More on chapter 5

Author: ctkelley

SIAMFANLCh5.ipynb

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     "\\lambda \\le 1 - \\sqrt{1-c} < 1.\n",
     "$$\n",
     "Hence $\\calf'(H)$ is nonsingular for $0 < c < 1$ on the lower branch.\n",
-    "This also proves the norm estimate from Chapter __4__.\n"
+    "This also proves the norm estimate from Chapter __4__.\n",
+    "    \n",
+    "For $c=1$, $\\calf'(H)$ is singular with a one-dimensional null space.\n",
+    "To complete the analysis of the singularity we need to make the dependence\n",
+    "in $c$ explicit and write the equation as\n",
+    "$$\n",
+    "\\calf(H, c) = 0.\n",
+    "$$\n",
+    "We let $\\calf'_H$ be the\n",
+    "Frechet derivative in $H$ and\n",
+    "$\\calf'_c(H,c)$ be the derivative in $c$. At a solution\n",
+    "$$\n",
+    "\\calf'_c(H,c) = - H^2 \\frac{1}{2} \\call H.\n",
+    "$$\n",
+    "and so, at $c=1$, using the equation.\n",
+    "$$\n",
+    "\\calf'_c(H,1) = -H^2( H - 1).\n",
+    "$$\n",
+    "    \n",
+    "The singularity at $c=1$ is a\n",
+    "__simple fold__ <cite data-cite=\"herb\"><a href=\"siamfa.html#herb\">(Kel87)</cite>.\n",
+    "This means that $\\calf'(H)$ has a null space\n",
+    "of dimension 1 and that $\\calf'_c(H,1) \\not \\in {\\cal R}(\\calf'_H(H))$.\n",
+    "Here $\\cal{R}$ denotes the range of an operator. In this case\n",
+    "$$\n",
+    "{\\cal R}(\\calf'_H(H)) = \\{ u \\in C[0,1] \\, | \\,\n",
+    "\\int_0^1 u(\\nu) \\psi(\\nu) \\dnu = 0 \\},\n",
+    "$$\n",
+    "where $\\psi$ is the eigenfunction of $\\calf'_H(H)^*$, the adjoint of\n",
+    "$\\calf'_H(H)$ corresponding to the zero eigenvalue. The reader can\n",
+    "verify that, for $c=1$,\n",
+    "$$\n",
+    "\\psi(\\mu) = (\\calg'(H)^* \\psi)(\\nu) = \\frac{1}{2} \\int_0^1\n",
+    "\\frac{\\nu}{\\nu+\\mu} H^2(\\nu) \\psi(\\nu) \\dnu\n",
+    "$$\n",
+    "is satisfied by $\\psi = H^{-1}$. Since $H \\ge 1$ and $\\calf'_c(H,1)$ does\n",
+    "not change sign, $\\calf'_c(H,1)$ cannot be in the range\n",
+    "of $\\calf'_H(H)$. We will show why this fact is important in the next\n",
+    "section.\n",
+    "\n"
    ]
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+    "### Section 5.2.2: Continuation\n",
+    "\n",
+    "The object of continuation methods is to solve\n",
+    "a parameter dependent set of nonlinear equations\n",
+    "$$\n",
+    "\\mf(\\vx, \\lambda) = 0.\n",
+    "$$\n",
+    "The output is a\n",
+    "__solution arc__\n",
+    "or\n",
+    "__solution path__ \n",
+    "$$\n",
+    "\\{ \\vx(\\lambda) \\, | \\, \\mf(\\vx(\\lambda), \\lambda) = 0,\n",
+    "\\lambda_0 \\le \\lambda \\le \\lambda_{max} \\}.\n",
+    "$$\n",
+    "The solution arc for the H-equation is a simple arc\n",
+    "with no bifurcations. Because of this we can use a\n",
+    "straightforward path following algorithm. The\n",
+    "objective of this section is to show how a nonlinear solver\n",
+    "can be modified in a simple way to resolve singularities\n",
+    "such as the one for the $H$-equation at $c=1$.\n",
+    "\n",
+    "We have seen other examples\n",
+    "of parameter-dependent nonlinear equations in\n",
+    "the buckling beam problem from Chapters __2__ and __3__\n",
+    "and the cubic polynomial example\n",
+    "in Chapter __1__. These examples have\n",
+    "pitchfork bifurcations <cite data-cite=\"herb\"><a href=\"siamfa.html#herb\">(Kel87)</cite>\n",
+    "and most path following codes\n",
+    "<cite data-cite=\"doedel\"><a href=\"siamfa.html#doedell\">(Doe97)</cite>,\n",
+    "<cite data-cite=\"govaerts\"><a href=\"siamfa.html#govaerts\">(Gov00)</cite>,\n",
+    "<cite data-cite=\"kuznetsov\"><a href=\"siamfa.html#kuznetsov\">(Kuz98)</cite>,\n",
+    "<cite data-cite=\"loca\"><a href=\"siamfa.html#loca\">(SBRP<sup>+</sup>02)</cite>,\n",
+    "<cite data-cite=\"pitcon\"><a href=\"siamfa.html#pitcon\">(Rhe86)</cite>,\n",
+    "<cite data-cite=\"bifkit\"><a href=\"siamfa.html#bifkit\">(Vel20)</cite>,\n",
+    "have machinery to detect intersecting solution arcs and\n",
+    "follow the different branches. Those methods are beyond\n",
+    "the scope of this book. The Julia package __BifurcationKit.jl__\n",
+    "<cite data-cite=\"bifkit\"><a href=\"siamfa.html#bifkit\">(Vel20)</cite>\n",
+    "is quite complete.\n",
+    "    \n",
+    "One simple approach to computing a solution path would be to\n",
+    "identify a solution $\\vx(\\lambda_0)$ for a particular\n",
+    "value of $\\lambda$ and then increment $\\lambda$ by a small amount\n",
+    "and use Newton's method with an initial iterate from the previous\n",
+    "solution to compute the solution for the next step. In the\n",
+    "case of the $H$-equation, where the parameter is $c$. One can\n",
+    "use $c_0 = 0$ and $H \\equiv 1$ to begin the continuation.\n",
+    "\n",
+    "A candidate for the algorithm is __natural parameter continuation__\n",
+    "where we increment the original parameter in the equation. Note that in\n",
+    "the discussion in this section, as is common practice, the parameter is\n",
+    "called $\\lambda$ in the context of a general method, but the actual\n",
+    "name of the parameter ($c$ in the case of the H-equation) is used when\n",
+    "talking about specific examples.\n"
+   ]
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