From bf3c13315e7a1536f242bae7baecc1eb64462c8f Mon Sep 17 00:00:00 2001 From: Bas Nijholt <basnijholt@gmail.com> Date: Tue, 17 Sep 2019 16:51:19 +0200 Subject: [PATCH] add line loss error fig --- figures.ipynb | 102 ++++++++++++++++++++++++++++++++++++ figures/line_loss_error.pdf | Bin 0 -> 201939 bytes paper.md | 19 ++++++- 3 files changed, 119 insertions(+), 2 deletions(-) create mode 100644 figures/line_loss_error.pdf diff --git a/figures.ipynb b/figures.ipynb index a7b1c46..81f605a 100644 --- a/figures.ipynb +++ b/figures.ipynb @@ -479,6 +479,13 @@ "plt.show()" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Line loss" + ] + }, { "cell_type": "code", "execution_count": null, @@ -550,6 +557,101 @@ "plt.savefig(\"figures/line_loss.pdf\", bbox_inches=\"tight\", transparent=True)\n", "plt.show()" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Line loss error" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import scipy\n", + "\n", + "\n", + "def f(x, offset=0.123):\n", + " a = 0.02\n", + " return x + a ** 2 / (a ** 2 + (x - offset) ** 2)\n", + "\n", + "\n", + "def g(x):\n", + " return np.tanh(x * 40)\n", + "\n", + "\n", + "def h(x):\n", + " return np.sin(100 * x) * np.exp(-x ** 2 / 0.1 ** 2)\n", + "\n", + "\n", + "funcs_1D = [\n", + " dict(function=f, bounds=(-1, 1), title=\"peak\"),\n", + " dict(function=g, bounds=(-1, 1), title=\"tanh\"),\n", + " dict(function=h, bounds=(-0.3, 0.3), title=\"wave packet\"),\n", + "]\n", + "\n", + "\n", + "def err(xs, ys, xs_rand):\n", + " ip = scipy.interpolate.interp1d(xs, ys)\n", + " abserr = np.abs(ip(xs_rand) - f(xs_rand))\n", + " return np.average(abserr ** 2) ** 0.5\n", + "\n", + "\n", + "def get_err(learner, N, bounds):\n", + " xs_rand = np.random.uniform(*bounds, size=100_000)\n", + "\n", + " xs = np.linspace(*bounds, N)\n", + " ys = f(xs)\n", + " err_lin = err(xs, ys, xs_rand)\n", + "\n", + " xs, ys = zip(*learner.data.items())\n", + " xs, ys = xs[:N], ys[:N]\n", + " ip = scipy.interpolate.interp1d(xs, ys)\n", + " err_learner = err(xs, ys, xs_rand)\n", + " return err_lin, err_learner\n", + "\n", + "\n", + "N_max = 10000\n", + "Ns = np.geomspace(4, N_max, 100).astype(int)\n", + "loss = adaptive.learner.learner1D.curvature_loss_function()\n", + "\n", + "fig, ax = plt.subplots(1, 1, figsize=(fig_width, fig_height))\n", + "\n", + "for i, d in enumerate(funcs_1D):\n", + " f = d[\"function\"]\n", + " bounds = d[\"bounds\"]\n", + " learner = adaptive.Learner1D(f, bounds, loss_per_interval=loss)\n", + " adaptive.runner.simple(learner, goal=lambda l: l.npoints >= N_max)\n", + " errs = [get_err(learner, N, bounds) for N in Ns]\n", + " err_hom, err_adaptive = zip(*errs)\n", + " color = f\"C{i}\"\n", + " label = \"abc\"[i]\n", + " ax.loglog(Ns, err_hom, ls=\"--\", c=color)\n", + " ax.loglog(Ns, err_adaptive, label=f\"$\\mathrm{{({label})}}$ {d['title']}\", c=color)\n", + "\n", + " d[\"err_hom\"] = err_hom\n", + " d[\"err_adaptive\"] = err_adaptive\n", + "\n", + "plt.legend()\n", + "ax.set_xlabel(\"$N$\")\n", + "ax.set_ylabel(r\"$\\text{Err}_{1}(\\tilde{f})$\")\n", + "plt.savefig(\"figures/line_loss_error.pdf\", bbox_inches=\"tight\", transparent=True)\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Error reduction\n", + "for d in funcs_1D:\n", + " print(d[\"err_hom\"][-1] / d[\"err_adaptive\"][-1])" + ] } ], "metadata": { diff --git a/figures/line_loss_error.pdf b/figures/line_loss_error.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6aeeccb01b7e0a336579f6ea16c2baa77792d805 GIT binary patch literal 201939 zcmaI7Q+Qt8_C6dnRvY_?%_p|)q_M5Wc4OOi8ry1Yt8vm;jjiwL`|kbw_R;@fUULqQ zaW5RK`RqT=+-46N|vmFs|tc6eq06Tr^M3Z9o2z^G_wZt4VJ0cn&0jH({?rT|6} zLuW%9J97X(KfI}}$zMm#{|OLtca~6bHgq-xu>Kn)ZfENZ0RDScG<0@0b+iSr{T1OE 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its triangle, see Fig. @fig:line_loss(b). The loss then becomes the average area of two adjacent traingles. By Taylor expanding $f$ around $x$ it can be shown that the area of the triangles relates to the contributions of the second derivative. -We can generalize this loss to $N$ dimensions +We can generalize this loss to $N$ dimensions, where the triangle is replaced by a $(N+1)$ dimensional simplex. {#fig:line_loss} +In order to compare sampling strategies, we need to define some error. +We construct a linear interpolation function $\tilde{f}$, which is an approximation of $f$. +We calculate the error in the $L^{1}$-norm, defined as, +$$ +\text{Err}_{1}(\tilde{f})=\left\Vert \tilde{f}-f\right\Vert _{L^{1}}=\int_{a}^{b}\left|\tilde{f}(x)-f(x)\right|\text{d}x. +$$ +This error approaches zero as the approximation becomes better. + +{#fig:line_loss_error} + +Figure @fig:line_loss_error shows this error as function of the number of points $N$. + <!-- https://bost.ocks.org/mike/simplify/ --> ## A parallelizable adaptive integration algorithm based on cquad -- GitLab