Articles of inverse problems

Reconstructing a functional from its Euler-Lagrange equations

Is it true that Euler-Lagrange equations associated to a functional determine the functional? Suppose I give you an equation and I claim that it is an Euler-Lagrange equation of some functional. Can you tell me what was the functional? Of course, there is always more than one functional whith prescribed E-L equations, since the critical […]

Find $J: S^2\rightarrow \mathbb{R}$, if given $I:S^2\rightarrow \mathbb{R}$, s.t. $I(\vec{a})=\int_{S^2}\vec{n}\cdot\vec{a} J(\vec{n}) ds$.

I thought of the following problem, when we were discussing radiation intensity in an astrophysics lecture. Suppose $\mathbb{R}^3$ is filled with uniform radiation, i.e. there is a function $J:S^2\rightarrow \mathbb{R}$, so that at any point in $\mathbb{R}$ the amount of radiation in direction $\vec{n}$ is $J(\vec{n})$. If we put a unit area with normal vector […]

References on Inverse Problems, Approximation theory and Algebraic geometry

For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined system. Gröbner bases provides a way to express a set of structure functions — however how can you know that the set of structure […]

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of three variables to functions of four variables. Then, “because the range function depends on 4 […]

When is $R \, A^{-1} \, R^t$ invertible?

In the context of a Gaussian model, I came across a matrix product $R \, A^{-1} \, R^t$ where $R$ is a $m \times n$ rectangular matrix and as implied $A$ is $n \times n$ and invertible. On which properties of $R$ does the existence of $(R \, A^{-1} \, R^t)^{-1}$ depend?

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve an unconstrained minimization of the least-squares penalty with $\alpha\|\beta\|^2$ added, where $\alpha$ is a constant (this is the Lagrangian form of the constrained problem). The above […]

Low-rank Approximation with SVD on a Kernel Matrix

I have very little experience in linear algebra so please bear with me. Here’s a little background of my issue. I’m working on a problem that utilizes a large kernel matrix, K. This matrix, when multiplied with 500 x 1 column vector A, results in a 500 x 1 column vector B as shown below: […]