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Numerical approximation of a class of constrained Hamilton-Jacobi equations

Benoît Gaudeul, Hélène Hivert

Abstract

In this paper, we introduce a framework for the discretization of a class of constrained Hamilton-Jacobi equations, a system coupling a Hamilton-Jacobi equation with a Lagrange multiplier determined by the constraint. The equation is non-local, and the constraint has bounded variations. We show that, under a set of general hypothesis, the approximation obtained with a finite-differences monotonic scheme, converges towards the viscosity solution of the constrained Hamilton-Jacobi equation. Constrained Hamilton-Jacobi equations often arise as the long time and small mutation asymptotics of population models in quantitative genetics. As an example, we detail the construction of a scheme for the limit of an integral Lotka-Volterra equation. We also construct and analyze an Asymptotic-Preserving (AP) scheme for the model outside of the asymptotics. We prove that it is stable along the transition towards the asymptotics. The theoretical analysis of the schemes is illustrated and discussed with numerical simulations. The AP scheme is also used to conjecture the asymptotic behavior of the integral Lotka-Volterra equation, when the environment varies in time.

Numerical approximation of a class of constrained Hamilton-Jacobi equations

Abstract

In this paper, we introduce a framework for the discretization of a class of constrained Hamilton-Jacobi equations, a system coupling a Hamilton-Jacobi equation with a Lagrange multiplier determined by the constraint. The equation is non-local, and the constraint has bounded variations. We show that, under a set of general hypothesis, the approximation obtained with a finite-differences monotonic scheme, converges towards the viscosity solution of the constrained Hamilton-Jacobi equation. Constrained Hamilton-Jacobi equations often arise as the long time and small mutation asymptotics of population models in quantitative genetics. As an example, we detail the construction of a scheme for the limit of an integral Lotka-Volterra equation. We also construct and analyze an Asymptotic-Preserving (AP) scheme for the model outside of the asymptotics. We prove that it is stable along the transition towards the asymptotics. The theoretical analysis of the schemes is illustrated and discussed with numerical simulations. The AP scheme is also used to conjecture the asymptotic behavior of the integral Lotka-Volterra equation, when the environment varies in time.
Paper Structure (26 sections, 13 theorems, 222 equations, 10 figures)

This paper contains 26 sections, 13 theorems, 222 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Construction of the scheme and main results
  3. An example: Lotka-Volterra integral equations
  4. Examples of schemes
  5. Asymptotic-preserving scheme
  6. Convergence of scheme \ref{['eq:scheme']}
  7. Proof of Prop. \ref{['thm:existence']}
  8. ${\boldsymbol{u}}^n$ satisfies property \ref{['it:pseudomon']}
  9. Existence of $I^{n+1}$
  10. Properties \ref{['it:Rpos']} and \ref{['it:I_bound']}
  11. Property \ref{['it:I_BVbound']}
  12. Property \ref{['it:sc']} - convex setting
  13. Conclusion
  14. Proof of Prop. \ref{['thm:limitconv']}
  15. Limits of ${u_{\Delta t}}$, and ${I_{\Delta t}}$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose that assumptions hyp:H_convex_superlinear to hyp:u0_minimum are satisfied, and that ${\mathcal{R}}$ does not depend on $t$.

Figures (10)

  • Figure 1: Scheme \ref{['eq:scheme']} with $T=0.5$, ${\Delta t}_{\mathrm{ref}} = 4\cdot 10^{-5}$, $S=2$, $b$, ${\mathcal{R}}$ and ${u^{\mathrm{in}}_{\mathrm{conv}}}$ defined in \ref{['eq:tests_b']}, \ref{['eq:tests_b']} and \ref{['eq:uin_conv']}. Convergence test for the solution of \ref{['eq:scheme']} with \ref{['eq:HP1']} and \ref{['eq:HCSS']}. Left: Error \ref{['eq:Lschemeconv_err']} on the component ${\boldsymbol{u}}$, as a function of ${\Delta t}$ (log scale). Right: Error \ref{['eq:Lschemeconv_err']} on the component ${\boldsymbol{I}}$, as a function of ${\Delta t}$ (log scale).
  • Figure 2: Scheme \ref{['eq:scheme']} with $T=0.5$, ${\Delta t} = 2\cdot 10^{-5}$, $S=2$, $b$, ${\mathcal{R}}$ and ${u^{\mathrm{in}}_\mathrm{not\;conv}}$ defined in \ref{['eq:tests_b']}, \ref{['eq:tests_R']} and \ref{['eq:uin_notconv']}. Qualitative properties of \ref{['eq:scheme']} with \ref{['eq:HP1']}. Left: $I_{\Delta t}$ as a function of $t$. Right: $u_{\Delta t}$ as a function of $x$ (trait) and $t$ (time). The solid lines represent respectively the local minima of $u_{\Delta t}$, and the optimal trait. Light colors emphasize small values of $u_{\Delta t}$.
  • Figure 3: Scheme \ref{['eq:scheme']} with $T=1.5$, $S=2$, ${\mathcal{R}}(0,x,I)$ defined in \ref{['eq:tests_R']}, and $b$ and ${u^{\mathrm{in}}_\mathrm{not\;conv}}$ defined in \ref{['eq:tests_b']} and \ref{['eq:uin_notconv']}. $I_{\Delta t}$ as a function of $t$. Left: with ${\Delta t}=10^{-2}$. Right: with ${\Delta t}=10^{-4}$.
  • Figure 4: With $T=0.5$, ${\Delta t}=10^{-3}$, $S=2$, $b$, ${\mathcal{R}}$ and ${u^{\mathrm{in}}_\mathrm{not\;conv}}$ defined in \ref{['eq:tests_b']}, \ref{['eq:tests_R']} and \ref{['eq:uin_notconv']}. AP property of scheme \ref{['eq:scheme_ep']}. Scheme \ref{['eq:scheme']} with \ref{['eq:HP1']}. Left: Error \ref{['eq:tests_AP_Err']} on the component ${\boldsymbol{v}}$ as a function of $\varepsilon$ (log scale). Right: Error \ref{['eq:tests_AP_Err']} on the component ${\boldsymbol{J}}$ as a function of $\varepsilon$ (log scale).
  • Figure 5: With $T=0.5$, ${\Delta t}=10^{-3}$, $S=2$, $b$, ${\mathcal{R}}$ and ${u^{\mathrm{in}}_\mathrm{not\;conv}}$ defined in \ref{['eq:tests_b']}, \ref{['eq:tests_R']} and \ref{['eq:uin_notconv']}. UA property of scheme \ref{['eq:scheme_ep']}. Left: For a range of ${\Delta t}$, error \ref{['eq:tests_UA_Err']} on the component ${\boldsymbol{v}}$ as a function of $\varepsilon$ (log scale). Right: For a range of ${\Delta t}$, error \ref{['eq:tests_UA_Err']} on the component ${\boldsymbol{J}}$ as a function of $\varepsilon$ (log scale).
  • ...and 5 more figures

Theorems & Definitions (50)

  • Theorem 1.1: CalvezLam2020
  • Remark 1.3
  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.2
  • Remark 2.4
  • Proposition 2.3: Scheme \ref{['eq:scheme']}: existence of solutions and qualitative properties
  • Remark 2.5
  • ...and 40 more