45 votes
Accepted
Mathematics of imaging the black hole
Essential elements$^*$ of the reconstruction algorithm were developed at MIT under the name CHIRP = Continuous High-resolution Image Reconstruction using Patch priors, as described in Computational ...
- 201k
9 votes
Accepted
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
Let $A$ be this matrix. Because of the formula $$\int_D\sigma\nabla u\cdot\nabla v\, dx=\sum_{i,j}a_{ij}U_iI_j,$$ ($U$ for voltages of $u$, $I$ for currents of $v$), we see three necessary conditions: ...
- 53.1k
9 votes
Automorphisms of projective spaces, and the Axiom of Choice
On abstract metamathematical grounds (having nothing to do with projective spaces), I claim that it is relatively consistent with ZFC and indeed with ZFC+Con(ZF) and much more that the answer to your ...
- 246k
8 votes
Accepted
Reconstruction of second-order elliptic operator from spectrum
In general, this will not be possible. For instance, the first non-trivial eigenfunction will have two nodal domains, so $f_1$ cannot be arbitrary. Furthermore, you expect to have some sort of ...
- 6,416
7 votes
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
In dimensions 3 and higher, and without any constraints on $\sigma$, one can apparently obtain any symmetric matrix $A = (a_{ij})$ such that $a_{ij} < 0$ when $i \ne j$ and $a_{ii} = -\sum_{j \ne i}...
- 17.4k
7 votes
Accepted
Inverting a function
Yes, you can use the Lehmer-Permutation to make a function that is suitable for cryptography, whose solution is just as hard as the Diffie-Helman problem. The relevant papers are: (1) Roberto Mantaci,...
- 31.5k
7 votes
Accepted
An explicit reconstruction of a matrix from its minors
Assume for simplicity that $kl=d+1$. Then I claim that $V \otimes \det V$ appears as a summand of $\left( \bigwedge^k V\right)^{\otimes l}$ with multiplicity $l-1$. The maps $\left( \bigwedge^k V\...
- 162k
6 votes
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
This question (for two dimensional domains) was answered by Curtis, Ingerman and Morrow, "Circular Graphs and planar Resistor Networks" (1998). Let $a$ be the $n \times n$ response matrix. As already ...
- 161k
5 votes
Accepted
Is there an English translation of Hadamard's classic French paper on well-posed problems?
No, there is no "official" English translation, however, Google translate should work just fine, here is the translation of the first paragraph, without any corrections from my side: The general ...
- 201k
4 votes
Accepted
Reference Request - Recovering a function from its definite integrals (inverse problem)
Here is how you make an inverse problem of this problem: Choose a space $X$ for the function $f$ you are looking for (e.g. $L^2(0,1)$ to work in Hilbert spaces, but other spaces may be more suitable, ...
- 13k
4 votes
Accepted
Inverse problem of the calculus of variations for autonomous second-order ODEs
Since the Euler-Lagrange equation for an autonomous Lagrangian $L(q,\dot q)$ is $$ \frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right) = L_q - \dot q L_{q\dot q}...
- 110k
4 votes
Is the real and imaginary part of the Dirichlet eta function invertible when viewed as single variable function?
Here are plots of $J(\alpha)$ for $\beta=3$ (left plot) and of $I(\beta)$ for $\alpha=1/2$ (right plot), as you can see these are not invertible functions.
- 201k
4 votes
Domains with discrete Laplace spectrum
This is more of a literature pointer than an answer. In the case of Dirichlet boundary conditions, the answer to the last question is yes: for example, the cross $$ C:=\{|x_1|\leq 1\}\cup \{x_2\leq 1\}...
- 9,865
4 votes
Accepted
On the equation $[U, V] - V_x = C(x)$
Yes, you can always do this, as follows: First, consider the equation $M_x = -M U$ with the initial condition $M(0) = I_n$. This linear equation with initial condition has a unique solution and $M(x)$...
- 110k
3 votes
Non-Fourier complete orthogonal basis?
(First, just for precision, your first point about "orthogonality" is morally correct, but not literally correct, because the exponentials are not in $L^2(\mathbb R)$...) In fact, there are ...
- 23.3k
3 votes
Reconstruction of second-order elliptic operator from spectrum
Just to show how overdetermined your general request is. It has be shown that in low dimensions the sphere with standard metric is uniquely determined by the spectrum of the Laplace-Beltrami operator. ...
- 697
3 votes
Is Sommerfeld radiation condition invariant under translations?
$\newcommand{\x}{\mathbf{x}} \renewcommand{\a}{\mathbf{a}} \renewcommand{\b}{\mathbf{b}} \renewcommand{\d}{\mathbf{d}} \newcommand{\0}{\mathbf{0}} \newcommand{\n}{\nabla} \newcommand{\R}{\mathbb R} \...
- 143k
3 votes
Accepted
Interesting questions for inverse parabolic problems
Inverse Problems for Partial Differential Equations (third edition, 2017) by Victor Isakov concludes each chapter with a collection of open research problems. Chapter 9 is specifically devoted to ...
- 201k
3 votes
Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?
Although I work exclusively on the nonnegative inverse eigenvalue problem (NIEP), I can assure you that IEPs are far from dead, both on the theoretical side and the applied side (there has been a ...
- 1,923
2 votes
Reference Request - Recovering a function from its definite integrals (inverse problem)
In general, it appears that hardly anything interesting can be said. E.g., let $A=\{(1/5,3/5),(2/5,4/5)\}$; here, it will be convenient to think of $A$ as a set of (say) open intervals, rather than a ...
- 143k
2 votes
Accepted
inverse interpolation
After having calculated an "explicit" interpolating function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$, satisfying $y_i=f (x_i)$, you can calculate the local inverse via evaluation of the implicit ...
- 13.9k
2 votes
Reconstructing the Green's function of an initial-value problem of partial differential equation
One can regard your problem as probing a medium by a known source $f_1$ generated at time $t=0$, recording the medium response $f_2$ at time $t=T$ (called the data) and the goal is the recovery of ...
- 21
2 votes
Can a bijection between function spaces be continuous if the space's domains are different?
As soon as your spaces $A$ and $B$ are Banach, and the only thing which you want from $\phi$ is continuity, you can use Kadets's theorem (implicitly mentioned by Nate), or its generalization by ...
- 902
2 votes
Accepted
Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$
$\newcommand{\R}{\mathbb{R}} \newcommand{\tto}{\underset{\text{onto}}\to}$ Let us answer the reformulated question: given a convex function $g\colon C\to\R$, when is it possible to find a decreasing ...
- 143k
2 votes
Accepted
Radon transform range theorem and radial functions
To answer Q1: There are trig identities at play. First, 0 is usually accepted under the definition of "homogeneous polynomial" (i.e., it's a polynomial whose coefficients are all zero) so ...
- 148
2 votes
Reconstruct the metric from partial information of the geodesic distance
Fix a point $p_0$ in your manifold and $U$ a small neighborhood of $p_0$. If $U$ is sufficiently small, then the function $\newcommand{\bR}{\mathbb{R}}$ $$ f:U\times U\to \bR,\;\;f(p,q)=d_g(p,p_0)^2, ...
- 35.8k
2 votes
Bayesian inverse problems on non-separable Banach spaces
$\newcommand\B{\mathscr B}\newcommand\C{\mathscr C}$There is hardly any particular intuition behind the concept of the cylindrical $\sigma$-algebra. This is just the smallest $\sigma$-algebra with ...
- 143k
1 vote
Accepted
Approximating the expectation of trace inverse of random Gaussian combination
The answer is similar to that of your earlier question: For $m,n\gg 1$, and $m/n\equiv r\in (0,1)$ fixed, an integration over the Marchenko–Pastur distribution gives (with $x_\pm=(1\pm\sqrt{r})^2$) $$\...
- 201k
1 vote
Accepted
How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation?
It is a separable problem. First you notice that the constant part of $f$ will play no role, since it would lead to a constant potential, so you may assume $$ \int_0^{2\pi} f(\theta)d\theta =0. $$ If ...
- 2,525
1 vote
Optimal transport: find cost function given observed transport
See "Inverse Optimal Transport" By Stuart and Wolfram, SIAM J. App. Math, 80(1), 2020, and "Learning to Match via Inverse Optimal Transport" by Li et al., JMLR 2019.
- 1,831
Only top scored, non community-wiki answers of a minimum length are eligible