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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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Sufficient conditions to guarantee that weakly convergent sequence also converges strongly

I know that any Hilbert space, $H$, satisfies the Radon-Riesz property: Any weakly convergent sequence, $x_n \rightharpoonup x$, such that $\lim_{n \rightarrow} \|x_n\| = \|x\|$, is also strongly ...
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First and second condition in Hadamard's Well-posedness

From, e.g., Wikipedia we have In mathematics, a well-posed problem is one for which the following properties hold: 1. The problem has a solution 2. The solution is unique 3. The solution's behavior ...
Why's user avatar
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Approximate a reaction-diffusion system by a "diffussion-then-reaction" system

Consider a 1D reaction-diffusion system with a scalar diffusion rate and a logistic reaction function: $$\frac{du}{dt} = D \nabla^2 u + \rho u(1-u).$$ Suppose the spatial domain is $\mathcal{X}=[0,10]$...
Miles N.'s user avatar
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If I can solve $Ax = b$ efficiently, is there a method to solve $(I+A)x=b$ efficiently?

Say if I already have the LU factorization of a square matrix $A$, is there an efficient way to get the LU factorization of $I+A$? (We may assume all the matrix I mentioned is invertible.) I know from ...
qdmj's user avatar
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Inverse Problems for Markov Models

Consider a Markov process with three states, whose transition scheme is represented as follows: The stated model includes four parameters, i.e., the transition rates $s_x$, $g_x$, $\mu^{N S}_x$, and $...
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Ways to invert complicated matrix formulas

I have two somewhat complicated matrix formulas that convert the mean vector and covariance matrix for a certain variable, $\mu \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{n \times n}$, into the ...
dherrera's user avatar
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uniqueness of the inversion to Riemann-Stieltjes integral equation

I believe that if, for a Riemann-Stieltjes integral with $h(s)$ of bounded variation, $$ \int_0^1 s^\alpha dh(s) = 0 \qquad\text{for any }\alpha\in(\alpha_0,\alpha_1) , \tag{1}\label{eq1}$$ then $h$ ...
Martin Lanzendörfer's user avatar
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Question about existence of solutions to integral equations of the first kind

We have three random variables $U, W, A$ and consider the integral operator. The integral operator $T$ is defined as $$Tf= \int f(w,u)p(w|a)dw = p(u|a). $$ for any fixed variable $u$, where $p(w|a)$ ...
叶心萤's user avatar
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How this variational derivative is calculated?

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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
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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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
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Linear elliptic problem inverse mapping is Lipschitz in log permeability

I am reading this paper and would like to check how they derive the inequality in (5) on page 4. Denotes $S^d$ the set of symmetric second order tensors on $\mathbb{R}^d$. Define the permeability ...
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6 votes
1 answer
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Inverse propagation of information from the PDF of $Y=f(X)$ to the PDF of $X$

Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\...
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find $D$ of $(D+A)$ for $diag((D+A)^-1)=k$

I am wondering whether it is possible to derive $D$ for $diag((D+A)^{-1})=k$ where $diag()$ produces a vector of diagonal elements of a squared matrix, $D$ is an unknown diagonal matrix with possible ...
user1407220's user avatar
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1 answer
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How can I find possible non-symmetric $A$ if $A^k$ is symmetric?

Assume $\bf A\in \mathbb R^{n\times n}$ If I know ${\bf A}^k$ and that it is symmetric, how can I systematically find the $\bf A$ which are not? Own work One approach I have considered is to assume a ...
mathreadler's user avatar
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1 answer
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Inverse Laplace transform of Dirac delta function

I am trying to understand how to identify, or at least derive some properties of the inverse Laplace transform of the Dirac delta function, i.e. a function $\eta$ s.t. $$ \int_0^\infty dt\, \eta(\...
knuth's user avatar
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Inverse kernel of a sine kernel

I’m not a mathematician and I’m working with some transforms in physical chemistry. I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and ...
PierT's user avatar
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0 answers
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Different functions/images with similar Radon Transform

A couple of days ago, I read somewhere (unfortunately can't find it anymore), that one cavet of inverse problems is that "similar looking" data $y$, given by $$ \mathcal{A}x = y, $$ where $\...
stish's user avatar
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How to calculate the adjoint operator of the projected intensity differential (based on the Fresnel propagator)?

Let $G(\phi,\epsilon) = 2Real \left ( (e^{-B+i\phi} * P_z \right ) \cdot ([-i\epsilon \cdot e^{-B-i\phi}]*\overline{P_z})$ with $\phi, \epsilon, B \in L^2(\Omega,\mathbb{R})$ and $P_z \in L^2(\Omega,\...
Loric's user avatar
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Can there be a Nim j, k which generalizes Nim k?

In Nim, players must remove objects from exactly $1$ heap, and the winning strategy involves converting all heap sizes to base $2$, and removing objects to manipulate to $0$ the digital sum in base $2$...
user10478's user avatar
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15 votes
0 answers
271 views

Recovering a binary function on a lattice by studying its sum along closed paths

I have a binary function $f:\mathbb N^2\rightarrow\{0,1\}$. While I do not known $f$ explicitly, I have a "device" located at the origin $(1,1)$ which can do the following: Given an even ...
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3 votes
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Writing a 2-form $\tau_{ij}$ as $\pi_{ijh}\wedge \Theta^h+\rho_{ijh}\wedge \Psi^h$ where $\pi_{ijh}$ and $\rho_{ijh}$ are totally symmetric 1-forms

I am currently studying a memoir written by Anderson and Thompson on the inverse problem of the calculus of variations [1]. In particular, I am working through Example 6.4, where they prove that if ...
projectilemotion's user avatar
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55 views

Is there a way to do the Hungarian algorithm in reverse?

The Hungarian algorithm is used to find the optimal choices for a given 'cost matrix' e.g. going from the output of the Hungarian algorithm ...
Machai's user avatar
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1 answer
129 views

Fixed-point of a function and an inverse problem

This is perhaps a theoretical computer science question. Please help redirect appropriately. The question is about mental processes involved in discovering a solution. Consider an iterative higher-...
Kedar Mhaswade's user avatar
4 votes
2 answers
239 views

Given a vector field $f:\mathbb R^3\to\mathbb R^3$, is there a mass distribution that generates $f$ as its Newtonian gravity?

Let $f:\mathbb R^3\to\mathbb R^3$ be a smooth bounded vector field. I want to produce a density $\rho:\mathbb R^3\to\mathbb R$ such that the Newtonian acceleration experienced by a particle in the ...
Derivative's user avatar
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1 vote
1 answer
69 views

Show that operator is equal to $-(\nabla -\tau\nabla f)\cdot(\nabla -\tau\nabla f) $

I have the following problem, I dont understand why is this equality. Let $f(x)$ be a function such that $f=\exp(g)$ with $g\geq0$ on a domain of interest. We are looking at the operator \begin{...
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Uniqueness of non-negative least squares solution

When solving linear problems, there are many different ways of assessing if $Ax=b$ has a unique solution for a given $A$ by looking at properties of $A$ like the rank or singular values. If I am ...
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Morozov Discrepancy for PDE-Constrained Optimization With Bound Constraints

$\textbf{Background and context}$ Let $\beta:\Omega\rightarrow\mathbb{R}$, be an unknown spatially distributed parameter (over the spatial domain $\Omega$). Let $\mathcal{F}$ represent a parameter-to-...
Tucker's user avatar
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How can I use the finite difference method to solve a simple unconstrained optimal control problem?

I want to solve this problem using finite difference method. \begin{equation} \begin{cases} &\min_{y, u} \quad \mathcal{J} = \frac{1}{2}\left\lVert y-z \right\rVert^{2} + \frac{\alpha}...
wc x's user avatar
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2 votes
1 answer
95 views

Does the flow identify the vector field?

Setup Let $d\in\mathbb N$. Let $f\colon\mathbb R^d\to\mathbb R^d$ be $L$-Lipschitz continuous and bounded by $C$ for some fixed constants $L,C\in(0,\infty)$, i.e., $\Vert f(x)\Vert \leq C$ and $\Vert ...
Froomfondel's user avatar
1 vote
0 answers
35 views

Eigenvalues of Parameter to Observable Map in Source Inversion Inverse Problem

TLDR: equation for eigenvalues of differential operator seem inverted based on what one would obtain from solving the eigenvector/-value problem. I don't understand where this equation comes from. Hi, ...
Riess's user avatar
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1 vote
0 answers
29 views

How do these definitions of regularization match?

I know regularization from the following point of view: You add a term R(f) to a loss function, e.g: $min_{f} \sum V(f(x_i),y_i) + \lambda R(f)$ where $\lambda$ is a parameter. I recently read about ...
melmo99's user avatar
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0 answers
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How does the multidimensional Fourier transform relate to gradients of functions?

The one-dimensional Fourier transform has a famous relationship with the derivative of a function $$\mathcal{F}\left\{\frac{d f}{d t}\right\} = iw\mathcal{F}\left\{f\right\}$$ Among other things this ...
mathreadler's user avatar
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0 votes
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Euler Lagrange equations in a two-dimensional setting

I am struggling to derive the Euler-Lagrange equations for the problem below. I have only recently been introduced to Euler-Lagrange equations and am struggling to understand them. I will also give ...
plan's user avatar
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3 votes
0 answers
37 views

find the dual pde-constrained optimization

Given a PDE-constrained optimization as \begin{equation} \min_{u\in U} J(y,u)=\frac{1}{2}\|y-y_d\|^2_{L^2(\Omega)}+\frac{\alpha}{2}\|u-u_d\|^2_{L^2(\Omega)} \end{equation} subject to \begin{...
79999's user avatar
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1 vote
0 answers
131 views

Constructing a tridiagonal matrix with some spectral information.

I am going to construct a matrix of the following form $$ T_n = \begin{bmatrix} a_1 & b_1 & 0 & 0 & 0 & 0\\ b_1 & a_2 & b_2 & 0 & 0 & 0 \\ 0& b_2 & a_2 &...
M a m a D's user avatar
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0 answers
28 views

How can I find a smoothest complex analytic function with a (finite) set of prescribed function values?

How can I find the smoothest complex analytic function $$x+yi \to u+vi$$ with a finite set of points having prescribed values $$\{x_1,\cdots,x_n\},\{y_1,\cdots,y_n\},\{u_1,\cdots,u_n\},\{v_1,\cdots,...
mathreadler's user avatar
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0 votes
1 answer
151 views

Diagonalization of a combination of circulant matrices

Write \begin{align} A &= \sum_{k=1}^r C_k E_k \in \mathbb{R}^{n \times n} \end{align} where $C_k$ are diagonal matrices and $E_k$ are circulant matrices for all $k \in \{1,2,\dots,r\}$. Given $x \...
mlbj's user avatar
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2 votes
1 answer
73 views

Compute $f(x)$ given $\sum_{n=0}^\infty f((n+\frac{1}{2})z)$

Let's say I have some function $$F(z) = \sum_{n=0}^\infty f\big(\big(n+\tfrac{1}{2}\big)z\big)$$ which I can evaluate for any $z>0$. Is there a way to recover $f$ purely by evaluations of $F$? I'm ...
Caleb Briggs's user avatar
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1 vote
0 answers
43 views

Looking for an inverse of an integral transform

I have an integral transform (motivated by a physics problem) $F (x) = \int_0^\infty dx' \frac {a x'} {(a x)^2 + (x - x')^2} f (x')$, where $x, x', a > 0$ real, $f : \mathbb{R}_+^0 \to \mathbb{R}$. ...
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0 answers
50 views

Phase retrieval for phase in range $[-2\pi,2\pi]$.

I have a phase retrieval problem, wherein the phase that needs to be determined lies in the range $[-2\pi, 2\pi]$. So, if we represent a phase distribution by $\phi$, such that $\phi$ belongs in the ...
A Khan's user avatar
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0 votes
1 answer
88 views

Does the Riemann-Lebesgue Lemma Apply to $L^1$ or $L^2$ Space?

In the literature on inverse problems, the Riemann-Lebesgue lemma is often used to demonstrate the ill-posedness of integral equations with square-integrable kernels. For example, in Groetsch (1984), ...
Jacob's user avatar
  • 155
1 vote
0 answers
53 views

Unique inversion for X-ray transform of vector field.

It is known that a part from the zero tensor field, we can only recover the solenoidal part of a tensor field field of arbitrary order from its X-ray transform. Given the vector field $F = (f_1,f_2),$ ...
Samuel's user avatar
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3 votes
0 answers
186 views

Approximating Green's function numerically by simply solving PDE with approximate Dirac delta right-hand side

Consider the following linear PDE with Dirichlet boundary condition. \begin{cases} \mathcal{L} u = f \text{ in } \Omega \\ u = u_{D} \text{ on } \partial \Omega \end{cases} I want to solve this PDE ...
TylerMasthay's user avatar
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0 answers
106 views

Construction of a possible Lagrangian for the ODE's system

Given: $\begin{cases} x''-(x-(y-z))=0 \\ y''+(x-(y-z))=0 \\ z''-f''y-f'y'=0 \end{cases}$ $x,y,z$ - variables, $f$ - arbitrary time-dependent function The Helmholtz conditions for this system are not ...
ayr's user avatar
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1 answer
215 views

Difference in Tikhonov regularization for linear and non-linear case?

the Tikhonov regularization for a linear operator $T: X \rightarrow Y, x \mapsto y$ means minimizing the least square problem $$\begin{align*} \lVert Tx - y \rVert^2_Y + \alpha \lVert x\rVert_X \...
milaking's user avatar
  • 183
2 votes
2 answers
109 views

Finding the values of a matrix multiplied between two unknown matrices

This is a slightly vague question I think, but I am wondering if there is any elegant way of solving this problem. Say I have a multiplication between three unitary matrices operating on a vector, ...
LDPC's user avatar
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1 vote
0 answers
43 views

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 ...
can't stop me now's user avatar
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0 answers
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how to show inverse of M-by-N data matrix using singular value decomposition?

The question is from the exercise questions from the EE4595 course being taught in TUDelft. The question is as follows... So far my attempted solution is like the following. The discretised data ...
cn_97's user avatar
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2 votes
1 answer
160 views

Linear Algebra: Solving the minimization with vector p-norm.

The course this question is from is called Wavefield imaging, It gives the vector p-norm of an N-by-1 vector $x=[x_1,x_2,...,x_N]^T$ is defined as $$ ||x||_p=\left(\sum_{n=1}^N |x_n|^p \right)^{1/p} $$...
cn_97's user avatar
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1 vote
1 answer
60 views

Analytic solution inverse kinematics - different solutions with different calculation steps

I have the following problem: $$5\cos \theta_1+3\sqrt{3}\sin \theta_1=4\qquad\qquad\textbf{(I)}$$ $$5\sin \theta_1-3\sqrt{3}\cos \theta_1=6\qquad\qquad\textbf{(II)}$$ I have two seemingly correct ...
meschi's user avatar
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