Auxiliary variables for numerically solving nonlinear equations with softly broken symmetries

General methods for solving simultaneous nonlinear equations work by generating a sequence of approximate solutions that successively improve a measure of the total error. However, if the total error function has a narrow curved valley, the available techniques tend to find the solution after a very large number of steps, if ever. The solver first converges rapidly to the valley, but once there it converges extremely slowly to the solution. In this paper we show that in the specific physically important case where these valleys are the result of a softly broken symmetry, the solution can often be found much more quickly by adding the generators of the softly broken symmetry as auxiliary variables. This makes the number of variables more than the equations and hence there will be a family of solutions, any one of which would be acceptable. We present a procedure for finding solutions in this case and apply it to several simple examples and an important problem in the physics of false vacuum decay. We also provide a Mathematica package that implements Powell's hybrid method with the generalization to allow more variables than equations.

Figure 1

(a) Error surface for a system of two equations that exhibits a narrow valley. (b) Trajectory that Powell's hybrid method took to find the solution. The initial guess is shown by the green asterisk. The solver first quickly converged to the valley, but then it moved very slowly along the valley (solid line) to the solution (green square) at the origin. If the valley has very sharp curves, it will take many more steps to find the solution.

Figure 2

(a) Error as a function of $x$ and $y$ for the functions defined in (4.1) with $\epsilon =0.1$. The problem has an approximate rotational symmetry. (b) To see the breakdown of the symmetry we zoomed in on the valley, which is slanted. (c) Path that the solver took without an extra variable. (d) Path in $x$ and $y$ taken when we use the extra variable $\varphi$ and choose $s=10$. In this case, the solver takes steps in $x^{\prime },y^{\prime }$, and $\varphi$, but we graph the points $x$ and $y$ given by (4.2) with $\theta =s\varphi$. In both cases the starting guess is $[-1.2cos(\pi /8),1.2sin(\pi /8\left)\right]$. The left panel took 14 steps and the right only 6 steps. We see in Fig. 3 that smaller $\epsilon$ increases the number of steps rapidly.

Figure 3

Steps taken by the solver for the example presented in (4.1) for $\epsilon ={10}^{-4}$. Now the rotational symmetry is broken very softly. (a) Steps taken without the extra variable and (b) steps taken with an extra variable $\varphi$ that generates the broken symmetry and $s=10$. In both cases the starting guess is $[-1.2cos(\pi /8),1.2sin(\pi /8\left)\right]$. The left panel took 535 steps and the right only 6 steps.

Figure 4

(a) Values of $F_1$ (blue circles) and $F_2$ (red squares) at steps taken by the solver before finding the solution. (b) Same as in (a) but with an extra variable $\delta$ (green triangles) added. The solution was found in ten steps.

Figure 5

(a) Potential with two minima at ${\varphi }_t=0$ and ${\varphi }_f$ in solid blue and the upside down potential as the dotted red graph. The minimum at ${\varphi }_f$ is a metastable minimum and can tunnel quantum mechanically to the other minimum. (b) Solution to the field equation.

Figure 6

Illustration of multishooting method. One has to adjust the values of $\{{\varphi }_1,{\varphi }_2,{\varphi }_2^{\prime },{\varphi }_4\}$ and evolve the field equations (5.1) along the blue curves such that the field and its derivative are continuous at $r_2$ and $r_3$. This produces a system of four equations of four variables.

Figure 7

Change of error after each step taken (a) without adding an auxiliary variable and (b) when including $\delta$, for the potential in (5.3) in order to solve (5.1).