This problem of uncertainty quantification in one dimension is depicted in Fig. 2a, where the solid blue curve denotes the known distribution at t0, and the broken green line indicates the unknown distribution at tf. In the first step, confinement of the region of interest at tf is done. This is the region in which the approximation of the unknown final PDF will be performed. This is done by locating the state space points at t0 whose PDF values match the boundary PDF value of interest. In this work, a PDF value equivalent to 6σ in a Gaussian distribution is specified as the boundary PDF value at t0. These state space points are then propagated to tf using the ordinary differential equations (ODEs) that govern the system. In this work, unless specified otherwise, all the ODEs are numerically integrated using MATLAB’s Runge-Kutta 45 integrator (ode45) with the tolerance values, both relative and absolute, set to 10⁻⁸. The region between these state space points, or their generalized surface in higher dimensions, is referred to as the region of extremal probability. This is depicted in Fig. 2b, where the red dashes on the x-axis mark the extremal bounds. Obtaining the extremal bounds at t0 is straightforward for well-characterized distributions (e.g., Gaussian or Cauchy). For more complex a priori distributions, a randomized searching method would be required. Unless stated otherwise, the initial distribution for all the cases presented in this work is Gaussian.

After locating the extremal bounds at tf, the space between them is populated with grid points whose PDF values will be computed from the a priori distribution and then used in the subsequent approximation process. Fig. 2c depicts cosine nodes distributed between the extremal bounds at tf. To achieve this, each evaluation node at tf is backpropagated to t0 using the system ODEs, with the state information of these nodes at tf serving as the initial condition. The PDF value at the backpropagated grid points is obtained by comparing the a priori distribution with the backpropagated state value of these nodes. According to Liouville’s theorem, these PDF values always remain unchanged for conservative systems. Hence, these PDF values are referenced to the grid points at tf, where they are utilized to generate a functional approximation of the final probability distribution using Chebyshev polynomials, f(x)≅∑k=0nakφk(x), where, n is the order of approximation, φ_k(x) is the Chebyshev polynomial of order k in x, and a_k is the approximation coefficient. Fig. 2d depicts the back-propagation of each of the grid points, and Fig. 2e shows the resulting approximation for the posterior PDF. It is important to note that the mapping of the known PDF values at the backpropagated grid points occurs from t0 to tf through the utilization of Liouville’s theorem. This step is also known as referencing the PDF values. The order of the approximation, n, is determined by the accuracy requirements of the PDF approximation, and the corresponding coefficients are estimated using a least-squares process. Since the Chebyshev polynomials φ_k(x) are orthogonal within the domain of approximation, the matrix inversion step in the least squares process can be executed more efficiently. The approximation process is performed using Chebfun, an open-source MATLAB toolbox for Chebyshev polynomial approximation [36]. Unless specified otherwise, all numerical results shown in this paper were obtained using the Chebfun toolbox.

The OPA technique outlined in Section 2.1 is based on Liouville’s theorem which states that the probability density value of a state space point, defined by an a priori distribution with initial conditions, remains constant as long as the point traverses a unique state space trajectory. This is encapsulated in Liouville’s equation [37] as, (1) dρdt=∂ρ∂t+∑i=1m(∂ρ∂qiqi˙+∂ρ∂pipi˙)=0where, q, p denote the state variables, and the number of random variables is indicated by m.

This is true only for conservative dynamical systems. For systems that involve non-conservative dynamics, energy may either be removed (e.g., friction and drag) or added (e.g., thruster, solar radiation input on satellite solar panels) to the system. Hence, Eq. (1) is no longer applicable, and the probability density of a given state space point changes with time. This variation can be quantified by the fact that, at any time instant, the integral of the probability density surface with respect to all the random variables must equal one. For example, in the case of dissipative systems, for the integral of the probability density surface to equal one at any instant, the PDF value of a state space point must increase as it moves forward in time. This increase continues until the system reaches a steady state, which has the maximum probability. Conversely, the opposite occurs in a system to which energy is added: the PDF value of a state space point decreases as it moves forward in time.

Fig. 3 illustrates the effect of energy dissipation in a damped Duffing oscillator which is a system with 2 random variables (i.e., position and velocity). As shown, the area covered by the extremal bounds at tf (green) is smaller than the area covered by the extremal bounds at t0 (red). This means that the associated PDF at tf increases in value in proportion to the area shrinkage of the extremal bounds at tf. In Fig. 2d, each back-propagated evaluation node picks up a PDF value based on its location in the state space and the a priori PDF. This PDF value is scaled or modified to reflect the area shrinkage. For problems with two random variables, a local scale factor is adopted for each evaluation node. The area of the triangle for which each evaluation node is a vertex is computed. The area change of each such triangle during back-propagation is tracked and used to compute the scale factor, which is then used to scale the PDF value. If A_fi and A_0i denote the area of the triangle associated with the i-th evaluation node at tf and t0, respectively, then the associated scale factor, SF_i is obtained by,

(2) A0i=12|x100x2001x110x2101x120x2201|i (3) Afi=12|x10fx20f1x11fx21f1x12fx22f1|i (4) A0i=12|a001a101a201|i (5) Afi=12|a0f1a1f1a2f1|i (6) SFi=A0iAfiwhere, a₀₀ (x₁₀₀, x₂₀₀), a₁₀ (x₁₁₀, x₂₁₀), a₂₀ (x₁₂₀, x₂₂₀) represent the vector locations of the triangle vertices/nodes at t0, and a_0f (x_10f, x_20f), a_1f (x_11f, x_21f), a_2f (x_12f, x_22f) represent the vector locations of the triangle vertices/nodes at t_f. This PDF modulation process for a single triangle is illustrated in Fig. 4. A systematic repetition of this process is done by constructing adjacent triangles to obtain the scale factors associated with all nodes (i.e., i = 0, 1, 2, ..., n). The referenced PDF value from t0, at each evaluation node is then multiplied/scaled by the scale factor SF_i before the Chebyshev approximation step.

Similarly, for a non-conservative dynamical system with 3 random variables (x₁, x₂ and x₃), a local scale factor is adopted for each evaluation node. The volume of the tetrahedron for which each evaluation node is a vertex is computed. The volume change of each such tetrahedron during back-propagation is tracked and used to compute the scale factor, which is then applied to scale the PDF value. If V_fi and V_0i denote the volume of the tetrahedron associated with the i-th evaluation node at tf and t0, respectively, and if SF_i is the associated scale factor, then, (7) V0i=13!|x110−x100x120−x100x130−x100x210−x200x220−x200x230−x200x310−x300x320−x300x330−x300|i (8) Vfi=13!|x11f−x10fx12f−x10fx13f−x10fx21f−x20fx22f−x20fx23f−x20fx31f−x30fx32f−x30fx33f−x30f|i (9) V0i=13!|a10−a00a20−a00a30−a00|i (10) Vfi=13!|a1f−a0fa2f−a0fa3f−a0f|i (11) SFi=V0iVfiwhere a₀₀, a₁₀, a₂₀, a₃₀ represent the vector locations of the tetrahedron vertices at t0 and a_0f, a_1f, a_2f, a_3f represent the vector locations of the tetrahedron vertices at tf. This process is illustrated for a single tetrahedron in Fig. 5. The process is systematically repeated to obtain the scale factors associated with each evaluation node, after which they are used to scale the referenced PDF values at each node before they are passed to the approximation step. For a non-conservative dynamical system with n random variables, the scale factor associated with each evaluation node is obtained by evaluating the hypervolume change associated with the corresponding n-dimensional simplex. Using linear algebra, the scale factor associated with the i-th evaluation node can be computed by tracking the hypervolume change of the corresponding n-dimensional simplex,

(12) V0i=1n!|x110−x100x120−x100…x1n0−x100x210−x200x220−x200…x2n0−x200⋮⋮⋮⋮xn10−xn00xn20−xn00…xnn0−xn00|i (13) Vfi=1n!|x11f−x10fx12f−x10f…x1nf−x10fx21f−x20fx22f−x20f…x2nf−x20f⋮⋮⋮⋮xn1f−xn0fxn2f−xn0f…xnnf−xn0f|i (14) V0i=1n!|a10−a00a20−a00…an0−a00|i (15) Vfi=1n!|a1f−a0fa2f−a0f…anf−a0f|i (16) SFi=V0iVfiwhere, a₀₀, a₁₀, ..., a_n0 represent the vector locations of the simplex vertices at t0, and a_0f, a_1f, ..., a_nf represent the vector locations of the simplex vertices at tf. Despite being a very precise way to scale the n-dimensional PDF, this local technique is computationally intensive. An alternative, albeit less accurate, approach is to estimate the scale factor by computing the overall hypervolume shrinkage associated with the n-dimensional extremal bounds. This can be done using the convexhull function in MATLAB.

An important step in OPA is the back-propagation of the grid points (Fig. 2d). This step presents two key challenges that need to be addressed: i) the curse of dimensionality, and ii) the system’s nonlinear dynamics. In lower-dimensional problems, up to 2D, even with an approximation order as high as 70, the time required for the back-propagation of, for example, 4900 (= 70 × 70) points is significantly less than the time required for a Monte Carlo simulation that achieves the same PDF approximation accuracy. However, in higher dimensional PDF approximation, the number of points created in a tensorial grid increases exponentially. But most of these grid points are clustered near the boundary where the PDF value is essentially zero. Hence, with each dimension, the number of grid points required for PDF approximation increases by nearly an order of magnitude, of which only a small fraction of them are in locations where the PDF values are non-zero. This phenomenon is known as the curse of dimensionality [38].

The second challenge is the divergence that can occur during the back-propagation of nonlinear dynamics (i.e., one or more state space variables become extremely large or approach infinity). In Fig. 6, every point on the extremal bounds at tf is obtained by forward propagation of the points on the extremal bounds at t0. Each point in the overlaid Chebyshev grid is a state space point that, in principle, can be backpropagated to t0. In theory, after back-propagation, every grid point that lies within the extremal bounds at tf (i.e., an internal grid point) will land within the extremal bounds at t0, and every grid point that lies outside the extremal bounds at tf (i.e., an external grid point) will land outside the extremal bounds at t0. However, in practice, for nonlinear systems, some of these external grid points may land far outside the extremal bounds at t0, and some may even diverge to infinity. Additionally, it is unnecessary to backpropagate these points, as the PDF value at these points is negligible (nearly zero) and not of interest to the problem. To address these issues, points outside the extremal bounds are filtered out, and only the points within the extremal bounds are backpropagated. This filtering improves the efficiency of OPA.

The grid filtering implemented in this paper uses dot products to determine which grid points lie within the extremal bounds. For the 2D case, in the rectangle ABCD shown in Fig. 7a, the points p₁ and p₂ are checked by first computing the unit normal vectors n^₁, . . ., n^₄ for the four sides of the rectangle ABCD. The next step is to ensure that all the obtained unit normal vectors are pointing inward. This is done by ensuring that the dot product between the normal vector and the vector from the geometric center to a point on the side is positive. Finally, the test point p_i is checked to see if it lies within the rectangle by evaluating the dot product between each of the normal vectors and the vector connecting p_i to any point on the side. This dot product value must be non-negative for p_i to lie inside ABCD.

Similarly, for the 3D case, in the tetrahedron ABCD shown in Fig. 7b, the points p₁ and p₂ are checked by first computing the unit normal vectors for all faces of the tetrahedron. After ensuring the inward pointedness of the unit normal vectors of each face, the dot product between each normal vector and the vector connecting the test point p_i to any point on the face is computed. If this dot product is non-negative, then p_i lies inside the tetrahedron ABCD. The detailed steps of the process are shown in Algorithm A1 in Appendix A. Algorithm A1 can be applied to any n-sided polygon or n-faced polyhedron. For dimensions higher than 3D, the same dot product inequality check is applied for grid filtering. However, this requires the computation of the null space to obtain the unit normal vectors for each simplex of the n-D convex hull. These procedures are documented and implemented in a user-created MATLAB function, inhull [39].

All the steps of OPA described so far are encapsulated in the flowchart shown in Fig. 1. When implementing OPA for nonlinear systems, Monte Carlo (MC) simulations are used to validate the outcomes of OPA. The first step is to create a significant number of sample/test points (i.e., initial conditions) based on the system’s initial known distribution. Typically, around 1 million samples are generated to achieve a reasonable approximation of 6σ extremal bounds. Each of these sample points is then propagated to tf, after which the kernel density estimator is used to generate the PDF from these propagated sample points. As in the OPA simulation, marginal PDFs and the cumulative probability integral are computed. The accuracy of the cumulative probability integrals from the OPA simulation and MC simulation are then compared. Because the initial PDF is truncated in these simulations, according to probability theory, the cumulative probability integral should remain constant between the initial and final PDFs and should be very close to one if the initial PDF is truncated at a 6σ distance from the mean. The approximation order and the number of samples are chosen such that both OPA and MC simulations achieve the same level of accuracy. Consequently, computational efficiency is quantified, for the same level of accuracy of the cumulative integral, by recording the runtime for each method. Algorithm A2 describes the sequence of steps required to implement OPA for a general n-dimensional dynamical system. All the numerical experiments shown in this paper were run on an HDD desktop computer with an Intel(R) Core(TM) i7 processor at a clock speed of 3.4 GHz and 16 GB RAM.

A simple undamped linear system with a unit mass is chosen as a representative of linear systems to validate the implementation of OPA, (17) X˙=[01−ωn20]X,X˙=AXwhere, X=[xx˙]T and ω_n is the natural frequency. The relevant OPA simulation parameters can be found in Table 1. In this dynamic system with two random variables, position, and velocity, the 6σ initial extremal bounds of the Gaussian distribution are defined using 100 points. The initial covariance for this dynamic system is given by P₀ = diag([0.03² 0.05²]). Here, Liouville’s theorem is used to place 100 points at the 6σ bounds from the mean values of position and velocity shown in Table 1. First, the eigenvalues and eigenvectors of the initial covariance matrix are obtained. Then, a set of random points on the surface of an n-dimensional unit sphere is generated. The dimension of this set of points corresponds to the dimensionality of the problem, and the cardinality of this set is the number of extremal bound points specified by the user. These random points are then scaled by the product of the initial standard deviation and the value of the extremal bounds (i.e., 6σ for the cases considered in this paper). Finally, these points are oriented using the eigenvectors and translated to lie on the 6σ surface of the distribution, whose mean is specified in Table 1. Nominal values for initial position and velocity, along with the 6σ initial extremal bounds, are propagated through the linear dynamics, Eq. (17), to a chosen final time, tf (Fig. 8).

For an approximation order of 30, the total number of grid points comes to 30² = 900. The probability distribution at any given time is a function of the state variables: position and velocity, which are also the random variables in the linear system. A 2D Chebyshev grid is formed using, (18) Xij={cos⁡(iπn1),cos⁡(jπn2)},0≤i≤n1,0≤j≤n2,i,j∈Z

For this case, the approximation order for both random variables is set to be equal, i.e., n₁ = n₂ = n. The grid created encompasses the extremal bounds at tf as shown in Fig. 8. Following the procedure outlined in Section 2.3, the external grid points are eliminated. Each grid point within the extremal bound is back-propagated to the initial time, t0 to obtain the corresponding PDF value (Fig. 9), which, according to Liouville’s theorem, remains unchanged for a conservative system. This set of referenced PDF values at the Chebyshev grid points is then used to approximate the PDF at tf, Fig. 10. The process of approximation is carried out using Chebfun, an open-source MATLAB package that implements Chebyshev polynomial approximations. In a linear dynamic system, the covariance matrix can be numerically propagated to t _f using the equation from linear error theory [40],

(19) P˙=AP+PAT (20) P(t0)=P0where, A is the system matrix, P is the covariance matrix and P₀ is the initial covariance matrix. Using the expression for the PDF value of a multivariate Gaussian distribution, (21) fanalytical=(2π)−k/2|Σ|−1/2exp⁡(−12(X−µ)TΣ−1(X−µ))and plugging in μ=[μxfμx˙f] (the propagated mean position and velocity), k = 2 (the cardinality of random variables), and Σ = P (the propagated covariance matrix), the analytical probability density values at the Chebyshev grid points at tf are obtained. Fig. 10 shows the final PDF obtained after approximation, which is compared with the PDF obtained from linear error propagation using Eqs. (19) and (20). The results show errors in the order of 10⁻⁷ (Fig. 11). Besides direct comparison of the PDFs, an important accuracy metric in PDF approximations is the evaluation of the total probability integral value, at tf, which is given by,

(22) (If)OPA=∫x∫x˙f(x,x˙)dx˙dxwhere, f(x, x˙) is the functional approximation of the PDF. The integral of f(x, x˙) within the 6σ extremal bounds at tf, evaluated using the Clenshaw-Curis quadrature, turns out to be 0.999999991460116 [41]. Since this linear system is assumed to have a Gaussian PDF at t0 that is truncated to the 6σ extremal bounds, the total probability integral at t0 can be computed by converting the distribution to polar coordinates and performing the integration as, (23) (I0)analytical=12π∫06(∫02πexp(−r22)dθ)rdr =∫06exp(−r22)rdr

Using integration by substitution with t=r2, it can be shown that, (24) I0=1−exp(−622) =0.999999984770020 (25) (I0)analytical=0.999999984770020

According to the fundamental theorem of probability, this value for the total probability integral should always remain constant. Without loss of accuracy, the analytic total probability integral value can be considered the true total probability integral value. It is seen that (I_f)_OPA − (I_f)_analytical is in the order of 10⁻⁹, which further demonstrates the accuracy of OPA.

In this problem, the oscillator mass is set to one, the damping coefficient is removed, and k is treated as the common stiffness coefficient for both the linear and nonlinear stiffness terms, Eq. (28). In this nonlinear conservative system, the PDF is a function of two random variables: position and velocity. Tables 2 and 3 provide the simulation parameters. In this case, there are 2025 (n² = 45²) Chebyshev grid points in total. The initial 6σ bound is defined by a set of 100 points that are generated using the procedure described in Section 2.4. The propagation time is set as three-fourths of the oscillator period. Fig. 12 depicts the morphing of a circular extremal bound at t0 to an irregularly shaped extremal bound at tf. A 2D Chebyshev grid is constructed, encompassing the extremal bounds, and the approximation nodes external to the extremal bounds are eliminated as shown in Fig. 13. (28) x¨+(1+k2)x−2k2x3=0

From the computation point of view, the back-propagation process is the most time-consuming section of OPA. The time required for the back-propagation depends on three factors: i) the number of points that lie within the extremal bounds, ii) the initial and the final time (i.e., time span) of the back-propagation and iii) the complexity of the ODE system that is numerically integrated. After the back-propagation of each filtered grid point to t0, the corresponding PDF value is referenced to tf (Fig. 13). Then, the approximation is done using the Chebfun package to result in the PDF shown in Fig. 14. As the dynamics of the system are no longer linear, the analytic linear error propagation theory cannot be used for validation. Hence, Monte Carlo simulation is performed with 1 million sample points and the corresponding PDF is obtained using the kernel density estimation function. Fig. 14 shows the PDF corresponding to the Monte Carlo simulation of 1 million evaluation points. Since the PDF is non-Gaussian, the total probability integral value, I, is used as the metric to evaluate the accuracy of the PDF approximation. This value should come close to the cumulative integral of the truncated probability density function at t0. Fig. 15 depicts the close agreement between the 1-dimensional marginal PDFs and the cumulative integral curves of OPA and MC simulations. Table 4 shows the cumulative integral value of the PDF and the corresponding computation time for the OPA and MC simulations. OPA is shown to achieve nearly more than 100 times speed-ups vs. Monte Carlo simulations, as shown in Table 4, where the speed-up is calculated using the computation time of OPA and MC simulations.

In this section, OPA is applied to a damped Duffing oscillator with uncertainty in the damping coefficient, Eqs. (29) and (30). In this non-conservative dynamic system, the PDF at any given time depends on three random variables: (1) position, (2) velocity, and (3) the damping coefficient, δ. The state space trajectory of the oscillator is depicted in Fig. 16. The simulation parameters can be found in Tables 5 and 6. Since the problem has three random variables, it requires a 3-dimensional Chebyshev grid. Choosing an approximation order of n=28, results in n³ = 28³ = 21,952 Chebyshev grid points before grid filtering. (29) x¨+δx˙+(1+k2)x−2k2x3=0 (30) δ˙=0

The stiffness coefficient k is treated as a constant, and the damping coefficient δ becomes the random variable whose variation with time is zero. Out of the 3 random variables in this problem, only position and velocity change with time. Fig. 17 shows the initial extremal bounds that are propagated to a final time tf using Eqs. (29) and (30) to obtain the final extremal bounds. As shown in Fig. 17, a 3-dimensional (3D) Chebyshev grid is then formed at the final time. Following the technique discussed in Section 2, the grid points within the final extremal bounds, Fig. 17, are filtered and then back-propagated. Because of the non-conservative nature of the system dynamics, the PDF values at the grid points referenced from the back-propagation need to be scaled before the approximation is done. Although the procedure described in Section 2.2 can be used to scale the backpropagated PDF values, there is a simple and efficient way to scale the PDF using the fact that the damping coefficient does not change with time in this simulation, δ˙=0. Using the nature of the Chebyshev grid, the selected grid points at tf can be grouped into several levels with each level defined by a constant value of δ (Fig. 18). A convex hull is formed using the grid points between two successive levels at tf. The volume of this convex hull is denoted as V_fi. Upon back-propagation, each grid point from a damping coefficient level at tf lands at different position and velocity coordinates at t0 but the value of the damping coefficient remains unchanged. A second convex hull is formed using the backpropagated grid points of the corresponding successive levels at t0, whose volume is termed as V_0i. The convex hulls with volumes V_fi and V_0i are shown in Fig. 19. The scale factor associated with this pair of levels is given by, (31) SFpairi=V0iVfi (32) SFleveli=SFpairi+1+SFpairi2

This process continues for each pair of successive levels until all the selected grid points are covered. For example, if there are 25 levels, then this method yields 24 scale factors. Since each level of grid points is paired with the level above and with the level below, the scale factor of that level is obtained by computing the average as shown in Eq. (32). This mean scale factor is then used to scale the referenced PDF values of that level. The procedure continues until the referenced PDF at all levels is scaled. After scaling, the 3D PDF is approximated using the Chebfun toolbox. Figs. 20 and 21 show the outcomes of integrating the approximated PDF with respect to one random variable at a time (i.e., damping coefficient and velocity, respectively). The sequence of integration is described in Eqs. (33) and (34), (33) g(x,x˙)=∫c1c2f(x,x˙,δ)dδ (34) h(x)=∫x1˙x2˙g(x,x˙)dx˙

The smoothness of the cumulative probability curve depends on the order of approximation used. In the 2D Duffing oscillator problem, Table 4 shows that an MC simulation with 1 million points is required to achieve a reasonably accurate cumulative integral value. Therefore, using 1 million points, Fig. 20, an MC simulation is used to validate the approximation process. Table 7 shows that both the OPA and Monte Carlo simulations yield cumulative integral values close to one (to three decimal places). However, OPA is more than 270 times faster than the conventional Monte Carlo approach.

A higher-dimensional application of OPA is propagating the PDF of a simulated orbit problem. When using Chebyshev polynomials for higher dimensions, it becomes computationally expensive to compute full multi-dimensional approximations for the PDF. Additionally, for most real-world problems, only the spatial dimensions are required for further computations. This means that a one-dimensional approximation can be used to integrate the non-spatial dimensions of the PDF so that only a 2D or 3D Chebyshev approximation is required. This is achieved by selecting a dimension for integration and then, for each of the combinations of the non-selected dimensions, integrating along a one-dimensional line of the selected dimension to compress its probability into a single point in a space defined by the non-selected dimensions. This process is then repeated for all non-spatial dimensions. A simple LEO planar orbit problem is considered with uncertainty in the initial position and velocity. Because of the nature of the coupled dynamics, the PDF becomes a function of four random variables: position (x, y) and velocity (Vx=x˙, Vy=y˙). In this problem with two spatial and two velocity dimensions, at each constant value of x, y, and x˙, a one-dimensional integral along y˙ will compress the 4D space into a 3D space. Repeating this along x˙ compresses the space to 2D, and then a 2D Chebyshev approximation can be used for the spatial dimensions.

The state vector for the planar orbit problem is written as, (35) X=[xyx˙y˙]T (36) X˙=[x˙y˙x¨y¨]Twith its dynamics given by, (37) x¨=−µer3x (38) y¨=−µer3y (39) r=x2+y2

Tables 8 and 9 present the orbital elements and other relevant parameters of the planar orbit chosen to demonstrate OPA, respectively. The 6σ initial extremal bound of the planar orbit problem is defined by 500 points, which are obtained using the initial values of the uncertainties for the four random variables given in Table 10 and the procedure described in Section 2.4. These points, with a constant value of the PDF, are shown in Fig. 22. Since there are four random variables, the 4D extremal bounds are shown in two separate plots. Upon propagating these extremal bounds to one-fourth (= 0.25 * T) of the period of the orbit, the final extremal bounds are obtained, as shown in Fig. 23. A 4D Chebyshev grid encompassing the final extremal bounds is then formed (Fig. 24), and the points within the extremal bounds are filtered (Fig. 25) using the procedure mentioned in Section 2.3. These points are backpropagated to t0 (Fig. 26), and the PDF value at each (x₀, y₀, x˙0, y˙0) is referenced to tf.

Although the number of grid points increases exponentially with the dimensionality of the problem, grid filtering makes the back-propagation tractable. From Table 10, the ratio of the standard deviations of the spatial and velocity dimensions is found to be 30/0.03 = 1000. From numerical experiments, it was found that maintaining the ratio around this value results in enough backpropagated grid points within the extremal bounds, which ensures a good-quality approximation of the PDF at tf. The 4D PDF is then approximated using the higher-dimensional approximation procedure described above. After approximation, a 2D and a 1D marginal PDF are obtained by the integration sequence given in Eqs. (30)–(32). Finally, the cumulative integral is also obtained by integrating the 1D marginal PDF with respect to the x position. (40) f2(x,y,x˙)=∫y1˙y2˙f1(x,y,x˙,y˙)dy˙ (41) f3(x,y)=∫x1˙x2˙f2(x,y,x˙)dx˙ (42) f4(x)=∫y1y2f3(x,y)dy

The equivalent 2D and 1D marginal PDFs, as well as the cumulative integral curves from a 1 million Monte Carlo simulation of the planar orbit problem, are also obtained by following the integration sequence given above. These comparisons can be seen in Figs. 27 and 28.

The smoothness of the curve and convergence of the cumulative integral value to one indicates the quality of the PDF approximation. Several trials of OPA with different approximation orders were run, and the outcomes are summarized in Table 11. The speed-up is obtained by considering the one million sample Monte Carlo simulation as a reference. Fig. 29 shows the plots of the error in the cumulative integral value and speed up with the approximation order. For the approximation order of 90, the cumulative integral value is comparable to that of the MC simulation, and the OPA simulation is at least eight times faster than the MC simulation. Although the computation time will vary according to the hardware architecture on which the simulation is performed, the speed up will remain constant. This shows the accuracy, efficiency, and reliability of OPA for approximating the PDF of the planar orbit problem.