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-{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-08-30T16:00:06","documenter_version":"1.6.0"}}
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dev/MDE_asymptotic_variances/index.html

@@ -9,11 +9,11 @@
   J(\vartheta, \Sigma) = \frac{3}{4 \beta} \left( \frac{\Sigma}{\vartheta^2} \right)^2 \sigma_2^5, \quad h(z) = \frac{\Sigma}{2 \vartheta^2 \beta} \left[ \sigma_1^3 (1 - \sigma_1^2 z^2) \exp \left( -\frac{\sigma_1^2 z^2}{2} \right) - \sigma_2^3 \right], \quad z \in \R,
 \end{aligned}\]</p><p>with</p><p class="math-container">\[\begin{aligned}
       \sigma_1^2 := \frac{\beta^2 \vartheta}{\vartheta + \Sigma\beta^2 }, \quad \sigma_2^2 := \frac{\beta^2 \vartheta}{\vartheta + 2 \Sigma\beta^2 }.
-\end{aligned}\]</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The running times for a single evaluation can take some time due to the involved integrations, but it usually needs to be computed only once.</p></div></div><hr/><p><strong>Arguments</strong></p><ul><li><code>ϑ::Real</code>:                    positive drift coefficient <span>$\vartheta$</span>.</li><li><code>Σ::Real</code>:                    positive diffusion coefficient <span>$\Sigma$</span>.</li></ul><hr/><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; Σ_∞_QdP(1.2, 0.6)</code></pre><hr/><p>See also <a href="../invariant_densities/#MDEforM.μ"><code>μ</code></a>, <a href="#MDEforM.Σ_∞_QrP"><code>Σ_∞_QrP</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/UQatKIT/MDEforM/blob/ed8649d5acb1fcdb515c99e523b590338044e30c/src/MDE_asymptotic_variances.jl#L60-L107">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MDEforM.Σ_∞_QrP" href="#MDEforM.Σ_∞_QrP"><code>MDEforM.Σ_∞_QrP</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">Σ_∞_QrP(ϑ, Σ)</code></pre><p>Return the asymptotic variance of the MDE in the multiscale overdamped Langevin drift parameter estimation problem with a quartic potential, drift parameter <code>ϑ</code>, and diffusion parameter <code>Σ</code>.</p><p>The asymptotic variance is given by</p><p class="math-container">\[\begin{aligned}
+\end{aligned}\]</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The running times for a single evaluation can take some time due to the involved integrations, but it usually needs to be computed only once.</p></div></div><hr/><p><strong>Arguments</strong></p><ul><li><code>ϑ::Real</code>:                    positive drift coefficient <span>$\vartheta$</span>.</li><li><code>Σ::Real</code>:                    positive diffusion coefficient <span>$\Sigma$</span>.</li></ul><hr/><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; Σ_∞_QdP(1.2, 0.6)</code></pre><hr/><p>See also <a href="../invariant_densities/#MDEforM.μ"><code>μ</code></a>, <a href="#MDEforM.Σ_∞_QrP"><code>Σ_∞_QrP</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/UQatKIT/MDEforM/blob/6d99b916c746a5cad07402c962f5e5e98dd4c007/src/MDE_asymptotic_variances.jl#L57-L104">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MDEforM.Σ_∞_QrP" href="#MDEforM.Σ_∞_QrP"><code>MDEforM.Σ_∞_QrP</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">Σ_∞_QrP(ϑ, Σ)</code></pre><p>Return the asymptotic variance of the MDE in the multiscale overdamped Langevin drift parameter estimation problem with a quartic potential, drift parameter <code>ϑ</code>, and diffusion parameter <code>Σ</code>.</p><p>The asymptotic variance is given by</p><p class="math-container">\[\begin{aligned}
   \frac{\tau^2(\vartheta, \Sigma)}{J(\vartheta, \Sigma)^2} = \frac{2\Sigma}{J(\vartheta, \Sigma)^{2}} \int_\R |\Phi&#39;(x)|^2 \mu(x, \vartheta, \Sigma, V) \, dx,
 \end{aligned}\]</p><p>where</p><p class="math-container">\[\begin{aligned}
   J(\vartheta, \Sigma) = \| \partial_\vartheta \mathscr{C}_{\vartheta} \|_{L^2(\varphi)}, \quad \Phi&#39;(x) = \frac{1}{\Sigma \mu(x, \vartheta, \Sigma, V)} \int_{-\infty}^x h(z) \mu(x, \vartheta, \Sigma, V) \, dz, \quad x \in \R.
 \end{aligned}\]</p><p>Here <span>$\mu(x, \vartheta, \Sigma, V)$</span> corresponds to <a href="../invariant_densities/#MDEforM.μ"><code>μ</code></a> with <span>$V(x) = x^4/4-x^2/2$</span>. In this case it holds</p><p class="math-container">\[\begin{aligned}
   &amp;J(\vartheta, \Sigma) = \int_\R (\partial_\vartheta \mu(\vartheta, \Sigma, V) \ast k_\beta)(x) \, \partial_\vartheta \mu(x, \vartheta, \Sigma, V) \, dx, \\[0.5cm]
   &amp;h(z) = (\partial_\vartheta \mu(\vartheta, \Sigma, V) \ast k_\beta)(z) - \int_\R (\partial_\vartheta \mu(\vartheta, \Sigma, V) \ast k_\beta)(x) \, \mu(x, \vartheta, \Sigma, V) \, dx.
-\end{aligned}\]</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The running times for a single evaluation can take some time due to the involved integrations, but it usually needs to be computed only once.</p></div></div><hr/><p><strong>Arguments</strong></p><ul><li><code>ϑ::Real</code>:                    positive drift coefficient <span>$\vartheta$</span>.</li><li><code>Σ::Real</code>:                    positive diffusion coefficient <span>$\Sigma$</span>.</li></ul><hr/><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; Σ_∞_QrP(1.2, 0.6)</code></pre><hr/><p>See also <a href="../invariant_densities/#MDEforM.μ"><code>μ</code></a>, <a href="../MDE_functionals/#MDEforM.k"><code>k</code></a>, <a href="#MDEforM.Σ_∞_QdP"><code>Σ_∞_QdP</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/UQatKIT/MDEforM/blob/ed8649d5acb1fcdb515c99e523b590338044e30c/src/MDE_asymptotic_variances.jl#L165-L207">source</a></section></article><h2 id="Index"><a class="docs-heading-anchor" href="#Index">Index</a><a id="Index-1"></a><a class="docs-heading-anchor-permalink" href="#Index" title="Permalink"></a></h2><ul><li><a href="#MDEforM.Σ_∞_QdP"><code>MDEforM.Σ_∞_QdP</code></a></li><li><a href="#MDEforM.Σ_∞_QrP"><code>MDEforM.Σ_∞_QrP</code></a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../MDE_optimizers/">« Optimization Task of the MDE</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Friday 30 August 2024 16:00">Friday 30 August 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The running times for a single evaluation can take some time due to the involved integrations, but it usually needs to be computed only once.</p></div></div><hr/><p><strong>Arguments</strong></p><ul><li><code>ϑ::Real</code>:                    positive drift coefficient <span>$\vartheta$</span>.</li><li><code>Σ::Real</code>:                    positive diffusion coefficient <span>$\Sigma$</span>.</li></ul><hr/><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; Σ_∞_QrP(1.2, 0.6)</code></pre><hr/><p>See also <a href="../invariant_densities/#MDEforM.μ"><code>μ</code></a>, <a href="../MDE_functionals/#MDEforM.k"><code>k</code></a>, <a href="#MDEforM.Σ_∞_QdP"><code>Σ_∞_QdP</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/UQatKIT/MDEforM/blob/6d99b916c746a5cad07402c962f5e5e98dd4c007/src/MDE_asymptotic_variances.jl#L162-L204">source</a></section></article><h2 id="Index"><a class="docs-heading-anchor" href="#Index">Index</a><a id="Index-1"></a><a class="docs-heading-anchor-permalink" href="#Index" title="Permalink"></a></h2><ul><li><a href="#MDEforM.Σ_∞_QdP"><code>MDEforM.Σ_∞_QdP</code></a></li><li><a href="#MDEforM.Σ_∞_QrP"><code>MDEforM.Σ_∞_QrP</code></a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../MDE_optimizers/">« Optimization Task of the MDE</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Friday 30 August 2024 16:15">Friday 30 August 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>