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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22 views

$(-1)/(2i/\{-2\lambda\})=-i\lambda$ - 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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1answer
104 views

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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1answer
39 views

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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16 views

Geometrical interpretation of back projection operator or adjoint of Radon transform.

If $f \in C_{c}^{\infty}\left(\mathbb{R}^{2}\right)$, the Radon transform of $f$ is the function $$R f(s, \omega):=\int_{-\infty}^{\infty} f\left(s \omega+t \omega^{\perp}\right) d t, \quad s \in \...
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32 views

Can compressive sensing reconstruction be linear?

Compressed sensing is a paradigm that allows us to sample a signal below the Nyquist rate and still be able to reconstruct it. This is done by taking a few random measurements from the signal. The ...
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17 views

How to understand the ill-posedness of Volterra equation of first kind and well-posedness of second kind?

Consider a Volterra equation of first kind as following: $$ f(t) = \int_0^t K(t,s) x(s) ds $$ We can change it to second kind by taking derivative of both sides: $$ \tilde f(t) = x(t) + \int_0^t \...
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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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15 views

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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1answer
39 views

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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31 views

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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22 views

Please help me understad this matrix operation. (Non-square matrix inversion, rank deficient matrix)

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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17 views

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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19 views

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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12 views

More than one model and operator in inverse problem

I am new to linear algebra and inverse problems. So, I want some help regarding the following problem. I am reading a paper and the inverse problem is stated as below problem: $d=L_0m_0+L_1m_1+L_2m_2=(...
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1answer
39 views

Intuitive implication of the fact that dual of $L_{p}$-norm space is $L_{q}$-norm space where $\frac{1}{p}+\frac{1}{q}=1$.

While studying the inverse problem theory (I am mainly concerning discrete variables), I learned the theorem that "the dual of $L_{p}$ where $1<p<\infty$ is $L_{q}$ provided that $\frac{1}{...
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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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1answer
36 views

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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1answer
47 views

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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1answer
26 views

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 ...
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2answers
54 views

Quadratic cost function solution [closed]

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: $$(L^Td+\mu W^T_m W_m)^{-1} L^Td $$
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Variance estimate of solution of inverse problem by nonlinear least squares with multi-dimensional model function?

The inverse problem I solve has the following basic outline: a number $n_g$ of Gaussian kernels ($\textbf{x}$) of some width $\sigma$ and each with an amplitude $s_j$ is propagated in n-d space to $\...
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1answer
30 views

Identifying specific pattern in a noisy dataset

I am working on a project where I have the following forward model. There are multiple point sources sparsely distributed in 3D space with coordinate $[x_0,y_0,z_0]$ for each of them. The detected ...
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51 views

Gaussian deconvolution for rapidly decreasing functions.

Gaussian convolution with variance $v$ is defined as $$ {\cal G}_v[f](x):=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi v}}f(y) e^{-\frac{(y-x)^2}{2v}}dx. $$ Given a function $g$, does there exist a a ...
3
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1answer
37 views

Error propagation in a simple linear model (asked by a non math-major researcher).

I have the following linear problem in a matrix form $Ax=Y$, where $A$ is a coefficient matrix, $x$ is a vector of unknown parameters and $Y$ is a vector of observed data. Matrix $A$ is usually over-...
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1answer
32 views

How can I find some $r\in\mathbb{R}$ such that the following matrix has a full rank?

I have to task to determine whether the following matrix $A_r\in\mathbb{R}^{39\times 39}$ defined as $$ A_r := \begin{pmatrix} 2+4!+r^2 & 2r & -1 & 0 & \ldots & \ldots & 0 \\ ...
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12 views

What is the main difference between interior and exterior inverse problem?

I wonder whether there is any difference between interior inverse problem and exterior inverse problem with respect to stability, numerical solution or anything.. Is it harder to solve interior ...
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74 views

Are there any particular reasons why the minimum norm solution should be preferred?

Let us consider a linear system of equations of the form: $$ Ax = b, $$ where $A \in \mathbb{R}^{m\times n}$ and $b \in \mathbb{R}^m$. Such a linear system naturally arises in many fields of ...
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18 views

if we represents $\varphi$ as trgonometric polynomial, do we make the solution smoother?

There is part in the article about inverse problem i could not understand. The integral operator S defined from $H^{\frac{1}{2}}(0,2\pi)\to L^2(0,2\pi)$ and $(S\varphi)(x)=f(x), x\in [0,2\pi]$. After ...
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0answers
34 views

Computing a least-squares least-norm solution to image deconvolution

I want to deconvolve an image $h$ by a kernel $f$. More precisely, let $$G = \operatorname*{argmin}_g \|f \ast g - h\|_2$$ be the set of least-squares solutions. I want to find the least-norm solution ...
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1answer
34 views

Wellposedness of differentiation.

A common example of an illposed problem is the following differentiation example: Let $y(t)\in C^1([0,1])$ with $y(0)=0$. We consider the problem of determining $x(t) = y'(t)$ and show why it is not ...
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0answers
12 views

Distribution of population size $n$ given binomial sample quantity $x$ and selection probability $\pi$

Given the number of draws $x$ from a binomial distribution with known probability parameter $\pi$, is there a function which gives distribution of likely $n$ from which these $x$ were sampled? For ...
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0answers
54 views

Imaginary zeros of quasipolynomial

In the context of DDEs of the type, $$\dot{x}(t)=f(x(t),x(t-\tau_1), ..., x(t-\tau_N)),$$ I am interested in the following question: For a fixed N, but arbitrary $\tau_n>0$ and $a_n\in\mathbb{R}$, ...
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68 views

How to extend the Radon transform to $L^2(\mathbb{R}^2)$?

The (2D) Radon transform $R$ is usually defined for functions in the Schwartz space $S(\mathbb{R}^2)$ or bump functions $C_c^\infty(\mathbb{R}^2)$ by \begin{align*} R\colon C_c^\infty(\mathbb{R}^2)&...
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1answer
58 views

How to find $q,\beta$ such that $\nabla\cdot[\gamma\nabla u]=0\Leftrightarrow(-\Delta +q)v=0$ for some $v=\beta u $?

Let $\Omega$ be an open subset of $\mathbb R^n$ . Let $\gamma\in C^1(\Omega)$ be bounded away from zero. Find $q,\beta\in C^1(\Omega)$ such that \begin{equation*} \nabla\cdot[\gamma\nabla u]=0\...
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18 views

Let $W$ be random matrix and $b$ be vector. Let $x$ be secret vector; let $(y(x))_j = \max(W_j^Tx+b,0)$. Find $x' \in B(x;r)$ s.t $y(x')=y(x)$ whp

Let $1 \le n \le p$ be large positive integers. $W$ be a random $n \times p$ matrix with entries drawn iid from $\mathcal N(0,1/p)$ and let $b$ be an $n$-dimensional random vector with coordinates ...
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16 views

Set of functions given integral conditions

Imagine I have the following set of equations which need to be satisfied for a set of complex functions $f_{1},f_{2}$: \begin{eqnarray} \int_{-\infty}^{\infty} dx e^{i(p-p' + (n-m-q)T)x}f_{1n(2n)}^{*}...
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0answers
18 views

Inversion of a modified Abel transform with higher order on denominator

I was doing a research about retrive element densities from emission lines intensity observed by spacecraft. Follow the symbols of https://en.wikipedia.org/wiki/Abel_transform, suppose that intensity ...
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1answer
69 views

Finding the complementary function from a boundary value problem

I have the following physics equation: $$a = \int_0^R K_1\frac{\delta \rho}{\rho} + K_2 \frac{\delta c^2}{c^2} \,\text{d}r$$ where $a$ is a real number, $R$ is a positive real number, and $K_1,K_2,\...
2
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1answer
31 views

Total variation regularization superior to classical quadratic choice

Where can I find literature on why the choice of regularization $$\int_{\Omega} |\nabla u| \,{\rm d} x\,{\rm d}y$$ is more effective than the classical choice $$\int_{\Omega} |\nabla u|^2 \,{\rm d} x\...
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0answers
35 views

Removing the complementary function from the solution of an ODE after transformation

I am trying to complete a computer program written by someone who is unfortunately now deceased. Sparing the details, the program computes a kernel $K$ intended to be used in the following integral ...
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0answers
24 views

Tikinov regularization and GCV-score

I am working on a problem where I have to model a system and its temporal evolution. The data provided is sampled with a temporal resolution of 1 minute. The physical description of the system does ...
2
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1answer
28 views

Statistical Inversion Problem $F = Ku + \mathcal{E}$ derive conditional probability density $p(f | u)$

Consider the following Inversion Problem $f = Ku + \varepsilon$ where $f \in \mathbb{R}^{m}$, $u \in \mathbb{R}^{n}$, $K \in \mathbb{R}^{m,n}$ and $\varepsilon$ is an additive, Gaussian noise. In the ...
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1answer
42 views

Please recommend some books/links for numerical solution of inverse source problem

Equation: $$ \frac{\partial f}{\partial t} = \frac{\partial^2 f}{\partial x^2} + k(x,t)\frac{\partial f}{\partial x} + \psi(t)\theta(x) $$ $$ x \in [0, 1], t \in [0, T] $$ boundary conditions: $$ f(0,...
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0answers
46 views

Approach to finding a function for a binary pattern

Below is a binary matrix of shape [64,63]. I want to find an equation to compute the values of this matrix as a function of rows and column indices. Are there any common strategies to attack such ...
4
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0answers
62 views

$L^2$ norm of inverse differential operator

This has come up in Lemma 1 of Mandache's 2001 paper on exponential instability for the inverse problem of the Schrodinger operator. Let $\Omega = B(0,1)$ in $\mathbb{R}^d$. Suppose $r_0\in (0,1)$ and ...
3
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1answer
120 views

Determine the structure of all finite sets $A$ of integers such that $|A| = k$ and $|2A| = 2k + 1$.

An exercise in Nathanson's text: Additive Number Theory, Inverse problems and the geometry of sumsets is the following (Excercise 16, P.No.37): Determine the structure of all finite sets $A$ of ...
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1answer
9 views

Condition inequality in perturbed LS

I have two matrices $A \in \mathcal{M}_{n,d}(\mathbb{R})$ and $B \in \mathcal{M}_{d,d}(\mathbb{R})$ with $B$ being symmetric definite-positive. I am trying to find a condition on $A$ for which I have ...
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0answers
28 views

Computing $L^2$ norm on a curved domain

I am currently working in a field related to elastography which tries to solve an inverse problem (of elasticity) similar to the one described below. Given a measured displacement field $u_{measured}$...
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0answers
35 views

References on parameter identification for ODE models from time-series data

Setting: Say I've got a system of ODEs, e.g. a generalized Lotka-Volterra system of the form: $$\dot x_i = x_i \left( r_i + \sum_{j=1}^n b_{ij} x_j \right), \quad 1 \leq i,j \leq n,$$ and I count ...

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