# American Institute of Mathematical Sciences

December  2020, 7(2): 183-208. doi: 10.3934/jcd.2020008

## Solving the inverse problem for an ordinary differential equation using conjugation

 1 Departamento de Ciência da Computação, Universidade Federal do Rio de Janeiro, Caixa Postal 68.530, CEP 21941-590, Rio de Janeiro, RJ, Brazil 2 Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-900, MG, Brazil 3 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago de Chile, Chile 4 Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil

* Corresponding author: Daniel G. Alfaro Vigo

Received  October 2019 Published  July 2020

Fund Project: The second author's work was partially supported by IMPA/CAPES. The third author was partially supported by FAPEMIG under Grant APQ 01377/15 and CNPq under grant 303245/2019-0. The fourth author was partially supported by DICYT grant 041933GM from VRIDEI-USACH

We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points $P = \{(t_i,x_i),\; i = 1,\dots,N\}$, find an ODE $x^\prime(t) = v (x)$ that admits a solution $x(t)$ such that $x_i \approx x(t_i)$ as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function $D(x)$ from the given data set. Afterwards, we determine the field $v(x)$, using the conjugate map defined by Schröder's equation and the solution of a related Julia's equation. Moreover, our approach also works for the inverse problems where one has to determine an ODE from multiple sets of data points.

We also study existence, uniqueness, stability and other properties of the recovered field $v(x)$. Finally, we present several numerical methods for the approximation of the field $v(x)$ and provide some illustrative examples of the application of these methods.

Citation: Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008
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Iteration function $D$ (solid line), points of the data set with a noise level $\sigma = 0.5$ (blue points) and the recovered iteration function (red circle) as discussed in Example 1 (Subsection 6.1). Notice that the exact and recovered iteration functions are basically indistinguishable
The exact field $v(x)$ (solid blue line) and the recovered field corresponding to Example 1 (Subsection 6.1), for different values of $\sigma$. For $\sigma = 0.1$ (dashed blue line), $\sigma = 0.5$ (dotted black), $\sigma = 0.9$ (dash-dot green line), $\sigma = 1.5$ (dashed red blue) and $\sigma = 2.9$ (dotted magenta line). Notice that for $\sigma<1$ exact and recovered fields are indistinguishable
Multiple sets of data points (upper plot) and the corresponding synthetic data (lower plot) used in Example 2 (Subsection 6.2). The set 1 (blue points) is used in both cases (a) and (b), whereas the other sets (red circles) are only used in case (b)
Exact and approximate iteration function and its derivative (upper plot) and the associated approximation errors (lower plot) in the interval $(0,1)$, corresponding to case (a) of Example 2 (Subsection 6.2). Notice that the exact functions and their approximations are indistinguishable
Exact and approximated field $v(x)$ (upper plot) and approximation error (lower plot) in the interval $(0,1)$ corresponding to case (a) of Example 2 (Subsection 6.2). Observe that the exact field and its approximation are indistinguishable
Exact and approximated iteration function and its derivative (upper plot) and the associated approximation errors (lower plot) corresponding to case (b) of Example 2 (Subsection 6.2). Notice that the exact functions and their approximations are almost indistinguishable
Exact and approximated field $v(x)$ (upper plot) and approximation error (lower plot) in the interval $(-2.04,2.04)$ corresponding to case (b) of Example 2 (Subsection 6.2). Notice that the exact field and its approximation are indistinguishable
Exact and approximated iteration function and its derivative (upper plot), and approximation errors for the iteration function and its derivative (lower plot); corresponding to Example 3 (Subsection 6.3)
Exact and approximated field (upper plot) and the approximation error (lower plot) corresponding to Example 3 (Subsection 6.3). Notice that the difference between the field and its approximation is only noticeable around $x = 1.2$
 Algorithm 1: Implementation of formula (35) Require: $x_0$, $\epsilon$, functions $D$ and $D^{\prime}$ Ensure: $g(x_0) = q_n$ 1: $x_n=x_0$, error=1, lim=1, $q_n=1$ 2: while error $> \epsilon$ do 3:    last=lim 4:    $q_n=q_n D(x_n)/(x_n D^{\prime}(x_n))$ 5:    $x_n=D(x_n)$ 6:    lim=$q_n$ 7:    error=$|$lim-last$|$/$|$last$|$ 8: end while 9: $q_n = x_o q_n$ 10: return $q_n$
 Algorithm 1: Implementation of formula (35) Require: $x_0$, $\epsilon$, functions $D$ and $D^{\prime}$ Ensure: $g(x_0) = q_n$ 1: $x_n=x_0$, error=1, lim=1, $q_n=1$ 2: while error $> \epsilon$ do 3:    last=lim 4:    $q_n=q_n D(x_n)/(x_n D^{\prime}(x_n))$ 5:    $x_n=D(x_n)$ 6:    lim=$q_n$ 7:    error=$|$lim-last$|$/$|$last$|$ 8: end while 9: $q_n = x_o q_n$ 10: return $q_n$
 Algorithm 2: Implementation of formula (37) Require: $x_0,\dots,x_m$, $\epsilon$, functions $D$ and $D^{\prime}$ Ensure: $g(x_0) = g_0, \dots, g(x_m) = g_m$ 1: $g_0 = \cdots = g_m = 1$, error=1, lim=1 2: $q_0 = D(x_0)/(x_0 D^{\prime}(x_0)), \dots, q_m = D(x_m)/(x_m D^{\prime}(x_m))$ 3: while error $> \epsilon$ do 4:    $gl_0 = g_0, \dots, gl_m = g_m$ 5:    Compute function $g(x)$ interpolating data : $(x_0,g_0), \dots, (x_m,g_m)$ 6:    $g_0= q_0 g(D(x_0)), \dots, g_m= q_m g(D(x_m))$ 7:    error=max($|gl_j-g_j|$)/max($|gl_j|$) 8: end while 9: $g_0 = x_o g_0, \dots, g_m = x_m g_m$ 10: return $g_0, \dots, g_m$
 Algorithm 2: Implementation of formula (37) Require: $x_0,\dots,x_m$, $\epsilon$, functions $D$ and $D^{\prime}$ Ensure: $g(x_0) = g_0, \dots, g(x_m) = g_m$ 1: $g_0 = \cdots = g_m = 1$, error=1, lim=1 2: $q_0 = D(x_0)/(x_0 D^{\prime}(x_0)), \dots, q_m = D(x_m)/(x_m D^{\prime}(x_m))$ 3: while error $> \epsilon$ do 4:    $gl_0 = g_0, \dots, gl_m = g_m$ 5:    Compute function $g(x)$ interpolating data : $(x_0,g_0), \dots, (x_m,g_m)$ 6:    $g_0= q_0 g(D(x_0)), \dots, g_m= q_m g(D(x_m))$ 7:    error=max($|gl_j-g_j|$)/max($|gl_j|$) 8: end while 9: $g_0 = x_o g_0, \dots, g_m = x_m g_m$ 10: return $g_0, \dots, g_m$
Values of the relative errors for the iteration function $D$, its derivative $D'$ and the field $v$, and the stability constant $C_v$ corresponding to example 6.1
 $\sigma$ $\epsilon_{D}$ $\epsilon_{D^{'}}$ $\epsilon_{v}$ $C_v$ [0.5ex] 0.1 2.51 0.026 0.019 0.0075 0.5 2.46 0.0883 0.06 0.026 0.9 2.65 0.45 0.25 0.080 1.9 2.3 0.67 0.49 0.1661 2.9 2.77 0.822 0.38 0.107 3.9 2.38 0.3137 0.27 0.10 4.5 2.59 0.263 0.1614 0.05
 $\sigma$ $\epsilon_{D}$ $\epsilon_{D^{'}}$ $\epsilon_{v}$ $C_v$ [0.5ex] 0.1 2.51 0.026 0.019 0.0075 0.5 2.46 0.0883 0.06 0.026 0.9 2.65 0.45 0.25 0.080 1.9 2.3 0.67 0.49 0.1661 2.9 2.77 0.822 0.38 0.107 3.9 2.38 0.3137 0.27 0.10 4.5 2.59 0.263 0.1614 0.05
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