. (A.6).

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Zhang, Y., Kougioumtzoglou, I.A. & Kong, F. Exploiting expansion basis sparsity for efficient stochastic response determination of nonlinear systems via the Wiener path integral technique. Nonlinear Dyn 107, 3669–3682 (2022).

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  • Sparse representations
  • Compressive sampling
  • Nonlinear systems
  • Stochastic dynamics
  • Path integral