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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Finding constants of non-linear inversible function through measurements.

We have a known non-linear inversible function $f \in R \to R$ with a known inverse. We know that $g$ has the following form: $$ g(x) = f(x*a + b)*c + d $$ We also have known values for $g(x_0), g(x_1)...
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how to drive the right Moore–Penrose inverse

I am trying to prove that HTW=I where HT is NxM matrix and W is MxN matrix and I is NxN identity matrix the conditions are : Full Column Rank (or simply Column Rank): Rank of a matrix of order MxN is ...
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Is pseudo inverse of a single vector element equal to the direct inverse of that element?

This question involves the second part of the previously asked question which can be found here. The question is as follows and my attempted solution is below the picture. My attempted solution: I ...
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how to show inverse of M-by-N data matrix using singular value decomposition?

The question is from the exercise questions from the EE4595 course being taught in TUDelft. The question is as follows... So far my attempted solution is like the following. The discretised data ...
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Derive adjoint advection-diffusion equation

Hello I am studying how to derive backward advection-diffusion equation from advection-diffusion equation by adjoint operator $\lambda$. I know that solving adjoint problem can lead the backward ...
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Linear Algebra: Solving the minimization with vector p-norm.

The course this question is from is called Wavefield imaging, It gives the vector p-norm of an N-by-1 vector $x=[x_1,x_2,...,x_N]^T$ is defined as $$ ||x||_p=\left(\sum_{n=1}^N |x_n|^p \right)^{1/p} $$...
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Analytic solution inverse kinematics - different solutions with different calculation steps

I have the following problem: $$5\cos \theta_1+3\sqrt{3}\sin \theta_1=4\qquad\qquad\textbf{(I)}$$ $$5\sin \theta_1-3\sqrt{3}\cos \theta_1=6\qquad\qquad\textbf{(II)}$$ I have two seemingly correct ...
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Least-norm problem with equality constraints

My inverse problems teacher said that the least norm problem can be resolved using equality constraints. How is that?
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If $\int_n^m{f(x,y)}dy=g(x)$, is there a way to find, or approximate $f(x,y)$ given $g(x)$

If I'm given $f(x,y)$, when $$\int_n^m{f(x,y)}dy=g(x)$$ then I know how that I can at least approximate $g(x)$ using a rieman sum, however if I am instead given $g(x)$ I don't know how to even ...
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Newton's Method for Nonlinear System with Constraints

I have a local solution of a dynamical system $\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x})$: \begin{equation} \mathbf{x}(t) = \mathbf{g}(t;\mathbf{A}), \end{equation} where $\bf{f},\bf{g}:\mathbb{R}^n\...
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One-dimensional heat conduction

I was reading a paper about one-dimensional heat conduction problem and I get stuck in one expression that I couldn't understand how to calculate. First, they define the heat conduction problem as ...
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Find optimal set of rows for matrix inversion

I am looking for regularized solutions to an over-determined system $$Ax=y$$ in the least-square sense, i.e. find $\hat{x}$, such that $$\gamma ||\hat{x}||^2 + ||A\hat{x} - y||^2 $$ is minimal, where $...
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Isomorphic Neural Nets

The following is from Reconstructing a neural net from its output by Fefferman. In here, I'm not sure about the notation. Are we fixing one $l$ such that $\gamma_l$ is identity, and the rest of the ...
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Best way to solve the inverse problem of an integral equation with uncertainties?

I wish to estimate the unknown functions $f$ and $g$ and their uncertainties according to $$ y_i = \int_0^1 F_i(x)\,f(x) \,\text{d}x \; + \int_0^1 G_i(x)\,g(x) \,\text{d}x, \quad i=1\ldots N $$ where ...
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What is the difference between regularization and preconditioning for a matrix?

The literature says "Preconditioning is a technique for improving the condition number of a matrix. Suppose that $M$ is a symmetric, positive-definite matrix that approximates $A$, but is easier ...
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Find a $2\times2$ matrix $B$ such that $(A^TBA)^{-1}A^T=I$, where I is the identity matrix

Problem Let $A=\begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}$. Find a $2\times2$ matrix $B$ such that $(A^TBA)^{-1}A^T=I$, where $I$ is the identity matrix. How does one go about solving this ...
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Inversion problem for coefficients in an infinite series

Is there a way to calculate the coefficients $a_{n}^{i,j}(x)$ (which correspond to functions defined in $\mathbb{R}$) in the following equation? \begin{eqnarray} \sum_{n\in\mathbb{Z}}a^{i,j}_{n}(x)\...
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Lower bound of convergence rate of Fourier transform

Usually, we know if the derivative of a function $f^{(m)} \in L^2(R^d)$, the Fourier transform $\hat{f}(\xi)$ of $f$ converges to zero with rate no slower than $O(|\xi|^{-m})$, i.e. $\liminf _{|\xi|\...
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Singular Value Decomposition vs iterative Methods for solving inverse Problems

Background: I am trying to motivate why I use an SVD instead of an iterative LMS solver for the solution of an equation of the form $$Ax=b$$ where $A\in\mathbb{C}^{M\times M}$,$x\in\mathbb{C}^{M}$ and ...
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3 votes
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Prove that the phase retrieval problem is non-convex

The phase retrieval problem consists of recovering phase information from given intensity measurements, as shown in the image below from Deep phase retrieval: analyzing over-parameterization in phase ...
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Why the more unkown variable(more underdetermined) will make non-convex optimization problem more difficult to solve?

Now I am try to solve an inverse problem. In the end it is to solve an non-convex optimization problem(we call it phase retrieval problem). The loss is $\lVert |f(x)|^2-I \lVert_2^2+\lVert x\lVert_1+...
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Is there a Lagrangian that produces these equations? How can I find one if it exists?

Consider the two differential equations \begin{align*} \ddot{x}_{A} - \gamma(x_{A} + x_{B}) &= 0, \\ \ddot{x}_{B} + \gamma(x_{A} + x_{B}) &= 0 \end{align*} where $\gamma$ is a constant. I am ...
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Solving a spatial pde with partial derivatives?

Assume I have the following function: $$0=a(\mathbf{x})h(\mathbf{x})+b(\mathbf{x})\nabla h(\mathbf{x})+c(\mathbf{x})$$ where $\mathbf{x}\in\mathbb{R}^{d}$ are points in some $d$-dimensional space, $a$,...
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Calculating SVD for some integral operators

I am currently trying to analytically find SVD for operators $T_1, T_2: L^2\big([0;1]\big) \to L^2\big([0;1]\big)$ using definition, where: $(T_1f)(x) = \int_0^x f(y) \ dy$ $(T_2f)(x) = \int_0^1 K(x,...
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Is the inverse of a symmetric block matrix ever a block matrix of the inverse blocks?

Background: Suppose that you have a $2n \times 2n$ block matrix $ \mathbf{M} = \begin{pmatrix} \mathbf{A} & \mathbf{B}\\ \mathbf{C} & \mathbf{D} \end{pmatrix} $ Where $\mathbf{A},\mathbf{B},\...
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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 ...
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Find an ODE with given functions as solutions

I am teaching a class on differential equations lately, and in the course of trying to invent problems for a midterm, I got to wondering how to generate an ordinary differential equation having ...
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Reconstructing function from integral transform

Physicist here, so forgive me if I'm being a bit sloppy. I was considering the integrals $$ \tau(s) = \int_{0}^{L}\frac{{\rm d}x}{\sqrt{1-f(x)/s}} $$ for all $s>\max\{f\}$, and I came to wonder ...
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A bilevel optimization problem

Consider the bilevel optimization problem: \begin{equation}\label{2} \min_{\lambda\geq0}\|x_{\lambda}-x_{ex}\|^{2} \ \ \ \mathrm{s.t.} \ \ \ x_{\lambda} = \arg \min_{x \in \mathbb{R}^{n}}\{\lVert ...
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What academic explanation could clarify a mathematical term?

I recently attended a class that is about inverse problems. The goal of course was to solve inverse problems based on Bayesian theory. In a section, the relationship Kolmogorov was described. Also, it ...
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Inversion Formula Gaussian Convolution

I am looking at the following 2004 paper by S. Saitoh, called "Approximate real inversion formulas of the Gaussian convolution": https://www.researchgate.net/publication/...
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2 answers
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Tikhonov regularization using Newton's method or gradient descent

In Tikhonov regularization an explicit solution denoted by $\hat {x}$, is given by $$ \hat{x} = (A^TA + \Gamma^T\Gamma)^{-1}A^Tb$$ How can we solve the same problem using Newton's method or gradient ...
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what makes a function invertible? [closed]

From what I understand invertible is if we can equate x = f(x) of some form.. Whereas Inverse is where the function can be reflected across the y=x axis. Are those related in any way? Is it true that ...
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Special function needed.

For the implementation of a dynamic Radontransform I'm looking for a set of functions $f_\alpha:\mathbb{R}\to\mathbb{R}$ with very special condtions, where $\alpha$ is in an arbitrary interval of $\...
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How to solve for the inverse of $f$ for a floor function? I really need to learn the method.

I was doing some exercises on discrete maths and came across this question I don't know how to solve, nor can I find any relevant examples in my book. Suppose $f: \mathbb R \to \mathbb R$ where $f(x) =...
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$(-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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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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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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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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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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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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3 votes
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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 ...
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2 votes
2 answers
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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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2 votes
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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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