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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 the domain shape, where the fundamental Helmholtz Eigenmode has certain properties at a boundary

The following is a problem about finding the shape of a domain such that the fundamental Helmholtz eigenmode has certain properties at the boundary of the domain. I search for an analytical or ...
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Can anyone help with the inverse problem and tuning parameters

For my final year project i want to model the population of London using the Verhulst logistic model. However, to gain more marks i wish to use the inverse problem to tune the parameters to make the ...
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Finding conditions such that the following inequality holds.

Let $ \boldsymbol{J} \in \mathbb{R}^{m \times n} $, with $ n>m $ be an invertible full row-rank matrix. Further: \begin{equation}\label{key} f= |\boldsymbol{J}^{-1} \boldsymbol{1}_{\pm}|_1 \...
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Radon transform expressed via delta distribution

The Radon transform for $f\in C_c^\infty(\mathbb{R}^n)$ is defined by \begin{equation} Rf(\theta , s) := \int _{\{ x^T\theta =s \}} f(x)dx. \ \ \ (1) \end{equation} Now it says, that an equivalent ...
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36 views

Construct operator with given spectrum. [duplicate]

The Spectrum of a bounded operator on a Banach space $X$ is always a compact subset of $\mathbb{C}$. What about the converse? Given any compact subset $K \subset \mathbb{C}$ is it always possible to ...
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On a nonlinear regression problem

Consider the function $f\colon \mathbb{R}^2\to \mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data $\{(x_1^{(i)},...
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What is the compactness uniqueness argument

I have come across the term "compactness uniqueness argument" in many inverse problems papers and books but the proof are usually omitted. Could anyone give such an example and relevant references?
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1answer
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Is there a reason for why some problems reduce to Caldéron's inverse conductivity equation?

Is there a reason for why some problems reduce to Caldéron's inverse conductivity equation? Or is it "coincindence"? Such problems include, optical tomography, inverse scattering. E.g. in https://...
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How is the Jacobian matrix computed in finite difference problems?

I have come across many papers which reference the Jacobian when solving certain finite difference inverse problems. And I have seen many articles and textbooks which discuss the mathematical ...
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Find the solution of an outer product induced system

Sorry if the question is lame, but I'm struggling to find the answer to the following problem: Given a matrix $A\in \mathbb{R}^{n,n}$ and a column vector $b\in \mathbb{R}^{n}$, how can one find the ...
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About $ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $

We know for $ u > 1 $ $$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$ Usually this is shown by using differentiation under the ...
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1answer
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Inverse function on matrices with help of Cayley-Hamilton theorem?

I have been thinking about inverse functions of matrices lately. (Yees yees, I know usually for anything more complicated than reals we need to define/select branch and for reals to select interval ...
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42 views

Integral equation in polar coodinate system

I need an inversion formula with the form $f(r)=\cdots$, from this integral relation: $$g(r)=\frac{1}{2\pi}\int_0^{2\pi}d\theta\,f\left(\sqrt{r^2+r_0^2-2rr_0\cos\theta}\right)$$ where $r_0\geq0$ is a ...
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Harmonic functions interpolation

Denote by $B = \{(x,y) \in \mathbb{R}^2|x^2 + y^2 < 1\}$ the open unit ball in $\mathbb{R}^2$, and by $S$ it's boundary, i.e. the unit sphere. For some $n > 0$ let $x_1,...,x_n \in B$, $y_1,...,...
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finite group realised as Galois group over $\mathbb{Q}(X_{1}, X_{2},\ldots, X_{n})$ can be realised a Galois group over $\mathbb{Q}$

I'm trying to prove this following statement: Prove that every finite group that can be realised as a Galois group of a Galois extension over $\mathbb{Q}(X_{1}, \ldots , X_{n})$ can be realised as ...
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Uniqueness of charge distribution

Some exercises in general physics ask for students to find specific charge distributions on the boundaries of given conductors in various of situations. Usual answers goes like follows; firstly using ...
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Interpolation with Regularized Linear Squares

I am fundamentally missing something and would appreciate some clarification. I am reading this text on Regularized Least Squares. This text has a problem where I have to find z, a set of unknown ...
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1answer
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How to make sense of operators' inverse and the derivative of operators themselves?

I'm a physics graduate student, it's when my professor for mathematical physics mentioned KdV equations that I encountered the following problem. This is what our professor wrote: $\mathbb{L}:=-d^2+...
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The formula of inverse Meijier-G?

I have an expression like $G^{\,2,1}_{1,3} \left(\begin{array}{c} a_1\\ b_1,b_2,b_3 \end{array} \middle\vert\ z\right) $, $a_1,b_1,b_2$ and $b_3$ are fixed number. ...
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Inverse Galois problem

The inverse Galois problem conjectures that every finite group is (isomorphic to) the Galois group of some Galois extension of $\mathbb Q$, however it is not known. My question is: what is the ...
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1answer
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Non-linear waves and solitons: verifying solution for Nonlinear Schrödinger Equation

If the initial solution is the unstable background $(2),$ the corresponding fundamental solution of the Lax pair is \begin{align*}\Psi_0(\lambda)&=\frac{1}{\sqrt{2\mu(\mu+\lambda)}}\,e^{it\sigma_3}...
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1answer
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Establishing this inequality in $L^2$

Let $ u \in C^{\infty}_{c}(\Omega) $ where $ \Omega $ is an bounded open set in $\mathbb{R}^{n}$ with smooth boundary. If $ 0 \leq h \leq 1 $ is it possible for us to establish the bound $$4h^2\|h\...
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How can I incorporate this specific a-priori sparsity information in a regularization approach to guide my inverse problem?

Note: I will be using capital letters to denote matrices, lowercase letters to denote vectors, and the Greek alphabet to denote any scalar quantities. Note 2: I have tried my best to find a topic ...
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Solving diag$ \left( \textit{A} \textit{X} \right) = \textbf{b}$ for first column of $\textit{X}$ under further constraints

$\newcommand{\matr}[1]{\textit{#1}} $ Consider an $n \times m$ matrix $\matr{A}$, an $m \times n$ matrix $\matr{X}$, an $m \times 1$ vector $\textbf{x}$, and an $n \times 1$ vector $\textbf{b}$, ...
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How to calculate the mean of two posterior probabilities using Bayes' theorem?

I'm doing atmospheric retrievals using non-linear optimal estimation, and I would like to calculate the associated uncertainty on the mean of two atmospheric profiles each starting from a different ...
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31 views

Green's function for heat equation with source

Suppose I have a PDE: $$\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(k(x)\frac{\partial T}{\partial x}\right)+Q(t)$$ with the following initial/boundary conditions: $T(0,x)=0$ and ...
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Solving an inverse problem $- \Delta u = f$ where we know that $f$ is harmonic?

Let $\Omega$ be the unit disk in $\mathbb{R}^2$ and consider the inverse problem $$ - \Delta u = f, $$ where we want to find $f$ when we are given $\partial_\nu u = g$ on $\partial \Omega$ and we know ...
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1answer
65 views

Ill-possedness of inverse problems

I'm working in image restoration so this is a inverse problem that usually it is ill-posed but I don't understand why inverse problems are usually ill-posed. Direct problem: given $x$ find $y$: $Kx =...
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Auto-correlation function, an inverse problem

$x[n]$ is a complex function $n=0,1,2,\cdots,L-1 $ we assume $x[n]$ is periodic in its index: $x[n+L]=x[n]$ Its auto-correlation function $C[n]$ is uniquely defined as: $$ C[n]=\sum_{i=0}^{L-1} x[i+...
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Why Inverse Eigenvalue Problems should have a unique solution? [duplicate]

When solving an IEP, The solution to the problem should be unique. For any Matrix $A$, there are many matrices with the same eigenvalues as $A$. Then why an IEP should have a unique solution while we ...
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Inverse Eigenvalue Problem for symmetric matrcies with no unique solution

I am solving an IEP but the form of the problem and the constraints is such that it has not unique solution. It may have many different solutions. I would like to know is there any problem if such ...
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2answers
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Help solving a specific sequence problem

I have tried for some time to solve this problem and I'm stuck, so any help would be greatly appreciated. I'm not a math guy, so I apologize if I am missing something basic. I have a function $f(x)$ ...
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Fourier Filtering

I have read TSVD used as regularization for ill-posed problems. But I also read something like fourier filtering. I want to know something about fourier filtering. Here is the explanation of TSVD ...
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1answer
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Well-posed Inverse Problems Definition

$X,Y$ Banach spaces, $F: X \to Y$. Then the operator-equation "$Fx = y$" is well-posed, if for all $y \in Y$ (1) there exists $x \in X: Fx = y$ (2) the solution is unique (3) the solution ...
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Proof in inverse scattering theory (regularization schemes)

I'm currently reading a book about inverse scattering theory and in this book there is a section about ill-posed problems and there's a proof I'm not completely sure I understand. There might be need ...
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How to utilize the right-hand side in inverse problems

Consider the inverse problem $A \, x = b$ with right-hand side $b$, using SVD: $\qquad A = \sum s_i \, U_i \otimes V_i \ $ — singular values $s_i, \ U_i$ and $V_i$ orthonormal bases $\qquad ...
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Solving deconvolution using Conjugate gradient

Source '''Levin, Anat, et al. "Deconvolution using natural image priors." Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory 3 (2007). APA ''' Generally 2-...
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1answer
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Solving minimization problem $L_2$ IRLS (Iteration derivation)

In the article ''' Chartrand, Rick, and Wotao Yin. "Iteratively reweighted algorithms for compressive sensing." Acoustics, speech and signal processing, 2008. ICASSP 2008. IEEE international ...
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1answer
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Bijective compact operator between $X$ and $Y$ with $\text{dim}(X)=\infty$?

In the lecture of Inverse Problems we had the following statement: "Let $X$ be an infinite dimensional space ($\text{dim}(X)=\infty$) and suppose that for $A\in K(X,Y)$ the inverse operator $A^{-1}$ ...
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1answer
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Can you hear the pins fall from bowling game scores?

Let $\mathbb T=\{1,\dotsc,10\}$ represent the ten pins in a standard game of bowling. Given two sets of pins $T\subseteq S\subseteq \mathbb T$, let's write $p_{S\to T}$ to represent the conditional ...
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Find coefficient that makes constraint non-binding

I have a problem in the following setting: We maximize a concave quadratic objective function over a polytope, so we have a QP. All the variables are non-negative and are required to sum up to one. ...
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solve $XA = B$ in MATLAB

I have an equation $XA = B$ in which $X$ and $B$ are $n \times m$ and $A$ is $m \times m$. And $n>m$ . $A$ is positive definite. How to calculate $X$. I used $X = B / A$ in MATLAB. but it ...
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1answer
71 views

Inverse Eigenvalue problem for star graph

This is the problem: Given $\lambda = [\lambda_1, \lambda_2, \cdots, \lambda_n]$ and $\mu = [\mu_1, \mu_2, \dots, \mu_{n-1}]$ build matrix $B$ such that its off-diagonal entries corresponds to a star ...
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does asymptotic behavior guarantee the uniqueness?

Suppose $w$ is solution of $$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$ with asymptotic condition $$\lim_{x\rightarrow \infty}w(x)e^{ikx}=1$$ and $u\in L^1_1(\mathbb{R})=\{f:\int_\mathbb{R}(1+|x|)|f|dx<\...
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Inverse conductivity problem - how to obtain the reconstruction numerically?

I'm working on an Electrical Impedance Tomography (or EIT) problem i.e. I wish to reconstruct an image based on information obtained at the boundary. The problem consists of the generalised LaPlace ...
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2answers
250 views

finding the kernel of Fredholm integral equation of the first kind

I have the following equation: $$F(x)=\int^{a}_{b}f(x')K(x,x')dx'$$ the f(x) and F(x) are known functions and I need to find only K(x,x'). Both functions F and f are smooth, integration limits are ...
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Well posed or ill posed inverse problem?

Consider the problem of finding a control parameter $p(t)$ in the following \begin{equation} \frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}+p(t)u(x,t)+f(x,t), \ ~~\ 0\leq x \leq1 ,...
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Applications of degenerate parabolic equations

I study inverse problems for degenerate parabolic equations, specifically cases when the unknown major coefficient in the equations vanishes at $t=0$, e.g.: $u_t = t^{\beta}a(t)\Delta u + f$ What ...
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1answer
39 views

Reference req : parameter estimation for hyperbolic PDEs

I would like to know some references (books, thesis,.. maybe recent) on "inverse problems" related to "hyperbolic PDEs" or what's commonly referred to as "Parameter estimation" in such equations. ...
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What type of minimization problem is $\arg \min_u \{\|s-u\|+\lambda\|u-L\left(u\right)\|\}$?

A few years ago, I came across the following minimization formulation $$J = \arg \min_u \{\|s-u\|+\lambda\|u-L\left(u\right)\|\}$$ $s$ is the measurement, $\lambda$ is the regularization parameter, ...