6

Two systematic ways of smoothing a function $h$ would be: 1. Join the piecewise smooth parts of your function using Hermite interpolation so that the derivatives are matched to your satisfaction. 2. Convolve your function $h(x)$ with a heat kernel of the form $f(x) = \frac{\exp\left\{-\frac{x^2}{2 \sigma^2}\right\}}{\sqrt{2 \pi \sigma^2}}$ so that instead ...


6

You won't like the answer, but it is in fact "This lies in the right functional-analytic framework" — which you have not given! In particular, you did not specify in which function space you are looking for $u$ in. If $u\in L^2$ as the formulation suggests, then asking for the value of $u$ at a single point makes no sense a priori. (Even if the ODE is based ...


5

This problem is actually more of an optimal control problem for a partial differential equation. As a starting point, I would recommend the following books: F. Tröltzsch, Optimal control of partial differential equations, AMS, 2010 M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE constraints, Springer The first one in particular is an ...


3

If you want a ten-line solution that is decently fast and stable, you can implement yourself the structured doubling algorithm: set up the coupled iteration $A_0 = A, G_0 = G = BR^{-1}B^T, H_0 = Q$ While $\frac{\|H_{k+1}-H_k\|}{\|H_{k+1}\|} \geq \varepsilon$: $\quad \quad A_{k+1} = A_k(I+G_kH_k)^{-1}A_k$ $\quad \quad G_{k+1} = G_k + A_k(I+G_kH_k)^{-1}G_kA_k^...


3

The m.MV() type has additional tuning parameters such as move suppression that is likely contributing to the difference in solution. Also, the m.MV() is adjustable at every time point in m.time instead of just a single value with an m.FV() over the entire time window. You can get similar results to an FV by making the following adjustments to the MV. Set ...


3

The problem you describe is an example of a nonlinear least squares problem. You can find documentation for the MATLAB function to solve such a problem along with several references describing the mathematics and numerical methods for solution here: lsqnonlin The idea behind least squares problems is that you have more data points (access point locations ...


3

In addition to Brian's reference, there was also a review article in SIAM Review a few years ago that summarized the state of research at the time: http://epubs.siam.org/doi/abs/10.1137/S0036144598342032


2

In the direct multiple shooting method, using the notation from the linked notes by Chachuat, the "extra decision variables" $\mathbf{\xi}_{0}^{k}$ are not present in the objective function, therefore the sensitivities of the objective function (i.e., the derivatives of the objective function) with respect to these parameters are all zero. You are correct ...


2

Unlike barrier methods, the affine scaling method doesn't use a barrier to push the iterates away from the boundaries of the feasible region. As a result, the iterates can very quickly approach the boundary of the feasible reason. Furthermore (and this can be a problem) the method can easily get "stuck" taking very short steps along the boundary of the ...


2

I am not sure about your application -- and we say the $L^2$ norm of a function and not a system. But for simplicity I will explain the concepts for real valued functions. Consider an open domain $\Omega$ and a function $f:\Omega \to \mathbb{R}$. We say that $f \in L^2(\Omega)$ if $||f||_{L^2(\Omega)} < \infty$ where \begin{equation} ||f||^2_{L^2(\Omega)} ...


2

It is true that $C$ depends on $x$ and $h$. This implies that if your function has a very poorly behaved second derivative at $x$, this method will be inaccurate. The dependance on $h$ is also to be expected, since finite differences are based on Taylor series, which converge locally, so our error estimate essentially depends on how well a Taylor series ...


2

You want to solve $$ \arg \min_{D(x),k(x)} \mathcal{G}(u;D,k) := \frac{1}{2} \int_{\Omega} \left(u(t=T) - {\bar{u}}(T) \right)^2 d\Omega + \frac{1}{2}\int_{\Omega} D^2 d\Omega + \frac{1}{2}\int_{\Omega} k^2 d\Omega $$ subject to \begin{align} \frac{\partial u}{\partial t} - \nabla \cdot (D \nabla{u}) - k u (1-u) &= 0 & \space \mathrm{on} \space \...


1

I haven't solved the HJB equations myself, but I can think of two things you can try based on my experience with other PDEs. Use an "upwinded" approximation for $J_x$, instead of a central difference scheme. The $x J_x$ term in the equation suggests that information is flowing from left to right in your system as you move forward in time (assuming $x > ...


1

Since you are solving inviscid equations, you will need some form of stabilization like SUPG or a DG scheme, to get a robust scheme. You wont need the pressure I presume. I would recommend solving the equations in vorticity-velocity form where the pressure is eliminated. See this paper for a DG scheme Jian-Guo Liu and Chi-Wang Shu, A High-Order ...


1

Have you thought of barycentric coordinates? There is a unique way to write $x=\alpha A + \beta B$ with $\alpha+\beta=1$. Barycentric coordinates are usually employer in larger dimensions, but seem to be the concept you are looking for. You can sort your observations according to either $\alpha$ or $\beta$, this will give you their "closeness" to either. If ...


1

There has been a lot of research in this area over the last 20 years. It's appropriate to start by using search engines such as Google Scholar and Web of Science to look for survey and review articles. For example, you might look at http://www.sciencedirect.com/science/article/pii/S2211692313000428


Only top voted, non community-wiki answers of a minimum length are eligible