# Questions tagged [inverse-problems]

Inverse problems involve for example reconstruction of an object based on physical measurements and finding a best model/parameters out of a family given observed data. Typically the corresponding "forward" problems are well-posed and can be solved straightforwardly, while the inverse problems are often ill-posed. Not to be confused with the (inverse) tag.

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### Given a vector field $f:\mathbb R^3\to\mathbb R^3$, is there a mass distribution that generates $f$ as its Newtonian gravity?

Let $f:\mathbb R^3\to\mathbb R^3$ be a smooth bounded vector field. I want to produce a density $\rho:\mathbb R^3\to\mathbb R$ such that the Newtonian acceleration experienced by a particle in the ...
1 vote
57 views

### Show that operator is equal to $-(\nabla -\tau\nabla f)\cdot(\nabla -\tau\nabla f)$

I have the following problem, I dont understand why is this equality. Let $f(x)$ be a function such that $f=\exp(g)$ with $g\geq0$ on a domain of interest. We are looking at the operator \begin{...
22 views

### Uniqueness of non-negative least squares solution

When solving linear problems, there are many different ways of assessing if $Ax=b$ has a unique solution for a given $A$ by looking at properties of $A$ like the rank or singular values. If I am ...
1 vote
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### Morozov Discrepancy for PDE-Constrained Optimization With Bound Constraints

$\textbf{Background and context}$ Let $\beta:\Omega\rightarrow\mathbb{R}$, be an unknown spatially distributed parameter (over the spatial domain $\Omega$). Let $\mathcal{F}$ represent a parameter-to-...
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### Literature on inverse problem where one approximates the forward operator by a surrogate model

Is there any literature about the case of inverse problems where the forward operator is being approximated by a surrogate model, especially the case where the surrogate model is a neural network, and ...
21 views

### How can I use the finite difference method to solve a simple unconstrained optimal control problem?

I want to solve this problem using finite difference method. \begin{equation} \begin{cases} &\min_{y, u} \quad \mathcal{J} = \frac{1}{2}\left\lVert y-z \right\rVert^{2} + \frac{\alpha}...
76 views

### How does one invert $e^{(k+i)\theta}+e^{(k-i)\theta}=e^{k\theta}+c$ to obtain $\theta$, given known $k,c$?
I need to find a real solution for $\theta$ in the following complex equation: $e^{(k+i)\theta} + e^{(k-i)\theta} = e^{k\theta} + c$ Here $k$ and $c$ are positive real numbers. The problem is related ...