Adaptive Sparse Grids with Nonlinear Basis in Interval Problems for Dynamical Systems

Abstract

Problems with interval uncertainties arise in many applied fields. The authors have earlier developed, tested, and proved an adaptive interpolation algorithm for solving this class of problems. The algorithm’s idea consists of constructing a piecewise polynomial function that interpolates the dependence of the problem solution on point values of interval parameters. The classical version of the algorithm uses polynomial full grid interpolation and, with a large number of uncertainties, the algorithm becomes difficult to apply due to the exponential growth of computational costs. Sparse grid interpolation requires significantly less computational resources than interpolation on full grids, so their use seems promising. A representative number of examples have previously confirmed the effectiveness of using adaptive sparse grids with a linear basis in the adaptive interpolation algorithm. The purpose of this paper is to apply adaptive sparse grids with a nonlinear basis for modeling dynamic systems with interval parameters. The corresponding interpolation polynomials on the quadratic basis and the fourth-degree basis are constructed. The efficiency, performance, and robustness of the proposed approach are demonstrated on a representative set of problems.

1. Introduction

Dynamic systems with interval uncertainties arise in various application areas [1]. It is often easier to determine the ranges in which the parameters of the problem are located than to measure their values precisely. In this connection, there is a necessity to construct an interval estimate of the solution by the known interval estimates of its parameters.

The existing methods for solving interval problems can be divided into several groups:

• Methods based on interval arithmetic [2,3,4];
• Methods representing the set of solutions to the problem using geometric primitives [5,6,7];
• Methods operating with symbolic expressions [8,9,10,11];
• Stochastic methods [12];
• Methods representing the solution as a polynomial of relative interval parameters [13,14,15].

Note that methods based on interval arithmetic are often subject to the wrapping effect manifesting itself in an unbounded growth of the width of the resulting interval estimates. This effect is discussed in detail in [16].

The adaptive interpolation algorithm proposed earlier by the authors [13] belongs to the latter group. Its essence is to construct a piecewise polynomial function that interpolates the dependence of the solution to the problem on point values of interval parameters. The algorithm has a theoretical justification [13,17] and has been successfully applied to rigid systems of ordinary differential equations (ODEs) [18], systems with dynamic chaos and bifurcations [19], applied problems of chemical kinetics [18], gas and molecular dynamics [20].

Unlike interval methods, this algorithm does not give guaranteed interval estimates but it obtains solution bounds with controlled accuracy, is not subject to the wrap effect, and has a high degree of parallelization.

The classical variant of the algorithm uses full grid interpolation. Due to this fact, it has exponential computational complexity and becomes practically inapplicable for problems with a large number of interval uncertainties. Sparse grid interpolation [21] requires significantly less computational resources than full grid interpolation, so their use seems promising. The effectiveness of using adaptive sparse grids with a linear basis in the adaptive interpolation algorithm has been previously confirmed by a representative number of examples [20,22]. In addition, in [23], the technology of tensor trains [24,25] was successfully used to reduce the computational complexity of the algorithm.

The purpose of this paper is to apply adaptive sparse grids with a nonlinear basis for modeling dynamic systems with interval parameters.

Sparse grid interpolation has a hierarchical basis [26]. The multidimensional basis is constructed using the product of one-dimensional bases. By not using all basis functions, it is possible to significantly reduce the number of nodes required to determine the weight coefficients and, as a consequence, reduce the computational cost.

The issues of nonlinear basis usage together with sparse grids were considered in [27,28,29]. Generally speaking, it is necessary to solve a system of linear algebraic equations (SLAE) to determine the unknown coefficients of the interpolation polynomial. However, due to a certain choice of basis functions it is possible to obtain expressions for the weight coefficients in explicit form. In [17] it was shown that increasing the degree of the interpolation polynomial in the classical adaptive interpolation algorithm leads to an increase in accuracy and a reduction in computational costs. In this regard, it seems promising to use sparse grids with a nonlinear basis. In this paper we describe the quadratic basis and the basis of the fourth degree. The expressions used for the weight coefficients are convenient for software implementation and adaptation.

Often, in practice, dynamic systems are described by means of ODE systems; therefore, they are considered in the paper. In the second section of the paper the Cauchy problem for an ODE system with interval uncertainties is formulated. The third section describes an adaptive sparse grid interpolation algorithm with a nonlinear basis. In the fourth section, the proposed approach is tested using a representative set of problems containing various amounts of interval uncertainties. The different bases are compared. Finally, the main results of the paper are formulated.

2. Problem Statement

We consider the Cauchy problem for the ODE system consisting of n equations with m interval initial conditions:

$$\left\{\begin{array}{c}\frac{d{y}_i\left(t\right)}{dt}=f_i(y_1\left(t\right),y_2\left(t\right),\cdots ,y_n\left(t\right)),1\le i\le n,\hfill \\ y_i\left(t_0\right)\in \left[\underline{y_i^0},\overline{y_i^0}\right],1\le i\le m,\hfill \\ y_i\left(t_0\right)=y_i^0,m<i\le n,\hfill \\ t\in \left[t_0,t_N\right].\hfill \end{array}\right.$$

(1)

The right part of the system $\mathbf{f}={\left(f_1,f_2,\cdots ,f_n\right)}^T$ satisfies all conditions ensuring uniqueness and the existence of a solution for all $y_i\left(t_0\right)\in \left[\underline{y_i^0},\overline{y_i^0}\right]$, $1\le i\le m$. By performing simple transformations, any ODE system containing parameters or not being autonomous can be reduced to the system of Equation (1).

It is required for each moment of time $t_k$ to build a function ${\mathbf{P}}^k\left(y_1^0,y_2^0,\cdots ,y_m^0\right)$, where $y_i^0\in \left[\underline{y_i^0},\overline{y_i^0}\right]$, $1\le i\le m$ interpolating a solution of the system of Equation (1) ${\mathbf{y}}^k\left(y_1^0,y_2^0,\cdots ,y_m^0\right)=\mathbf{y}\left(y_1^0,y_2^0,\cdots ,y_m^0,t_k\right)$ with controllable accuracy. Further definition of interval estimation of the solution (finding of left and right bounds of intervals) is reduced to $2n$ conditional optimization problems for explicit function, which can be solved for instance by means of methods presented in [30].

3. Adaptive Sparse Grid Interpolation Algorithm with Nonlinear Basis

At the initial moment of time $t_0$ the vector-function ${\mathbf{P}}^0$ is defined in a simple way:

$${\mathbf{P}}^0\left(y_1^0,y_2^0,\cdots ,y_m^0\right)={\left(y_1^0,y_2^0,\cdots ,y_m^0,y_{m+1}\left(t_0\right),\cdots ,y_n\left(t_0\right)\right)}^T.$$

Let us use induction to construct a solution for an arbitrary time moment. Suppose that at the moment of time $t_k$ we know ${\mathbf{P}}^k\left(y_1^0,y_2^0,\cdots ,y_m^0\right)$. The derivation of ${\mathbf{P}}^{k+1}\left(y_1^0,y_2^0,\cdots ,y_m^0\right)$ is reduced to the interpolation of the implicit function ${\widehat{\mathbf{y}}}^{k+1}\left(y_1^0,y_2^0,\cdots ,y_m^0\right)$ given in the form of an ODE system:

$$\left\{\begin{array}{c}\frac{d{\widehat{y}}_i\left(t\right)}{dt}=f_i({\widehat{y}}_1\left(t\right),{\widehat{y}}_2\left(t\right),\cdots ,{\widehat{y}}_n\left(t\right)),\hfill \\ {\widehat{y}}_i\left(t_k\right)=P_i^k(y_1^0,y_2^0,\cdots ,y_m^0),1\le i\le n,\hfill \\ t\in \left[t_k,t_{k+1}\right],\hfill \\ {\widehat{\mathbf{y}}}^{k+1}={\left({\widehat{y}}_1\left(t_{k+1}\right),{\widehat{y}}_2\left(t_{k+1}\right),\cdots ,{\widehat{y}}_n\left(t_{k+1}\right)\right)}^T.\hfill \end{array}\right.$$

Usually, the interpolation polynomial ${\mathbf{P}}^k$ is constructed according to a certain set of nodes, which form a grid. Therefore, firstly, the solutions that correspond to the nodes are transferred to the $k+1$ time layer and, then, depending on the value of the interpolation error, adaptation is performed. Where the error is large, new nodes are added and, where the error is small, the grid is sparse. This is the adaptive interpolation algorithm for modeling dynamical systems with interval parameters. Note that, in the general case, the dynamical system can be discrete and then ${\widehat{\mathbf{y}}}^{k+1}\left(y_1^0,y_2^0,\cdots ,y_m^0\right)$ will be specified explicitly.

The interpolation polynomial $\mathbf{P}$ can be arbitrary; it is only necessary to be able to control the interpolation error. The classical version of the algorithm uses interpolation on dense grids grouped in a hierarchical structure (kd-tree). This paper uses sparse grid interpolation.

Sparse grids are based on a hierarchical basis. Interpolation on a hierarchical basis can be considered as a sequential construction of polynomials interpolating a disparity.

First, a description of sparse grid interpolation with a linear basis is given. It is based on the hat function:

$${\phi }_1\left(x\right)=\left\{\begin{array}{c}1-\left|x\right|,x\in \left[-1,1\right]\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

(2)

In the one-dimensional case, the basis functions of l level are defined as

$${\phi }_{l,i}\left(x\right)={\phi }_1(2^{l}x-i)$$

(3)

and the interpolation polynomial has the form:

$$P\left(x\right)=a_{0,0}{\phi }_{0,0}\left(x\right)+a_{0,1}{\phi }_{0,1}\left(x\right)+\sum _{l=1}^{n_g}\sum _{i=1}^{2^l-1}{a}_{l,i}{\phi }_{l,i}\left(x\right),i\mathrm{odd},$$

(4)

$$a_{l,i}=f\left(x_{l,i}\right)-\left\{\begin{array}{c}0,l=0,\hfill \\ \frac{1}{2}\left[f\left(x_{l,i-1}\right)+f\left(x_{l,i+1}\right)\right],l>0,\hfill \end{array}\right.$$

(5)

where $n_g$ is the level of the corresponding grid, $x_{l,i}=i{2}^{-l}$ are nodes of the grid, and $f:\left[0,1\right]\to R$ is a smooth continuous function with bounded second derivatives: $\frac{{\partial }^{2}f}{\partial x^2}<\infty$. The basis functions of zero level ${\phi }_{0,0}\left(x\right)$ and ${\phi }_{0,1}\left(x\right)$ are necessary to account for the boundary values.

The coefficients $a_{l,i}$ are equal to the disparity at the point $x_{l,i}$ between the function f and the interpolation polynomial, which corresponds to the grid of level $l-1$. In addition, $a_{l,i}$ are characterized by the second derivatives and l tend to zero as $O\left(2^{-2l}\right)$.

Figure 1 shows the basis functions up to level 3 and shows the interpolation process of the function:

$$f\left(x\right)=\frac{3}{2}+3sin\left(\pi x\right)-x.$$

(6)

$P_l\left(x\right)$ denotes the sum of the basic level l functions with their respective weighting coefficients:

$$P_l\left(x\right)=\sum _i{a}_{l,i}{\phi }_{l,i}\left(x\right),i\in \left\{\begin{array}{c}\{0,1\},l=0,\hfill \\ \left\{2p-1|p={1,\cdots ,2}^{l-1}\right\},l>0.\hfill \end{array}\right.$$

D-dimensional basis functions are constructed using the product of one-dimensional basis functions (3):

$${\phi }_{\mathbf{l},\mathbf{i}}\left(\mathbf{x}\right)=\prod _{k=1}^d{\phi }_{l_k,i_k}\left(x_k\right)$$

(7)

where $\mathbf{l}=(l_1,l_2,\cdots ,l_d)$ is a multi-level, $\mathbf{i}=(i_1,i_2,\cdots ,i_d)$ is a multi-index, and $\mathbf{x}=(x_1,x_2,\cdots ,x_d)$ is a vector of variables. Depending on the constraints for the multi-level, the corresponding grid will be either full ($max{l}_k\le n_g$) or sparse ($\sum l_k\le n_g+d-1$).

The interpolation polynomial (4) in the multivariate case is written as follows:

$$P\left(\mathbf{x}\right)=\sum _{\mathbf{l},\mathbf{i}}{a}_{\mathbf{l},\mathbf{i}}{\phi }_{\mathbf{l},\mathbf{i}}\left(\mathbf{x}\right),\sum _{k=1}^d{l}_k\le n_g+d-1,i_k\in \left\{\begin{array}{c}\{0,1\},l_k=0,\hfill \\ \left\{2p-1|p={1,\cdots ,2}^{l_k-1}\right\},l_k>0.\hfill \end{array}\right.$$

(8)

The weighting coefficients $a_{\mathbf{l},\mathbf{i}}$ are determined recurrently by generalizing the relation (5):

$$\begin{array}{c}{a}_{\mathbf{l},\mathbf{i}}=a_{\mathbf{l},\mathbf{i}}^{\left(1\right)},\hfill \\ a_{\mathbf{l},\mathbf{i}}^{\left(k\right)}=a_{\mathbf{l},\mathbf{i}}^{(k+1)}-\left\{\begin{array}{c}0,l_k=0,\hfill \\ \frac{1}{2}\left(a_{\mathbf{l},i_1\cdots i_k-1\cdots i_d}^{(k+1)}+a_{\mathbf{l},i_1\cdots i_k+1\cdots i_d}^{(k+1)}\right),l_k>0,\hfill \end{array}\right.\hfill \\ a_{\mathbf{l},\mathbf{i}}^{(d+1)}=f\left({\mathbf{x}}_{\mathbf{l},\mathbf{i}}\right),\hfill \end{array}$$

(9)

where ${\mathbf{x}}_{\mathbf{l},\mathbf{i}}=\left(x_{l_1,i_1},x_{l_2,i_2},\cdots ,x_{l_d,i_d}\right)$ and $f:{\left[0,1\right]}^d\to R$ is a smooth continuous function with bounded mixed derivatives:

$$\frac{{\partial }^{|\alpha {|}_1}f}{\partial x_1^{{\alpha }_1}\partial x_2^{{\alpha }_2}\cdots \partial x_d^{{\alpha }_d}}<\infty ,|\alpha {|}_{\infty }\le 2$$

(10)

where $|\alpha {|}_{\infty }=max{\alpha }_k$, $|\alpha {|}_1=\sum {\alpha }_k$. The coefficients $a_{\mathbf{l},\mathbf{i}}$ characterize the derivatives of (10).

In practice, the adaptive version of interpolation is usually used. Initially, the basis contains only functions (7), whose multi-level components are 0 or 1. Adaptation is performed using the value of $\left|a_{\mathbf{l},\mathbf{i}}\right|$. If $\left|a_{\mathbf{l},\mathbf{i}}\right|>\epsilon$, then functions are added to the basis at one level more

$$\begin{array}{c}{\phi }_{(l_1+1,l_2,\cdots ,l_d),(2i_1-1,i_2,\cdots ,i_d)},{\phi }_{(l_1+1,l_2,\cdots ,l_d),(2i_1+1,i_2,\cdots ,i_d)},l_1>0,\hfill \\ {\phi }_{(l_1,l_2+1,\cdots ,l_d),(i_1,2i_2-1,\cdots ,i_d)},{\phi }_{(l_1,l_2+1,\cdots ,l_d),(i_1,2i_2+1,\cdots ,i_d)},l_2>0,\hfill \\ \cdots \hfill \\ {\phi }_{(l_1,l_2,\cdots ,l_d+1),(i_1,i_2,\cdots ,2i_d-1)},{\phi }_{(l_1,l_2,\cdots ,l_d+1),(i_1,i_2,\cdots ,2i_d+1)},l_d>0.\hfill \end{array}$$

In addition, when calculating $a_{\mathbf{l},\mathbf{i}}$, functions ${\phi }_{\mathbf{l},\mathbf{i}}$, which correspond to $a_{\mathbf{l},\mathbf{i}}^{(d+1)}$, are automatically added to the basis.

When using a nonlinear basis, the difference is in defining the basis functions ${\phi }_{l,i}\left(x\right)$ and the corresponding weighting coefficients $a_{l,i}$.

Let us consider the transition to the quadratic basis. The hat function (2) is complemented by the quadratic function

$${\phi }_2\left(x\right)=-\left\{\begin{array}{c}(x-1)(x+1),x\in \left[-1,1\right],\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(11)

and (3) takes the following form:

$${\phi }_{l,i}\left(x\right)=\left\{\begin{array}{c}{\phi }_1(2^{l}x-i),l=0,\hfill \\ {\phi }_2(2^{l}x-i),l>0.\hfill \end{array}\right.$$

(12)

To calculate the weighting coefficients $a_{l,i}$, it is necessary to perform grid interpolation one level less. The interpolation polynomial (4) can be regarded as a piecewise discontinuous function which interpolates the original dependence with a certain order on each segment. Therefore, only a few nodes with a lower level in the vicinity of $x_{l,i}$ are needed for determination of $a_{l,i}$ (Figure 2):

$$a_{l,i}=f\left(x_{l,i}\right)-\left\{\begin{array}{c}0,l=0,\hfill \\ \frac{1}{2}\left[f\left(x_{l,i-1}\right)+f\left(x_{l,i+1}\right)\right],l=1,\hfill \\ \frac{1}{8}\left[3f\left(x_{l,i-1}\right)+6f\left(x_{l,i+1}\right)-f\left(x_{l,i+3}\right)\right],l>1,imod4=1,\hfill \\ \frac{1}{8}\left[3f\left(x_{l,i+1}\right)+6f\left(x_{l,i-1}\right)-f\left(x_{l,i-3}\right)\right],l>1,imod4=3.\hfill \end{array}\right.$$

Figure 3 shows the basis functions (12) and the process of interpolation of function (6). Compared to the linear basis interpolation (Figure 1), the error decreases faster with increasing level.

In the multivariate case, the coefficients $a_{\mathbf{l},\mathbf{i}}$ are also determined using recurrence relations:

$$\begin{array}{c}{a}_{\mathbf{l},\mathbf{i}}=a_{\mathbf{l},\mathbf{i}}^{\left(1\right)}\hfill \\ a_{\mathbf{l},\mathbf{i}}^{\left(k\right)}=a_{\mathbf{l},\mathbf{i}}^{(k+1)}-\left\{\begin{array}{c}0,l_k=0,\hfill \\ \frac{1}{2}\left(a_{\mathbf{l},i_1\cdots i_k-1\cdots i_d}^{(k+1)}+a_{\mathbf{l},i_1\cdots i_k+1\cdots i_d}^{(k+1)}\right),l_k=1,\hfill \\ \frac{1}{8}\left(3a_{\mathbf{l},\cdots i_k-1\cdots }^{(k+1)}+6a_{\mathbf{l},\cdots i_k+1\cdots }^{(k+1)}-a_{\mathbf{l},\cdots i_k+3\cdots }^{(k+1)}\right),l_k>1,i_{k}mod4=1,\hfill \\ \frac{1}{8}\left(3a_{\mathbf{l},\cdots i_k+1\cdots }^{(k+1)}+6a_{\mathbf{l},\cdots i_k-1\cdots }^{(k+1)}-a_{\mathbf{l},\cdots i_k-3\cdots }^{(k+1)}\right),l_k>1,i_{k}mod4=3,\hfill \end{array}\right.\hfill \\ a_{\mathbf{l},\mathbf{i}}^{(d+1)}=f\left({\mathbf{x}}_{\mathbf{l},\mathbf{i}}\right).\hfill \end{array}$$

(13)

Let us move on to considering the construction of the fourth-degree basis; (2) and (11) are complemented by the following functions:

$$\begin{array}{c}{\phi }_4^1\left(x\right)=-\frac{1}{6}\left\{\begin{array}{c}(x+1)(x-1)(x-2)(x-3),x\in \left[-1,3\right],\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\hfill \\ {\phi }_4^2\left(x\right)=-\frac{1}{6}\left\{\begin{array}{c}(x+3)(x+2)(x+1)(x-1),x\in \left[-3,1\right],\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}$$

(14)

and ${\phi }_{l,i}\left(x\right)$ is defined as

$${\phi }_{l,i}\left(x\right)=\left\{\begin{array}{c}{\phi }_1(2^{l}x-i),l=0,\hfill \\ {\phi }_2(2^{l}x-i),l=1,\hfill \\ {\phi }_4^1(2^{l}x-i),l>1,imod4=1,\hfill \\ {\phi }_4^2(2^{l}x-i),l>1,imod4=3.\hfill \end{array}\right.$$

Similarly to the quadratic basis, several nodes in the neighborhood of $x_{l,i}$ should be considered to calculate the coefficients $a_{l,i}$ (see Figure 4):

$$\begin{array}{c}{a}_{l,i}=f\left(x_{l,i}\right)-\left\{\begin{array}{c}0,l=0,\hfill \\ \frac{1}{2}\left(f_{l,i-1}+f_{l,i+1}\right),l=1,\hfill \\ \frac{1}{8}\left\{\begin{array}{c}3f_{l,i-1}+6f_{l,i+1}-f_{l,i+3},imod4=1,\hfill \\ 3f_{l,i+1}+6f_{l,i-1}-f_{l,i-3},imod4=3,\hfill \end{array}\right.,l=2,\hfill \\ \frac{1}{128}{z}_{l,i},l>2,\hfill \end{array}\right.\hfill \\ \\ z_{l,i}=\left\{\begin{array}{c}35f_{l,i-1}+140f_{l,i+1}-70f_{l,i+3}+28f_{l,i+5}-5f_{l,i+7},imod8=1,\hfill \\ -5f_{l,i-3}+60f_{l,i-1}+90f_{l,i+1}-20f_{l,i+3}+3f_{l,i+5},imod8=3,\hfill \\ 3f_{l,i-5}-20f_{l,i-3}+90f_{l,i-1}+60f_{l,i+1}-5f_{l,i+3},imod8=5,\hfill \\ -5f_{l,i-7}+28f_{l,i-5}-70f_{l,i-3}+140f_{l,i-1}+35f_{l,i+1},imod8=7.\hfill \end{array}\right.\hfill \end{array}$$

Here, to shorten the record, $f\left(x_{l,i}\right)$ means $f_{l,i}$.

Figure 5 shows the hierarchical basis and interpolation process for function (6).

In the d-dimensional case, $a_{\mathbf{l},\mathbf{i}}=a_{\mathbf{l},\mathbf{i}}^{\left(1\right)}$ and

$$\begin{array}{c}{a}_{\mathbf{l},\mathbf{i}}^{\left(k\right)}=a_{\mathbf{l},\mathbf{i}}^{(k+1)}-\left\{\begin{array}{c}0,l_k=0,\hfill \\ \frac{1}{2}\left(a_{\mathbf{l},i_1\cdots i_k-1\cdots i_d}^{(k+1)}+a_{\mathbf{l},i_1\cdots i_k+1\cdots i_d}^{(k+1)}\right),l_k=1,\hfill \\ \frac{1}{8}\left\{\begin{array}{c}3a_{\mathbf{l},\cdots i_k-1\cdots }^{(k+1)}+6a_{\mathbf{l},\cdots i_k+1\cdots }^{(k+1)}-a_{\mathbf{l},\cdots i_k+3\cdots }^{(k+1)},i_{k}mod4=1,\hfill \\ 3a_{\mathbf{l},\cdots i_k+1\cdots }^{(k+1)}+6a_{\mathbf{l},\cdots i_k-1\cdots }^{(k+1)}-a_{\mathbf{l},\cdots i_k-3\cdots }^{(k+1)},i_{k}mod4=3,\hfill \end{array}\right.,l_k=2,\hfill \\ \frac{1}{128}{z}_{\mathbf{l},\mathbf{i}},l_k>2,\hfill \end{array}\right.\hfill \\ \\ z_{\mathbf{l},\mathbf{i}}=\left\{\begin{array}{c}35a_{\mathbf{l},i_k-1}^{(k+1)}+140a_{\mathbf{l},i_k+1}^{(k+1)}-70a_{\mathbf{l},i_k+3}^{(k+1)}+28a_{\mathbf{l},i_k+5}^{(k+1)}-5a_{\mathbf{l},i_k+7}^{(k+1)},i_{k}mod8=1,\hfill \\ -5a_{\mathbf{l},i_k-3}^{(k+1)}+60a_{\mathbf{l},i_k-1}^{(k+1)}+90a_{\mathbf{l},i_k+1}^{(k+1)}-20a_{\mathbf{l},i_k+3}^{(k+1)}+3a_{\mathbf{l},i_k+5}^{(k+1)},i_{k}mod8=3,\hfill \\ 3a_{\mathbf{l},i_k-5}^{(k+1)}-20a_{\mathbf{l},i_k-3}^{(k+1)}+90a_{\mathbf{l},i_k-1}^{(k+1)}+60a_{\mathbf{l},i_k+1}^{(k+1)}-5a_{\mathbf{l},i_k+3}^{(k+1)},i_{k}mod8=5,\hfill \\ -5a_{\mathbf{l},i_k-7}^{(k+1)}+28a_{\mathbf{l},i_k-5}^{(k+1)}-70a_{\mathbf{l},i_k-3}^{(k+1)}+140a_{\mathbf{l},i_k-1}^{(k+1)}+35a_{\mathbf{l},i_k+1}^{(k+1)},i_{k}mod8=7,\hfill \end{array}\right.\hfill \\ a_{\mathbf{l},\mathbf{i}}^{(d+1)}=f\left({\mathbf{x}}_{\mathbf{l},\mathbf{i}}\right).\hfill \end{array}$$

(15)

Note that functions (2), (11) and (14) on which the basis is constructed can be considered as the basis Lagrangian polynomials since they are equal to 1 only in a single node and equal to 0 in other nodes.

The constructed bases are only one set of possible bases. The obtained expressions for weight coefficients are convenient for further software implementation and adaptation.

With regard to the solution of the original problem (1), expression (8) will take the form:

$$\begin{array}{c}{\mathbf{P}}^{k+1}(y_1^0,\cdots ,y_m^0)={\displaystyle \sum _{\mathbf{l},\mathbf{i}}}{\mathbf{a}}_{\mathbf{l},\mathbf{i}}{\phi }_{\mathbf{l},\mathbf{i}}\left(\frac{y_1^0-\underline{y_1^0}}{\overline{y_1^0}-\underline{y_1^0}},\frac{y_2^0-\underline{y_2^0}}{\overline{y_2^0}-\underline{y_2^0}},\cdots ,\frac{y_m^0-\underline{y_m^0}}{\overline{y_m^0}-\underline{y_m^0}}\right),\hfill \\ {\displaystyle \sum _{j=1}^m}{l}_j\le n_g+m-1,i_j\in \left\{\begin{array}{c}\{0,1\},l_j=0,\hfill \\ \left\{2p-1|p={1,\cdots ,2}^{l_j-1}\right\},l_j>0.\hfill \end{array}\right.\hfill \end{array}$$

And when calculating the weighting coefficients ${\mathbf{a}}_{\mathbf{l},\mathbf{i}}$, the value at the last iteration will be determined as

$${\mathbf{a}}_{\mathbf{l},\mathbf{i}}^{(m+1)}={\widehat{\mathbf{y}}}^{k+1}\left(\underline{y_1^0}+\left[\overline{y_1^0}-\underline{y_1^0}\right]x_{l_1,i_1},\cdots ,\underline{y_m^0}+\left[\overline{y_m^0}-\underline{y_m^0}\right]x_{l_m,i_m}\right)$$

The norm of the vector ${\mathbf{a}}_{\mathbf{l},\mathbf{i}}$, for example, the maximum one, can be used as an adaptation criterion.

4. Results

The solutions of several ODE systems with interval uncertainties are found and the results are compared. To estimate the a posteriori absolute error at the initial moment, a test set of points from the uncertainty region of the system parameters is randomly generated and the corresponding solutions are constructed and compared at the final moment:

$$error=\underset{{\mathbf{y}}^{ch}\left(t_N\right),{\mathbf{y}}^{ch}\left(t_0\right)\in \chi }{max}{∥{\mathbf{P}}^N\left(y_1^{ch}\left(t_0\right),y_2^{ch}\left(t_0\right),\cdots ,y_m^{ch}\left(t_0\right)\right)-{\mathbf{y}}^{ch}\left(t_N\right)∥}_{\infty },$$

where

$$\chi =\left\{{\mathbf{y}}^{ch}\left({\mathbf{y}}^{ch}\left(t_0\right),t_N\right)\left|{\mathbf{y}}^{ch}\left(t_0\right)={\left(\mathrm{rand}\left[\underline{y_1^0},\overline{y_1^0}\right],\cdots ,\mathrm{rand}\left[\underline{y_m^0},\overline{y_m^0}\right],y_{m+1}^0,\cdots ,y_n^0\right)}^T\right.\right\}.$$

During construction of interval estimation (determination of interval bounds), the solution of $2n$ conditional optimization problems for explicit vector-function $\mathbf{P}$ is performed:

$$\begin{array}{c}{P}_i\left(\mathbf{x}\right)\underset{\mathbf{x}\in \left[\underline{y_1^0},\overline{y_1^0}\right]\times \cdots \times \left[\underline{y_m^0},\overline{y_m^0}\right]}{\to }min,1\le i\le n,\hfill \\ P_i\left(\mathbf{x}\right)\underset{\mathbf{x}\in \left[\underline{y_1^0},\overline{y_1^0}\right]\times \cdots \times \left[\underline{y_m^0},\overline{y_m^0}\right]}{\to }max,1\le i\le n.\hfill \end{array}$$

(16)

If the solution error of problems (16) is much smaller than the interpolation error, which is typical in practice, then the $error$ value can be used to judge the accuracy of the bounds of the resulting interval estimate. Note that, in the case of a linear basis, it is sufficient to enumerate all values in the nodes of the grid to find the exact solution of (16).

To estimate the computational costs, we use the criterion I introduced in [13,20] and numerically equal to the number of solved non-interval systems (1) at specific point values of interval parameters:

$$I=\frac{1}{N}\sum _{k=1}^N{C}_k,$$

where $C_k$ is the number of nodes in the grid at the k-th step.

Earlier, the authors of [17] analyzed computational costs for the classical variant of the algorithm. As a rule, the costs of integration for non-interval systems (1) significantly exceeds other costs (including interpolation costs). Therefore, the criterion I is indicative and characterizes the main computational costs.

Then, for each problem, computational cost I is compared for different bases and different values of $\epsilon$. The parameter $\epsilon$ defines the condition of grid adaptation: if ${∥{\mathbf{a}}_{\mathbf{l},\mathbf{i}}∥}_{\infty }>\epsilon$, then the basis functions are added to the basis by one level more (and the nodes that correspond to the new basis functions are added). Since the weight coefficients ${\mathbf{a}}_{\mathbf{l},\mathbf{i}}$ characterize the disparity, they actually determine the local $\epsilon$ error at each step of the algorithm.

Consider the first nonlinear ODE system:

$$\left\{\begin{array}{c}{x}^{\prime }=-y{\left(1+\sqrt{x^2+y^2}\right)}^{-1},\hfill \\ y^{\prime }=x{\left(1+\sqrt{x^2+y^2}\right)}^{-1},\hfill \\ x\left(0\right)=x_0\in \left[-1,1\right],y\left(0\right)=y_0\in \left[0,1\right],\hfill \\ t\in \left[0,25\right].\hfill \end{array}\right.$$

(17)

The analytical solution to the system of Equation (17) has the following form:

$$\left\{\begin{array}{c}x(x_0,y_0,t)=\sqrt{x_0^2+y_0^2}cos\left[t{\left(1+\sqrt{x_0^2+y_0^2}\right)}^{-1}+arctan\left(\frac{y_0}{x_0}\right)\right]\hfill \\ y(x_0,y_0,t)=\sqrt{x_0^2+y_0^2}sin\left[t{\left(1+\sqrt{x_0^2+y_0^2}\right)}^{-1}+arctan\left(\frac{y_0}{x_0}\right)\right]\hfill \end{array}\right.$$

(18)

Figure 6 shows the set of solutions for the ODE system (17) and the resulting grid in the course of algorithm operation at various moments of time when using the basis of the fourth degree with parameter $\epsilon ={10}^{-3}$. The points in the upper figures correspond to the points in the lower figures.

In the vicinity of the points $x_0=0$ and $y_0=0$, there is a densification of the grid; this is partly due to the fact that this point is a singular point of the center type.

Table 1. Comparison of different methods for the system of Equation (17).

Methods	Interpolation Order	$\mathit{ϵ}={10}^{-3}$		$\mathit{ϵ}={10}^{-5}$
		$\mathit{I},\times {\mathbf{10}}^{\mathbf{3}}$	$\mathit{Error}$	$\mathit{I},\times {\mathbf{10}}^{\mathbf{3}}$	$\mathit{Error}$
Adaptive sparse grid	1	$2.3$	$6.6\times {10}^{-3}$	$31.8$	$9.3\times {10}^{-5}$
	2	$1.1$	$4.2\times {10}^{-3}$	$7.0$	$7.1\times {10}^{-5}$
	4	$0.7$	$3.9\times {10}^{-3}$	$3.4$	$5.3\times {10}^{-5}$
Classic algorithm	1	$35.7$	$7.5\times {10}^{-3}$	–	–
	2	$3.9$	$1.2\times {10}^{-2}$	$82.1$	$1.7\times {10}^{-4}$
	4	$1.6$	$1.4\times {10}^{-2}$	$9.7$	$1.1\times {10}^{-4}$

shows the results of different methods for two values of $\epsilon$. A variant of the adaptive interpolation algorithm with sparse grids works at least two to three times faster (using the fourth-degree basis) than the classical variant of the algorithm.

Consider an ODE system with two initial interval conditions and two interval parameters:

$$\left\{\begin{array}{c}{x}^{\prime }=a\left(bx+\sqrt{1-b^2}y\right),\hfill \\ y^{\prime }=a\left(-\sqrt{1-b^2}x+by\right),\hfill \\ x\left(0\right)=x_0\in \left[-1,1\right],y\left(0\right)=y_0\in \left[-1,1\right],\hfill \\ a\in \left[0.01,0.99\right],b\in \left[0.01,0.99\right],\hfill \\ t\in \left[0,2\right].\hfill \end{array}\right.$$

(19)

The analytical solution has the form:

$$\left\{\begin{array}{c}x(x_0,y_0,a,b,t)=exp\left(abt\right)\left[y_{0}sin\left(a\sqrt{1-b^2}t\right)+x_{0}cos\left(a\sqrt{1-b^2}t\right)\right],\hfill \\ y(x_0,y_0,a,b,t)=exp\left(abt\right)\left[y_{0}cos\left(a\sqrt{1-b^2}t\right)-x_{0}sin\left(a\sqrt{1-b^2}t\right)\right].\hfill \end{array}\right.$$

(20)

Figure 7 shows the set of solutions to the system of Equation (19) at various moments of time and the projection of the resulting adaptive sparse grid onto a two-dimensional plane. The important feature of this system is that its solution (20), in contrast to (18), is represented in the form of the linear combination with respect to initial conditions. As a result, grid compaction will take place only on the subsets corresponding to parameters a and b (Figure 7, grey color). At the initial point $t=0$ the set of solutions on the phase plane is a square. At points $t>0$ the parameters a and b will already influence the solution and the structure of the set will change. It will be a projection of a four-dimensional deformed rectangular parallelepiped onto the phase plane.

Table 2. Comparison of different methods using the system of Equation (19).

Methods	Interpolation Order	$\mathit{ϵ}={10}^{-3}$		$\mathit{ϵ}={10}^{-5}$
		$\mathit{I},\times {\mathbf{10}}^{\mathbf{3}}$	$\mathit{Error}$	$\mathit{I},\times {\mathbf{10}}^{\mathbf{3}}$	$\mathit{Error}$
Adaptive sparse grid	1	$2.5$	$5.1\times {10}^{-3}$	$32.3$	$8.9\times {10}^{-5}$
	2	$1.0$	$2.3\times {10}^{-3}$	$6.1$	$2.3\times {10}^{-5}$
	4	$0.7$	$3.2\times {10}^{-3}$	$2.0$	$1.3\times {10}^{-5}$
Classic algorithm	2	$6.4$	$9.6\times {10}^{-3}$	$120.3$	$1.2\times {10}^{-4}$
	4	$4.3$	$1.1\times {10}^{-2}$	$14.2$	$6.6\times {10}^{-5}$

shows the results of different methods. At $\epsilon ={10}^{-5}$, the use of sparse grids with the fourth-degree basis allows us to reduce computational costs by 16 times in comparison with the linear basis.

Further, the constructed bases are approbated for the ODE system, which describes the motion of bodies with uncertainties in the initial velocities under the action of gravitational forces around a massive body. Using dimensionless variables, we can write the following equations [20]:

$$\left\{\begin{array}{c}{\left(v_i^x\right)}^{\prime }={\displaystyle \sum _{j=1,j\ne i}^4}{m}_j\frac{x_j-x_i}{r_{i,j}^3},{\left(v_i^y\right)}^{\prime }={\displaystyle \sum _{j=1,j\ne i}^4}{m}_j\frac{y_j-y_i}{r_{i,j}^3},\hfill \\ {\left(v_i^z\right)}^{\prime }={\displaystyle \sum _{j=1,j\ne i}^4}{m}_j\frac{z_j-z_i}{r_{i,j}^3},\hfill \\ {x^{\prime }}_i=v_i^x,{y^{\prime }}_i=v_i^y,{z^{\prime }}_i=v_i^z,i=\overline{1,4},\hfill \\ x_1\left(0\right)=y_1\left(0\right)=z_1\left(0\right)=v_1^x\left(0\right)=v_1^y\left(0\right)=v_1^z\left(0\right)=0,\hfill \\ x_{2,3}\left(0\right)=\pm 1,y_{2,3}\left(0\right)=z_{2,3}\left(0\right)=0,v_{2,3}\left(0\right)={(0,\pm 316.23,0)}^T+\Delta v_{2,3}^T,\hfill \\ y_4\left(0\right)=1,x_4\left(0\right)=z_4\left(0\right)=0,v_4\left(0\right)={(0,0,316.23)}^T+\Delta v_4^T,\hfill \\ t\in [0.0,0.02],\hfill \end{array}\right.$$

(21)

where $r_{i,j}$ is the distance between the bodies with the indices i and j, $m_1={10}^5$, $m_{\overline{2,4}}={10}^{-5}$ are the masses of the bodies, and $\Delta v_{\overline{2,4}}=\left(\left[-2,2\right],\left[-2,2\right],\left[-2,2\right]\right)$ are the interval uncertainties in the velocities of the bodies.

Figure 8 shows the sets of positions of all four bodies at different times. The most massive body is in the center and is marked with a large gray circle. It does not actually move in space. The remaining three bodies rotate in circular orbits around the massive body and fly half the circle of their trajectory in a period of time $t\in [0,0.01]$ (shown in Figure 8). It can be seen that the region of position uncertainty for those bodies increases with time. Note that the presence of interval uncertainties in the initial velocities makes bodies’ positions also interval.

Due to a large number of uncertainties, the application of the classical adaptive interpolation algorithm to this problem becomes difficult. Therefore, all calculations are performed using sparse grids.

Table 3. Comparison of different bases for the system of Equation (21).

Interpolation Order	$\mathit{ϵ}={10}^{-3}$		$\mathit{ϵ}={10}^{-5}$
	$\mathit{I},\times {\mathbf{10}}^{\mathbf{3}}$	$\mathit{Error}$	$\mathit{I},\times {\mathbf{10}}^{\mathbf{3}}$	$\mathit{Error}$
1	$144.3$	$1.6\times {10}^{-2}$	–	–
2	$34.2$	$3.4\times {10}^{-3}$	$97.9$	$8.0\times {10}^{-5}$
4	$27.1$	$7.1\times {10}^{-5}$	$42.0$	$6.4\times {10}^{-6}$

shows that an increase in the degree of interpolation makes it possible to significantly reduce computational costs and obtain a more accurate solution. Obtaining an $error$ value smaller than $\epsilon$ is associated with the fact that, when using a large degree of interpolation, grid compaction leads to a sharp increase in accuracy, which is substantially less than the given value $\epsilon$. And even in spite of the error accumulation at each step, at the final moment of time, the total error does not exceed $\epsilon$.

Figure 9 shows some projections of the resulting nine-dimensional adaptive sparse grid during the integration of system (21) using different bases. Figure 9b shows the structure of the grid: three-dimensional subsets (cubes) are distinguished. This is primarily due to the fact that the uncertainty in the initial velocity of a particular body mainly affects only the position and velocity of the same body. Therefore, the grid compaction mainly occurs only in subsets that correspond to interval uncertainties within a single body.

5. Discussion

According to the values given in Table 1, Table 2 and Table 3 it can be concluded that the use of adaptive sparse grids with nonlinear basis allows significantly reducing the computational costs in modeling of dynamic systems with interval parameters. Criterion I is the main characteristic of computational costs. Compared to the classical version of the adaptive interpolation algorithm, the sparse mesh version of the algorithm works 3–7 times faster for $p=4$ and 10–20 times faster for $p=2$ with $ϵ={10}^{-5}$. It is important to note that problem (21) was solved only using an algorithm based on sparse grids, since the application of the classical adaptive interpolation algorithm to this problem is difficult due to the large number of interval uncertainties. The results obtained confirm the advantage of using sparse grids in terms of reducing the computational load. Note that criterion value I significantly decreases at transition from linear to quadratic basis and less significantly decreases at the transition from quadratic to fourth-degree basis. In particular, this can be related to both the peculiarities of the ODE system and the dimensionality of the uncertainty area (the number of interval parameters), and the fact that the nonlinear basis begins to be involved only from a certain level l. In general, the use of a nonlinear basis in comparison with a linear basis for sparse grids reduces computational costs by 3–16 times. The expressions for the weighting coefficients (9), (13) and (15) have a recursive form and are convenient for further software implementation, since they make it possible to avoid complex nested loops. The weight coefficients allow one to immediately estimate the discrepancy and in accordance with their values effectively adapt the grid.

6. Conclusions

This paper considers a modification of the adaptive interpolation algorithm for modeling dynamic systems with interval parameters using sparse grids with a nonlinear basis. The classical version of the algorithm uses full grid interpolation and consequently has exponential complexity with respect to the number of interval parameters. Sparse grid interpolation requires significantly less computational resources than full grid interpolation, so its use seems promising. Earlier, the effectiveness of sparse grids with a linear basis for solving this class of problems was shown. In this paper, a combination of sparse grids with a nonlinear basis is used. Construction of a quadratic basis and a basis of the fourth degree is executed. The obtained expressions for weight coefficients are convenient for further programming and adaptation. The efficiency and robustness of the sparse grid-based adaptive interpolation algorithm with a nonlinear basis have been demonstrated using a representative set of problems. In certain cases, the computational cost reduction by several orders of magnitude is obtained.