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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Radon transform range theorem and radial functions

(UPDATED for rapid decay considerations + new question) In dimension 2, the Radon transform range theorem states that a rapidly decaying (Schwartz) function $g(t,\theta)$ can be represented as a ...
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Math calculators to find inverse laplace transform of slightly complicated equation [migrated]

I have found out the inverse laplace of $\log\frac{s+4}{s-4}$ which is $\frac{2 sinh(4t)}{t}$ But I do not know how to proceed. The multiplication by s property does not seem to hold good here as the ...
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1answer
85 views

Optimal transport: find cost function given observed transport

Could you advise me please on what to read on the "inverse" problem: suppose I have a source measure, a target measure and I observe the solution to optimal transport problem -- can I "back out" the ...
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Radon transform on complex Grassmannians

Let $Gr_{i,n}$ denote the Grassmannian of complex linear $i$-dimensional subspaces in the Hermitian space $\mathbb{C}^n$. Let $1\leq i<n/2$. Consider the Radon transform between space of functions ...
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1answer
138 views

Interesting questions for inverse parabolic problems

I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
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1answer
223 views

Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$

Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function. Question Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \...
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Mathematics of imaging the black hole

The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 . It has been claimed that ...
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1answer
184 views

Can a bijection between function spaces be continuous if the space's domains are different?

It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\...
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Kernel of Radon transform in $\mathbb{R}^3$

Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$: $$(Rf)(H):=\int_{l\subset ...
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506 views

An explicit reconstruction of a matrix from its minors

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ $\newcommand{\Cof}{\operatorname{cof}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
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558 views

Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
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Reconstructing the Green's function of an initial-value problem of partial differential equation

Consider a partial differential equation that is of the following form: \begin{equation} (-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t) \end{equation} where $g(x)$ is a real function. Suppose that $f(...
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How to numerically invert a bilateral (two-sided) Laplace transform?

For one-sided Laplace transforms I can find many algorithms to invert them numerically (e.g. algorithms named after: Talbot, Stehfest, Euler, ...). However, I am interested in numerical inversion of ...
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298 views

Reference Request - Recovering a function from its definite integrals (inverse problem)

I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int_{a}^{b} f$ for some ...
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How often does the gradient of a solution to elliptic equation vanish on the boundary?

This question is motivated by an inverse coefficient problem, for which it is useful to find solutions to a particular PDE so that the gradient of the solution does not vanish at all, or at least too ...
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1answer
55 views

inverse interpolation

Given data points $(x_i,y_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y_i=f (x_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ ...
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149 views

Is there an English translation of Hadamard's classic French paper on well-posed problems?

This paper by Hadamard is often cited as being the source of the definition of well-posed and ill-posed problems. However, it is in French so I cannot verify that claim. Is there an English ...
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Is $\delta(df \wedge df)=0$ an Euler-Lagrange equation?

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\tr}{\...
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109 views

Solution of Poisson equation vanishing at the boundary of any order

Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and $\Delta u=f$ in $\Omega$ such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \...
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On the Integration and Manipulation of Expressions Involving Hypergeometric Functions

I would like to ask the following two: For the integral: \begin{equation} \int_{0}^{t}\left( 1-s^p \right)^{\frac{1-p}{p}}ds \end{equation} I know that it is reduced to the following product ...
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468 views

Is this result on an unconstrained inverse quadratic programming problem new or known already?

Is this problem and solution actually new, or has someone done this earlier? The details can be found in the preprint: arxiv:1701.01477. Let us consider a direct quadratic programming problem: $$ \...
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Approximating Uniform Distribution with Mixture of Gaussians

Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$. Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$ \...
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A question in inverse Galois Theory

Let $\mathbb{G}= \{g_1,\cdots,g_n\}$ be a finite group ,$\rho$ be the regular representation. Let $x_1,\cdots,x_n$ be indeterminates and let $x = (x_1,\cdots,x_n)^\top$. Let the matrix $G$ be defined ...
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How to find Suleimanova's work on the Nonnegative Inverse Eigenvalue Problem?

Many papers cite the work of Suleimanova when studying inverse eigenvalue problems - in particular, the nonnegative inverse eigenvalue problem (NIEP). However, I cannot seem to find her work anywhere....
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163 views

Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?

I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational ...
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246 views

Question related to Galois covering of Projective line over rational numbers

Suppose that we have Galois covering $$X\longrightarrow P^{1}_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group....
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59 views

The jump set of $SBV$ function for different value of parameter in image denoising problem

The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary, $$ u_\...
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1answer
496 views

Inverting a function

I posted this question on crypto.SE but got no answer: Let $w = a_0 \cdot a_1 \cdots a_{n-1} $ be a word from $ \{0,1\}^n $, $|w| = n$ Let $m = \sum_{i=0}^{n-1}{ a_i \cdot 2 ^ {n-1-i} } $ be the ...
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Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black holes one billion light-years away. One of the black holes had 36 times the mass of the sun, and the other 29. After the merging the ...
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Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions: All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix); all principal minors are $>1$, except ...
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Question on Inverse Galois Theory

Let $G$ be a finite group, $n=|G|$. Let $\rho:G\rightarrow GL(n,\mathbb{C})$ be the regular representation. Let $G \le H \le S_n$ be another group. Then we have $\mathbb{Q}[x_1,\cdots,x_n]^H \le \...
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2answers
874 views

Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...
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1answer
97 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet. Link:https://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded ...
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1answer
179 views

Radon transform between affine grassmannians

Let $\overline{GR}(n,k)$ be the manifold of all affine k-dimensional subspaces in $R^n$, and let $R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l))$, $0\le k<l\le n-1$, be the ...
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Combining microlocal Helgason's support and Holmgren's theorem to prove injectivity of limited-angle Radon transform

This questions is slightly related to Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems, in which I asked for some references. Now I ...
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1answer
195 views

Injectivity of the Funk transform for nonsmooth functions

Let $S^{n-1}$ be the unit sphere in $\mathbb R^n$ and $\Gamma_n$ the collection of great circles on it. Assume $n\geq3$. The Funk transform of a function $f:S^{n-1}\to\mathbb R$ is a map $Ff:\Gamma_n\...
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7answers
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What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$. $P$ could be a polyhedron, or a smooth shape, or an arbitrary shape; I'll assume below that $P$ is a (non-degenerate, perhaps non-...
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Counting variables to look for invariances/range conditions

A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...
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1answer
364 views

An unusual metric reconstruction problem

$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have ...
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2answers
557 views

Certain inverse problem related to moments

Suppose $D\subset \mathbb C$ is a smoothly bounded domain and it contains the origin. Let $ds$ denote the arc length measure on $\partial D.$ I am interested in the following two inverse problems (...
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1answer
120 views

Fredholm integral with functions constrained to [0;1]

I am trying to feed information about the solution when solving an inverse problem given by a Fredholm integral of the form $$ g(t)=\int_{a}^{b}K(t,s)f(s)ds. $$ Say I know $g(t)$ and $K(t,s)$, and ...
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186 views

Applications of Freiman's theorem?

What are some interesting applications of Freiman's theorem or, better-yet, its recent generalizations (eg Green-Ruzsa) that could be included in a graduate course in additive combinatorics? I'm ...
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How to check if a manifold can be foliated by strictly convex hypersurfaces?

Let $M$ be a compact Riemannian manifold with boundary. How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces? An exact definition is given below. If the ...
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Sample based inversion of the Radon transform

I have a classic tomography problem in which I would like to infer the internal density $p_0: \mathbb R^2 \to \mathbb R$ from external Radon projections. The internal density however is viewed as a ...
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43 views

Uniquenss of domain with given interior newtonian potential

The newtonian potential of a domain $\Omega$ is defined by $\Gamma*(\chi_{\Omega})$ ($\Gamma$ is the fundamental solution of laplacian operator $\Delta$), i.e. the convolution of indicator function ...
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236 views

A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
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128 views

Do position and momentum measurements determine a wave function?

Suppose we have a function $f\in L^2(\mathbb R^n)$ and we know the functions $x\mapsto|f(x)|$ and $p\mapsto|\hat f(p)|$, where $\hat f$ is the Fourier transform of $f$. Can we reconstruct the function ...
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1answer
101 views

General Radon-type inverse problem

Let $f : \mathbb R^n \to \mathbb R$ be a density which is sufficiently smooth and can also be restricted to have compact support for now. Let $t \ge 0$ and $F_t : \mathbb R^n \to \mathbb R$, i.e. $(...
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1answer
218 views

Invertibility of an inverse problem

Let $p$ be a scalar field $p: \mathbb R^n \to \mathbb R$. I encountered the problem of reconstructing an unknown density $p$ from its integral values $$I(t,z) = \int_{V_t} p(x) dS$$ along a one-...
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1answer
1k views

Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...