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.
95 questions
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Single source-sink, Inverse shortest path problem with bounds, complexity and algorithms
Given a network $G=(V,E,w)$, two vertices $s$ and $t$ as source and sink, and a designated path $P_0$ from $s$ to $t$, the inverse shortest path problem (ISP) asks to adjust new weights $w^*$ at ...
- 55
-1
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0
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46
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Solving inverse triangle-center problems
I came across this problem by trying to construct a "nice" triangle, e.g. for an illustration, from 3 distances $\lbrace\|A-O\|, \|B-O\|, \|C-O\|\rbrace$ and the fuzzy requirement that $O$ ...
- 14k
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68
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Automatic balancing of multiple loss terms in polynomial PDE optimization
I'm solving an inverse PDE problem by fitting a signed distance function (SDF) to a 2D shape using polynomial approximation (Chebyshev basis). The loss consists of:
a boundary loss: $ u = 0 $ on the ...
- 21
2
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1
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225
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Reconstruct the metric from partial information of the geodesic distance
Suppose in a Riemannian manifold $M$ we know the geodesic distance function $l(p_1,p_2)$ between any two points $p_1$ and $p_2$ (at least locally), we can reconstruct the metric as follows. We ...
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For which $X$ has the matrix $M = (I + X^T X A)^{-1} X^T X$ dominant diagonal coefficients?
Let $A \in \mathbb{R}^{n \times n}$ be a Laplacian matrix (i.e. symmetric such that $A_{ii} = - \sum_{j \neq i} A_{ij} > 0$ for any $i$ and $A_{ij} \leq 0$ for any $i \neq j$), and $X \in \mathbb{R}...
- 2,602
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120
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inverse Galois problem on cyclic groups
It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...
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Inverse problem of the calculus of variations for autonomous second-order ODEs
Consider the following particular case of the inverse problem of the calculus of variations: given a system of second-order equations
$$
\ddot{q}^i = f^i(q, \dot{q}, t), \quad i = 1, \dots, n, \label{...
- 1,065
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130
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Inverse Galois problem for G with a derived sequence of length 2
For a finite group G with a derived sequence of length 2, please tell me how to specifically construct a field that is a Galois extension of $\mathbb{Q}$ and whose Galois group is G.
I made some edits ...
- 49
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Is the real and imaginary part of the Dirichlet eta function invertible when viewed as single variable function?
If we examine $\Re(\eta(\alpha + \beta i))$ as a function of $\alpha$ or only $\beta$ is $\eta$ invertible? That is, if we define that map $J:\mathbb{R}\rightarrow \mathbb{R}$ as
\begin{equation}\...
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How can we calculate the Euler-lagrange equations?
In this paper https://arxiv.org/pdf/1907.09605.pdf \
let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
- 11
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0
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191
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Recovering a binary function on a lattice by studying its sum along closed walks
I recently posted this question on MSE. While it attracted interest, no answers were submitted, so I thought to try and post it here.
I have a binary function $f:\mathbb N^2\rightarrow\{0,1\}$. While ...
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Is there any sufficient or equivalent condition for the invertibility of a regular map, i.e. a self map of $\mathbb{R}^m$ with polynomial components?
Let $P:\mathbb{R}^m\to \mathbb{R}^m$ be a regular map, i.e. a map whose components are polynomials. I was wondering whether we can say anything about the the component polynomials, their degrees or ...
- 1,691
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2
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Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
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Implicit function theorem / Implicit selections when Jacobian not invertible
I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
- 413
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107
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Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
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Domains with discrete Laplace spectrum
Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
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515
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Automorphisms of projective spaces, and the Axiom of Choice
It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...
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Resconstructing finite planar point sets from projections
What is the smallest length $m$ of a sequence $u_1,\ldots,u_m$ of $d$-dimensional vectors with real entries such that every finite set $X$ of $d$-dimensional vectors with real entries can be ...
- 4,132
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What are the limitations for calculating the inverse of a polynomial with the Lagrange inversion theorem?
I have been attempting to produce a series expression for the roots of high degree polynomial using the Lagrange Inversion theorem. I am curious about the statement from the Wikipedia page on Bring ...
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175
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Functional inverse problem based on a variational principle
I am trying to solve an inverse problem based on variational principle.
I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently ...
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145
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Analytically compare two 3D heatmaps of the brain
I have two heatmaps of a 3D model of the brain, with the color of each pixel being an intensity of the response to a stimulus, and I want to get a metric of how "alike" those two heatmaps ...
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129
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Approximating the expectation of trace inverse of random Gaussian combination
Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E_{A}(\mathrm{Tr}( (A^T ...
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134
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Proving some properties of the Landweber–Fridman iterates
$\newcommand\norm[1]{\lVert#1\rVert}$Let $B\in \mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Assume that $x\in \mathbb R^n$ is the solution of $Bx=w$ for some given $w\in \mathbb ...
- 21
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121
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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
...
3
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1
answer
339
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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-...
0
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93
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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 ...
- 41
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92
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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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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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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 \...
- 241
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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 $$R f(s, \omega):=\int_{-\infty}^{\infty} f\left(s \omega+t \omega^{\perp}\right) d t, \quad s \in \...
- 309
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On the equation $[U, V] - V_x = C(x)$
While considering the zero curvature equation $U_t - V_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given $U(x)$ and $C(x)$, find $V(x)$ ...
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How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation?
I have encountered the following problem.
Let $\chi:=\chi_{B(0,1/2)}$ be characteristics function i.e it take $1$ if $x\in B(0,1/2)$ otherwise $0$.
$\nabla\cdot ((1+\chi_{B(0,1/2)})\nabla u )=0 $ in $...
- 309
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0
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Reference request: diffusion approximation to the radiative transport
I'm looking for a good, modern reference for the diffusion approximation to the radiative transport problem. I'm aware of the text of Dautray and Lions, as well as the monograph by Bensoussan, Lions, ...
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Is Sommerfeld radiation condition invariant under translations?
A smooth function $U:\mathbb{R}^3\setminus B_{r_0}(0)\to\mathbb{C}$ (for some $r_0>0$) satisfies the Sommerfeld Radiation Condition with index $k$, denoted $U\in \texttt{SRC}$, whenever
$$
\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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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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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 ...
- 1,839
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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):\...
- 545
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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 ...
- 23.1k
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1
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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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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{...
- 542
3
votes
1
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297
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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(...
- 197
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1
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294
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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 ...
- 1
6
votes
3
answers
605
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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 ...
- 517
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0
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139
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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 ...
- 648