# 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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### $(-1)/(2i/\{-2\lambda\})=-i\lambda$ - this can't be true?

Maybe I am confused but in the screenshot below, there is a line saying that $$c_1=(-1)/(2i/\{-2\lambda\})=-i\lambda,\qquad \hat{c_1}=(-1)/(2i/2\lambda)=i\lambda.$$ Isn't that false? Shouldn't it be ...
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### Convergence of approximation of pseudo-inverse

Let $X, Y$ be Hilbert spaces and $A\in L(X,Y)$ be a bounded linear map. Let $\{ R_t\}_t$ a family of functions $Y \rightarrow X$ and $\gamma: \mathbb{R}_+\times Y\rightarrow \mathbb{R}_+$ with the ...
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### Why minimize squared L2 norm and not only the L2 norm? [closed]

I'm studying Inverse Problems and usually, they minimize the squared of the L2 norm($L_2, L_0, L_ \infty$), why don't minimize only the norm? if the goal is to have a measure of the distance between 2 ...
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### If all order of normal derivatives on some nbhd of point $x_0$ is same then all derivatives on the boundary at $x_0$ is also same.

I was reading https://doi.org/10.1002/cpa.3160370302. In that author wanted to prove $$D^k\gamma_1(x_0)=D^k\gamma(x_0)$$ for all $k=(k_1,\cdots,k_n)$. But he says that it is enough to show for each $k$...
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### How Bayesian solution of inverse problem could be validated?

Recently, I read a paper about the inverse problem and parameter estimation. The main approach of the paper is based on the Bayesian method. The answer in this method is a posterior probability ...
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### How to correct a set of data based on new mean, covariance and weights

I have a set of discrete data points {xi}, sampled from a specific gaussian mixture. Then I update the mixture parameters $mean^{old}, covariance^{old}, weights^{old}$ using Optimal transport ...
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### How to analyse a vector transformed by a special symmetric matrix?

I use a sampling matrix $\mathbf{\Phi }\in \mathbb{R}^{M\times N} (M\ll N)$ to obtain a measurement vector $\mathbf{y }\in \mathbb{R}^{M}$ from the original signal $\mathbf{x }\in \mathbb{R}^{N}$ by ...
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I encountered very strange matrix operation while reading a scientific paper on modelling of unsteady aerodynamics (https://doi.org/10.1016/j.enganabound.2011.12.007). The wake is discretized and the ...
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### how to solve an inverse problem with more than one operator and model?

If for an inverse problem, the data be constructed from more than one model and operator, like written below: $d=L_0m_0 + L_1m_1$ in which, $d$ be the data, $L$ the operator and $m$ model. How can I ...
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### What is the relation of fractional calculus to fractional function decomposition?

Background : Fractional function decomposition can be defined like $$x\to f^{\circ (1/k)}(x) \text{ s.t. } x\to (f^{\circ (1/k)})^{\circ k}(x) = f(x)$$ This is in general a hard inverse problem unless ...
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### Can we say $\frac{i}{4}H_0^{(1)}(k|x−x_0|),\frac{i}{4}H_0^{(1)}(k|x−x_1|),\frac{i}{4}H_0^{(1)}(k|x−x_2|)$ are linear independent?

$\frac{i}{4}H_0^{(1)}(k|x-x_0|$ is fundamental solution of Helmholtz equation for $x\neq x_0$ where $H_0^{(1)}(k|x-x_0|$ is Hankel function. Assume that we have three different points $x_0,x_1,x_2$ ...
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### Finding the adjoint operation of a black-box differential operator

I have a black-box code (automatic differentiation) that computes $Du$, where $D \in \mathbb{R}^{n \times n}$ and $u \in \mathbb{R}^n$. Note that I do not have the matrix $D$ explicitly and neither do ...
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### Additive Inverse and integer modulo

I am not completely sure how inverses work with sets of integer modulo. I have just started to learn about them. I have tried some practice problems, though I am not sure if my approach is correct in ...
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### Inverse Rubik's Cube

If I am given five faces of a rubik's cube, is it possible to a) Determine if these are five sides of an actually solvable cube b) Extend this to the sixth face in a unique way Assuming one eliminated ...
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### Discrete regularisation

Consider the following least squares problem in $X$: $||AX-B||_2^2\rightarrow\min$, where $A$ and $B$ are known, real-valued matrices. Is it there a regularisation method which ensures that the ...
Why the solution to the following cost function: $$\frac{1}{2}\|Lm-d\|^2 + \frac{1}{2} \mu \|W_m m\|^2_2$$ the below equation: $$(L^Td+\mu W^T_m W_m)^{-1} L^Td$$