0
$\begingroup$

I am looking for references on the implementation of computational optimization methods within the context of differential equations.

As an example, let's say I have several coupled non-linear ordinary differential equations describing a biogeochemical system where some living organism $X$ eats chemical compounds $c_1$, $c_2$ and $c_3$ that contain some energy for living and reproduction ($\Omega_1$, $\Omega_2$ and $\Omega_3$). This energy is variable in time and depends on the concentrations of $c_1$, $c_2$ and $c_3$:

$$\begin{aligned} \dot c_1 &= - X \cdot c_1 \cdot \alpha\\ \dot c_2 &= - X \cdot c_2 \cdot \beta \\ \dot c_3 &= - X \cdot c_3 \cdot \gamma\\ \dot X &= \Omega_1 (c_1, c_2 ,c_3)\cdot X \cdot c_1 \cdot \alpha + \Omega_2(c_1, c_2 ,c_3)\cdot X \cdot c_2 \cdot \beta + \Omega_3(c_1, c_2 ,c_3)\cdot X \cdot c_3 \cdot \gamma \end{aligned}$$

where

$$\alpha + \beta + \gamma = 1$$

At every timestep, an optimization algorithm finds the maximum

$$\max \bigg( \Omega_1 (c_1)\cdot X \cdot c_1 \cdot \alpha + \Omega_2(c_2)\cdot X \cdot c_2 \cdot \beta + \Omega_3(c_3)\cdot X \cdot c_3 \cdot \gamma \bigg)$$

i.e., to find the best value of $\alpha, \beta, \gamma$ to ensure maximum growth of $X$.

The problem I am facing is that the solution is not smooth, i.e., on some time step living organism picks best energy-yielding product $c_1$, but the next time step it picks $c_2$, so it produces non-smooth solutions for $\alpha, \beta, \gamma$ and it kind of jumps back and forth between $c_1$, $c_2$ and $c_3$ in time, which is not very realistic. So I need some kind of optimization algorithms that has smooth switching functions, e.g., sigmoid functions, but for multiple variables $\alpha, \beta$, and $\gamma$.

This is a toy example. In the real example, I am solving PDE of nonlinear equations and have more than $30$ variables of $\alpha$, $\beta$, and $\gamma$.

I found some information about receding horizon algorithm for the system that is very close to the system I described above (link). However, the description is very general and no information on the implementation.

$\endgroup$
  • $\begingroup$ What can you tell about functions $\Omega_i$? $\endgroup$ – Rodrigo de Azevedo Apr 27 at 15:27
  • $\begingroup$ Are $\alpha, \beta, \gamma$ parameters or time-varying control inputs? $\endgroup$ – Rodrigo de Azevedo Apr 27 at 15:30
  • $\begingroup$ $\Omega$ is the function of $c$. Something like $\Omega = A + RT ln (c_1^m)/(c_2^n)$ $\endgroup$ – Igor Markelov Apr 28 at 12:23
  • $\begingroup$ In the question, there are functions $\Omega_1, \Omega_2, \Omega_3$, each depending on a single input, namely, $c_1, c_2, c_3$, respectively. Yet, your function $\Omega$ (where is it used?) contains two inputs. $\endgroup$ – Rodrigo de Azevedo Apr 28 at 12:33
  • $\begingroup$ For example, $\Omega_1$ will have $c_1$ for sure, but It may also have other $c_i$. In the real example, it is Gibbs Free Energy, $G = G_0 + RT ln A^mB^n/C^jD^k$, where $A$, $B$, $C$, $D$ are the concentrations in the chemical reaction $A$ + $B$ -> $C$ + $D$ and $G_0$, R and T are constants. In other words $\Omega = f(c_1, c_n)$ $\endgroup$ – Igor Markelov Apr 28 at 12:42
0
$\begingroup$

Hint: I will assume that $\alpha$, $\beta$ and $\gamma$ are constants. You can simplify your problem by eliminating $c_2$ and $c_3$ by dividing the first and second and the first and the third differential equation.

$$\dfrac{dc_1}{dc_2}=\dfrac{\alpha}{\beta}\dfrac{c_1}{c_2}\implies \dfrac{dc_1}{c_1}=\dfrac{\alpha}{\beta}\dfrac{dc_2}{c_2}$$ $$\implies \ln c_1 -\ln c_1(t=0)=\dfrac{\alpha}{\beta}[\ln c_2 -\ln c_2(t=0)]$$ Now, solve for $c_2$ to obtain

$$c_2=c_2(0)\left[\frac{c_1}{c_1(0)} \right]^{\beta/\alpha}.$$

Do the same procedure for the first and third equation. If the parameters are not constant then you will have to solve a time-variant system. Then you will only be left with a second order system. You should be able to use a similar method for the resulting system.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.