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New answers tagged numerics

0

I think for time reversibility we need to see if we can predict velocity and position at time $t$ if the values at $t+\Delta t$ is known. Verlet Algorithm The integrator equation is given as: $r_{t+\Delta t} = 2r_t -r_{t-\Delta t} + \Delta t^2 a_t$ Given we know the positions at $t-\Delta t$ and $t$ we can predict the position at $t+\Delta t$. So if this is ...

0

Let's recap what you want to do: You have some set $V \subset \mathbb{R}^5$ and want to approximate the integral of some function $f\colon V \to \mathbb{R}$: $$\int_V f(x) \,\mathrm{d}x$$ The "mean value method" sounds like a Monte Carlo-type approximation of the form  \int_V f(x) \,\mathrm{d}x \approx \frac{\operatorname{vol}(V)}{N} \sum_{i=1}^...

5

Traditionally, the convergence of optimization algorithms has been analyzed in terms of the asymptotic rate of convergence. A quadratically convergent algorithm has $x_{k} \rightarrow x^{*}$ and $\lim_{k \rightarrow \infty} \frac{\| x_{k+1}-x^{*} \|}{\| x_{k}-x^{*} \|^{2}}=L< \infty$ while a linearly convergent algorithm has \$\lim_{k \rightarrow \infty} \...

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From your description, exponential integrators seem to be a good fit. They are based on variation of parameters and "bake in" an exact solution for the linear part of the ODE. There are many flavors of exponential methods that are determined by the discretization of the integral term in variation of parameters. Exponential Runge--Kutta methods ...

1

It sounds to me there are two things you want to model/simulate that live on different length scales. Molecular diffusion of any drug agent or signaling molecule can indeed be simulated discretley via molecular dynamics (MD) simulations. There, you simulate the positions, velocities and interactions of single molecules and may observer how they diffuse or ...

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The virtual strain energy should be $$\delta U = \int\limits_V {\delta {{\bf{\varepsilon }}^T}{\bf{C}\left(\varepsilon - {\bf{\bar \varepsilon }} \right)dV}}$$ where $${\bf{\bar \varepsilon }} = \alpha \Delta T\left\{ \begin{array}{l} 1\\ 1\\ 1\\ 0\\ 0\\ 0 \end{array} \right\}$$ ...

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