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.
92 questions
0
votes
1
answer
152
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| \...
1
vote
0
answers
121
views
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 ...
- 111
3
votes
1
answer
552
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:
$$
\...
6
votes
2
answers
965
views
Approximating Uniform Distribution with Mixture of Gaussians
Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$.
Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$:
$$
\...
- 181
4
votes
0
answers
514
views
A question in inverse Galois Theory
Let $\mathbb{G}= \{g_1,\dots,g_n\}$ be a finite group and $\rho$ its regular representation. Let $x_1,\dots,x_n$ be indeterminates and let $x = (x_1,\dots,x_n)^\top$. Let the matrix $G$ be defined ...
user6671
12
votes
3
answers
737
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....
- 121
5
votes
1
answer
375
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 ...
- 1
-1
votes
1
answer
261
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....
- 591
1
vote
0
answers
71
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_\...
- 679
8
votes
1
answer
579
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 ...
user6671
57
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
...
- 50.8k
3
votes
0
answers
182
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 ...
- 2,479
8
votes
1
answer
706
views
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 \...
user6671
3
votes
2
answers
947
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 ...
- 1,583
4
votes
1
answer
137
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
...
- 1,465
6
votes
1
answer
196
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 ...
2
votes
0
answers
108
views
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 ...
- 251
4
votes
1
answer
271
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\...
- 8,086
26
votes
7
answers
3k
views
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-...
- 151k
1
vote
0
answers
29
views
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 ...
- 203
6
votes
1
answer
393
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 ...
- 34.7k
1
vote
2
answers
572
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 (...
- 1,583
1
vote
1
answer
140
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 ...
- 13
4
votes
0
answers
238
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 ...
4
votes
0
answers
324
views
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 ...
- 8,086
4
votes
0
answers
75
views
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 ...
user45183
1
vote
0
answers
70
views
Uniqueness 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 the indicator function ...
- 41
2
votes
0
answers
246
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 ...
- 4,270
4
votes
0
answers
167
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 ...
- 8,086
3
votes
1
answer
131
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. $(...
user45183
4
votes
1
answer
236
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$$
along a one-...
user45183
23
votes
1
answer
1k
views
Geodesics in finite groups
It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for ...
- 8,086
7
votes
1
answer
251
views
Solving for a set of points from projections
Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
S = \{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let $v \in \mathbb R^n$ and consider the image set (not counting ...
user45183
0
votes
2
answers
200
views
Solving a functional equation
I would like to consider the following simple problem. I want to find two functions $f,g : \mathbb R \to \mathbb R$ such that, being given a collection $(h_v)_{v\in V}$ of real functions indexed by ...
user45183
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
5
votes
2
answers
204
views
Reconstructing set of points from one-dimensional images
Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
\{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let us be given a finite family of non-injective matrices
\begin{...
user45183
3
votes
1
answer
217
views
Partial recovery from Radon transform
Let $f : \mathbb R^3 \to \mathbb R$ be an integrable function. Let $\eta$ be a one-dimensional subspace of $\mathbb R^3$. We denote $p+\eta$ the affine subspace (a line) which is obtained by ...
user45183
2
votes
1
answer
266
views
"Limited angle" in n-dimensional Radon transform?
The Radon transform in two-dimensions is well studied. It maps a sufficiently nice function $f: \mathbb R^2 \to \mathbb R$ to its line integral along a certain line $L$, i.e.
\begin{align*}
Rf(L)...
- 41
4
votes
1
answer
258
views
Interpretation of the integral "with respect to a plane wave" in terms of Radon transform
This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform:
$\newcommand{\R}{\mathbb{R}}$
$$
Rf(\varphi,s)=\int_{-\infty}^\infty f(s\...
- 203
2
votes
2
answers
4k
views
Choosing the order of Tikhonov regularization of an inverse problem
This question is migrated from math.stackexchange.
Let me first describe the problem I am trying to solve and then the question I have. I greatly appreciate anyone who can shine some light on it.
...
- 123
7
votes
1
answer
478
views
Inversion of Radon transform by incomplete data: specific case
Let $R[f](p,t)$ denote the Radon transform of smooth function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ with compact support in $\mathbb{R}^n_+$:
$$
R[f](p,t) = \int\limits_{x \cdot p = t} f(x) dx.
...
- 1,329
-2
votes
1
answer
890
views
Determine noise distribution [closed]
I'm trying to solve the following least squares problem:
$\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$
where $Ax = b$ and $\tilde{b} = b + w$
Question:
How do I determine which probability ...
- 35