Predictors - Correctors

The continuation method works with the following pattern (see [Allgower1990]):

1. compute tangent
2. call predictor (based on tangent, mostly)
3. call corrector

There are several couples predictor-tangent/corrector which can be used in BifurcationKit.jl as we now explain. The tangent computation is formally included in the predictor whereas it is a distinct function in the code.

Corrector

Note that setting the predictor also sets the corresponding corrector: it selects the couple predictor-corrector. You don't have (in fact cannot) set them independently.

1. Natural, zeroth order predictor

This is the dumbest predictor based on the formula $(x_1,p_1) = (x_0, p_0 + ds)$ with Newton corrector ; it fails at Turning points. This is set by the algorithm Natural() in continuation. For matrix based jacobian, it is not faster than the pseudo-arclength predictor because the factorisation of the jacobian is cached. For Matrix-free methods, this predictor can be faster than the following ones until it hits a Turning point.

2. First order predictor

This predictor is based on a computation of the tangent $\tau = (dx,dp)$ to the curve of solutions, it is given by $(x_1,p_1) = (x_0,p_0) + ds\cdot \tau$. This predictor passes Turning points when used with PALC Newton corrector. BifurcationKit.jl provides two ways to compute the tangent $(dx, dp)$.

2a. Secant predictor

This predictor is called secant and is parametrized by the algorithm PALC(tangent = Secant()) in continuation with Secant . It is computed with $\tau = (x_1, p_1) - (x_0, p_0)$ and normalized with the norm $\|(x, p)\|^2_\theta := \frac{\theta}{length(x)} \langle x,x\rangle + (1 - \theta)\cdot p^2$ for some $0<\theta<1$.

Parameter θ

The parameter θ in the norm above (see also the struct ContinuationPar) is very important. It should be tuned for the continuation to work properly especially in the case of large problems where the $\langle x - x_0, dx_0\rangle$ component in the [Pseudo arclength continuation

](@ref) constraint might be favored too much. Also, large θs favour p as the corresponding term in the constraint $N$ involves the term $1-θ$.

2b. Bordered predictor

This predictor departs from the previous one in the way the tangent $\tau$ is estimated. It computes $\tau:=(dx, dp)$ by solving solving the bordered linear system $\begin{bmatrix} F_x & F_p \\ \frac{\theta}{length(x)}dx_0 & (1-\theta)dp_0\end{bmatrix}\begin{bmatrix}dx \\ dp\end{bmatrix} =\begin{bmatrix}0 \\ 1\end{bmatrix}$

where $\tau_0:=(dx_0, dp_0)$ is the tangent at the previous continuation step.

The predictor is set by the option PALC(tangent = Bordered()) in continuation with Bordered. The linear solver for the linear problem in $(dx, dp)$ is set by the option bls in PALC: it is one of Bordered linear solvers (BLS).

3. Polynomial predictor

The polynomial predictor is based on a fit (least square regression) of an $n$th-order polynomial $P$ on the last $k$ solution vectors, where $n < k$. The arclength $s$ is used for the polynomial which then fits the solution $(x_i,p_i,s_i)$ as $P(s_i)\approx (x_i,p_i)$. To keep $s$ in suitable range (see [Waugh]), we rescale it as $s\to \frac{s-\bar s}{\sigma}$ where $\sigma$ is the standard deviation of the $s_i$.

This algorithm is parametrized by alg = Polynomial(Fred, n, k, v0) where pred::AbstractTangentComputation is the tangent predictor used only for the first $k$ solutions before the polynomial predictor is operational and v0 is an example of guess. More information is available in Polynomial.

4. Multiple predictor (aka pmcont in pde2path)

The predictor is designed [Uecker2014] to avoid spurious branch switching and pass singular points especially in PDE where branch point density can be quite high. It is based on the use of many predictors with increasing "jumps" $(x_i,p_i) = (x_0,p_0) + i\cdot ds\cdot \tau,\ i\leq nb$ and use a corrector (PALC Newton) with the following twist. The criterion is that in each Newton step, the residual has to decrease by a factor $0<\alpha<1$:

$$$\| F(u_n,p_n)\|\leq \alpha \| F(u_{n-1},p_{n-1}) \|$$$

otherwise the corrector fails. The solution that is returned is the one for the highest $i$. We refer to [Uecker2014] for an exposition of the step size adaption strategy.

This algorithm is parametrized by alg = Multiple(pred, x0, α, nb) where τ is an initial tangent vector (used to set the types) and pred::PALC is a predictor. The default value is pred = PALC(). More information is available in Multiple.

References

• Allgower1990

Allgower and Georg, Numerical Continuation Methods, 1990

• Uecker2014

1.Uecker, H. pde2path - A Matlab Package for Continuation and Bifurcation in 2D Elliptic Systems. NMTMA 7, 58–106 (2014).

• Waugh

Waugh, Illingworth, and Juniper, “Matrix-Free Continuation of Limit Cycles for Bifurcation Analysis of Large Thermoacoustic Systems.”