100 lines
3.6 KiB
Markdown
100 lines
3.6 KiB
Markdown
# Homotopy Continuation
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### Introduction
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The aim of a homotopy method consists in solving a system of N nonlinear
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equations in N variables \[1, p.1\]:
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$$
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F(\bm{x}) = 0, \hspace{5mm} F: \mathbb{R}^N \rightarrow \mathbb{R}^N
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$$
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This is achieved by defining a homotopy, or deformation,
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$H : \mathbb{R}^N \rightarrow \mathbb{R}^N$ such that
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$$
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H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x},1) = F(\bm{x}),
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$$
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where $G: \mathbb{R}^N \rightarrow \mathbb{R}^N$ is trivial, having known zero
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points. One can then trace an implicitly defined curve
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$\bm{c}(s) \in H^{-1}(0)$ from $(\bm{x}_1,0)$ to $(\bm{x}_2, 1)$ [2, p.122].
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Typically, a convex homotopy such as
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$$
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H(\bm{x}, t) := tG(\bm{x}) + (1-t)F(\bm{x})
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$$
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is chosen [1, p.2].
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### Implicit Definition of Solution Curve
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"In the construction of the homotopy H(x, t) = 0, it may seem natural to use t
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as the designated parameter for the solution curve [...]. However, this
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parametrization has a severe limitation, as t cannot be used as a smooth
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parameter in certain situations. For example, [...] at points where ∂H/∂x is
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singular the solution curve of H(x, t) = 0 cannot be parametrized by t
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directly. Actually, one may avoid such difficulties by considering both x and t
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as independent variables and parametrize the smooth curve γ by the arc-length."
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[2, p. 125-126].
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We can show that the solution curve can be implicitly defined as
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$$
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\begin{align}
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DH(\bm{y}(s))\cdot \dot{\bm{y}}(s) = 0 \\[.5em]
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\text{det}\left(\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)\end{array}\right) = \sigma_0 \\[.5em]
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\lVert \dot{\bm{y}}(s) \rVert = 1 \\[.5em]
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\bm{y}(0) = (\bm{x}_0, 0)
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.
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\end{align}
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$$
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where $DH(y)$ is the Jacobian of $H(y)$ [2, p.127] [1, p.9]. Equation (1)
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corresponds to $\dot{y}(s)$ being tangent to the solution curve, the purpose of
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(2) is to fix the direction in which the curve is traced.
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### Tracing solution curves
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"In principle, any of the available ODE solvers capable of integrating the
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above system can be used to trace the curve [...]. However, due to numerical
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stability concerns, a more preferable method to trace the curve is the
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'prediction-correction scheme'. [...] Among many different choices for the
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'predictors' as well as 'correctors', we shall present here a commonly used
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combination of the (generalized) Euler’s method for the predictor and Newton’s
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method for the corrector." [2, p.127]
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#### Euler's Predictor
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$$
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\hat{\bm{y}} = \bm{y}_0 + \Delta s \cdot \sigma \cdot \Delta \bm{y},
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$$
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with $\Delta s$ denoting the step size and $\Delta y$ the numerical
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approximation of $\dot{y}(s)$ [2, p.128].
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#### Newton's corrector
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$$
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\bm{y} = \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm} \mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} - (DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}}).
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$$
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" [...] the shrinking distance
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$\lVert \mathcal{N}^j(\hat{\bm{y}}) - \mathcal{N}^{j-1} (\hat{\bm{y}})\rVert$
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between successive points produced by the iterations can be used as a criterion
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for convergence. Of course, if the iterations fail to converge, one must go
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back to adjust the step size for the Euler’s predictor." [2, p.130]
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______________________________________________________________________
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## References
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\[1\]: E. L. Allgower and K. Georg, Introduction to numerical continuation
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methods. in Classics in applied mathematics, no. 45. Philadelphia, Pa: Society
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for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6,
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Philadelphia, PA 19104), 2003. doi: 10.1137/1.9780898719154.
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\[2\]: T. Chen and T.-Y. Li, “Homotopy continuation method for solving systems
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of nonlinear and polynomial equations,” Communications in Information and
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Systems, vol. 15, no. 2, pp. 119–307, 2015, doi: 10.4310/CIS.2015.v15.n2.a1.
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