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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Alternative to the Sampling Theorem / Invertible transform with sampling criteria

I seek a transform $T$ that operates on real-valued $x(t)$, that Is perfectly invertible Has discrete counterpart with continuous reconstructor Provides conditional reconstruction guarantees ...
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1 answer
80 views

Non-Fourier complete orthogonal basis?

The Fourier Transform (FT) Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero Is invertible: info-preserving, has inverse function Is energy-...
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Reconstructing a function from its residue and discontinuity

I am trying to work through the Zakharov and Shabat paper on inverse scattering for the nonlinear Schrodinger equation . I am stuck on section 2, specifically this problem. I need to know how to ...
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Identity principle of solutions of SL-problems with matching values on open set

Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, &...
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Bayesian inverse problems on non-separable Banach spaces

I am now studying Bayesian inverse problems. In the note of Dashti and Stuart https://arxiv.org/abs/1302.6989, they mentioned that "... when considering a non-separable Banach space $B$, it is ...
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Optimal distribution on $k$-dimensional subspaces of $R^n$ which maximizes $\mathbb E_{V \sim D}\|Proj_V(x)\|^2$, given side informaiton on fixed $x$

Let $d$ be a large integer and let $ k \in [1,d]$ be another integer. Let $V$ be uniform over the grassmannian $G_{k,d}$ of $k$-dimensional subspaces of $\mathbb R^n$, and let $P_V:\mathbb R^n \to \...
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Second-order envelope theorem for linear programming

Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
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1 vote
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Stable deconvolution of a band-limited function from its convolution with a Gaussian

Suppose that $f : \mathbb R \to \mathbb C$ is a band-limited function, i.e. its Fourier transform $\hat f$ has support in a compact interval $[-a,a]$. Let $\phi(t) = e^{-\frac{t^2}{2\sigma^2}}$ be a ...
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5 votes
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Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$

Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to $$ \Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
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Inverse Laplace transform through contour integration

How can I prove that in formal way, this function doesn't have inverse Laplace transform. $$ F(s)=\frac{\sin(s)}{\sqrt{s}} $$ Strictly it should be in Bromwich contour method. Could you please tell ...
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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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On the equation $[U, V] - V_x = C(x)$

While considering the zero curvature equation $U_t - V_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given $U(x)$ and $C(x)$, find $V(x)$ ...
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How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation?

I have encountered the following problem. Let $\chi:=\chi_{B(0,1/2)}$ be characteristics function i.e it take $1$ if $x\in B(0,1/2)$ otherwise $0$. $\nabla\cdot ((1+\chi_{B(0,1/2)})\nabla u )=0 $ in $...
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Reference request: diffusion approximation to the radiative transport

I'm looking for a good, modern reference for the diffusion approximation to the radiative transport problem. I'm aware of the text of Dautray and Lions, as well as the monograph by Bensoussan, Lions, ...
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Is Sommerfeld radiation condition invariant under translations?

A smooth function $U:\mathbb{R}^3\setminus B_{r_0}(0)\to\mathbb{C}$ (for some $r_0>0$) satisfies the Sommerfeld Radiation Condition with index $k$, denoted $U\in \texttt{SRC}$, whenever $$ \lim_{r\...
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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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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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3 votes
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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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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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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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52 votes
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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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2 votes
1 answer
242 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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17 votes
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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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11 votes
3 answers
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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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3 votes
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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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6 votes
3 answers
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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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2 votes
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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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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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3 votes
1 answer
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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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15 votes
1 answer
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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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0 votes
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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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1 answer
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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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6 votes
2 answers
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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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12 votes
3 answers
650 views

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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5 votes
1 answer
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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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-1 votes
1 answer
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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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1 vote
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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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8 votes
1 answer
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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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54 votes
2 answers
5k views

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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3 votes
0 answers
168 views

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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8 votes
1 answer
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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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3 votes
2 answers
911 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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3 votes
1 answer
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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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6 votes
1 answer
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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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2 votes
0 answers
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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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4 votes
1 answer
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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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