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.
287
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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 ...
- 1,294
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1
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57
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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{...
- 31
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0
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22
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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 ...
- 101
1
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0
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14
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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-...
- 2,080
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41
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Literature on inverse problem where one approximates the forward operator by a surrogate model
Is there any literature about the case of inverse problems where the forward operator is being approximated by a surrogate model, especially the case where the surrogate model is a neural network, and ...
- 60
0
votes
0
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21
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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}...
- 1
2
votes
1
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76
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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 ...
- 65
1
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0
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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, ...
- 11
1
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0
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26
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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 ...
- 107
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0
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30
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Confusion with implementation of PDE constrained Bayesian 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 $$Y = \Phi(...
- 101
1
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0
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76
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How does the multidimensional Fourier transform relate to gradients of functions?
The one-dimensional Fourier transform has a famous relationship with the derivative of a function
$$\mathcal{F}\left\{\frac{d f}{d t}\right\} = iw\mathcal{F}\left\{f\right\}$$
Among other things this ...
- 25.4k
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0
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41
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Euler Lagrange equations in a two-dimensional setting
I am struggling to derive the Euler-Lagrange equations for the problem below. I have only recently been introduced to Euler-Lagrange equations and am struggling to understand them. I will also give ...
- 21
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0
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12
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Tightening error bounds backwards through multiplication operation
Assume I have three scalar random variables $a,b,c \in \mathbb{R}$ related to each other through a multiplication operation:
$$ab=c$$
Let us further assume that we have three real-valued bounds for ...
- 929
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0
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22
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Inverting an integral operator that looks like a fractional integral
I was dealing with the integral operator
$$\tilde{I}^{\alpha}_{-, 2} f \left( r \right) = \int\limits_{r}^{\infty} \frac{f (s, r)}{(s^2 - r^2)^{1 - \alpha}} s \mathrm{d}s,$$
for "nice" ...
- 3,957
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find the dual pde-constrained optimization
Given a PDE-constrained optimization as
\begin{equation}
\min_{u\in U} J(y,u)=\frac{1}{2}\|y-y_d\|^2_{L^2(\Omega)}+\frac{\alpha}{2}\|u-u_d\|^2_{L^2(\Omega)}
\end{equation}
subject to
\begin{...
- 135
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0
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92
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Constructing a tridiagonal matrix with some spectral information.
I am going to construct a matrix of the following form
$$
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 &...
- 663
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26
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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 $$\{x_1,\cdots,x_n\},\{y_1,\cdots,y_n\},\{u_1,\cdots,u_n\},\{v_1,\cdots,...
- 25.4k
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Diagonalization of a combination of circulant matrices
Write
\begin{align}
A &= \sum_{k=1}^r C_k E_k \in \mathbb{R}^{n \times n}
\end{align}
where $C_k$ are diagonal matrices and $E_k$ are circulant matrices for all $k \in \{1,2,\dots,r\}$. Given $x \...
- 131
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0
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30
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Does the transpose of the comatrix of a Jacobian equal the adjoint of the Jacobian?
In my lecture, my teacher wrote that for a Jacobian matrix $J_{ij}$ and with determinant $J \neq 0$, we have $J_{ij}^{-1} = \frac{1}{J} J_{ij}^\dagger$.
However, we know that the inverse of $A$ is $$ ...
- 177
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votes
1
answer
73
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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 ...
- 1,043
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0
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40
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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, $f : \mathbb{R}_+^0 \to \mathbb{R}$.
...
0
votes
0
answers
37
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 ...
- 21
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votes
1
answer
49
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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),
...
- 155
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37
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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 $F = (f_1,f_2),$ ...
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0
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84
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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 ...
- 643
0
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0
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34
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estimating confidence interval and covariance with singular design matrix
I would like to obtain the confidence interval for a linear system $Gm=d$, except that the design matrix $G$ is singular. To solve for the model, I used the Moore-Penrose inverse. To estimate the ...
- 3
0
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0
answers
76
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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 ...
- 717
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0
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23
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Calculate the constants (transmission rate etc.) in the SIRV model without knowing S, I values calculated in the model
I want to model the pandemic using the SIRV model below:
$$\frac{dS}{dt}=\beta S(t)I(t)-\gamma I(t)$$
$$\frac{dR}{dt}=rI(t)+u(t)$$
$$\frac{dS}{dt}=-\beta S(t)I(t)-u(t)$$
Since there is no data about ...
- 21
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0
answers
28
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What is the current best method to approximate a HUGE precision matrix?
I'm wondering how efficiently estimate a precision matrix whose size would be $10^5\times 10^5$ (or even larger). Suppose we cannot naively invert the covariance matrix due to the large matrix size. ...
0
votes
0
answers
27
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What's the meaning of correlated observation in regression?
I have a model:
$$y=f(x)+\epsilon$$
where $\epsilon \sim N(0,\Sigma)$.Then we have
$$y|x\sim N(f(x),\Sigma)$$
My question is:
1.When will y is called correlated observation, such that $\Sigma$ is not ...
- 1
0
votes
1
answer
116
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
$$\begin{align*}
\lVert Tx - y \rVert^2_Y + \alpha \lVert x\rVert_X \...
- 163
2
votes
2
answers
104
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Finding the values of a matrix multiplied between two unknown matrices
This is a slightly vague question I think, but I am wondering if there is any elegant way of solving this problem.
Say I have a multiplication between three unitary matrices operating on a vector, ...
- 25
1
vote
0
answers
40
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Functional inverse problem based on a variational principle
I am trying to solve an inverse problem based on variational principle.
I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently ...
0
votes
0
answers
62
views
how to show inverse of M-by-N data matrix using singular value decomposition?
The question is from the exercise questions from the EE4595 course being taught in TUDelft. The question is as follows...
So far my attempted solution is like the following.
The discretised data ...
- 23
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votes
1
answer
108
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Linear Algebra: Solving the minimization with vector p-norm.
The course this question is from is called Wavefield imaging, It gives the vector p-norm of an N-by-1 vector $x=[x_1,x_2,...,x_N]^T$ is defined as
$$
||x||_p=\left(\sum_{n=1}^N |x_n|^p \right)^{1/p}
$$...
- 23
1
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1
answer
54
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Analytic solution inverse kinematics - different solutions with different calculation steps
I have the following problem:
$$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)}$$
I have two seemingly correct ...
- 11
1
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0
answers
37
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Least-norm problem with equality constraints
My inverse problems teacher said that the least norm problem can be resolved using equality constraints. How is that?
- 49
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1
answer
25
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If $\int_n^m{f(x,y)}dy=g(x)$, is there a way to find, or approximate $f(x,y)$ given $g(x)$
If I'm given $f(x,y)$, when $$\int_n^m{f(x,y)}dy=g(x)$$ then I know how that I can at least approximate $g(x)$ using a rieman sum, however if I am instead given $g(x)$ I don't know how to even ...
- 165
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0
answers
145
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Newton's Method for Nonlinear System with Constraints
I have a local solution of a dynamical system $\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x})$:
\begin{equation}
\mathbf{x}(t) = \mathbf{g}(t;\mathbf{A}),
\end{equation}
where $\bf{f},\bf{g}:\mathbb{R}^n\...
- 11
1
vote
1
answer
68
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One-dimensional heat conduction
I was reading a paper about one-dimensional heat conduction problem and I get stuck in one expression that I couldn't understand how to calculate.
First, they define the heat conduction problem as ...
- 23
0
votes
1
answer
35
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Isomorphic Neural Nets
The following is from Reconstructing a neural net from its output by Fefferman.
In here, I'm not sure about the notation. Are we fixing one $l$ such that $\gamma_l$ is identity, and the rest of the ...
- 1,570
1
vote
1
answer
221
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Find a $2\times2$ matrix $B$ such that $(A^TBA)^{-1}A^T=I$, where I is the identity matrix
Problem
Let $A=\begin{pmatrix}
2 & 0 \\
1 & 2
\end{pmatrix}$. Find a $2\times2$ matrix $B$ such that $(A^TBA)^{-1}A^T=I$, where $I$ is the identity matrix.
How does one go about solving this ...
- 123
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votes
0
answers
204
views
Singular Value Decomposition vs iterative Methods for solving inverse Problems
Background: I am trying to motivate why I use an SVD instead of an iterative LMS solver for the solution of an equation of the form
$$Ax=b$$
where $A\in\mathbb{C}^{M\times M}$,$x\in\mathbb{C}^{M}$ and ...
- 45
4
votes
2
answers
190
views
Prove that the phase retrieval problem is non-convex
The phase retrieval problem consists of recovering phase information from given intensity measurements, as shown in the image below from Deep phase retrieval: analyzing over-parameterization in phase ...
- 51
1
vote
0
answers
30
views
Why the more unkown variable(more underdetermined) will make non-convex optimization problem more difficult to solve?
Now I am try to solve an inverse problem. In the end it is to solve an non-convex optimization problem(we call it phase retrieval problem). The loss is $\lVert |f(x)|^2-I \lVert_2^2+\lVert x\lVert_1+...
- 51
14
votes
2
answers
742
views
Is there a Lagrangian that produces these equations? How can I find one if it exists?
Consider the two differential equations
\begin{align*}
\ddot{x}_{A} - \gamma(x_{A} + x_{B}) &= 0, \\
\ddot{x}_{B} + \gamma(x_{A} + x_{B}) &= 0
\end{align*}
where $\gamma$ is a constant.
I am ...
- 2,417
0
votes
0
answers
43
views
Solving a spatial pde with partial derivatives?
Assume I have the following function:
$$0=a(\mathbf{x})h(\mathbf{x})+b(\mathbf{x})\nabla h(\mathbf{x})+c(\mathbf{x})$$
where $\mathbf{x}\in\mathbb{R}^{d}$ are points in some $d$-dimensional space, $a$,...
- 929
4
votes
1
answer
184
views
Calculating SVD for some integral operators
I am currently trying to analytically find SVD for operators $T_1, T_2: L^2\big([0;1]\big) \to L^2\big([0;1]\big)$ using definition, where:
$(T_1f)(x) = \int_0^x f(y) \ dy$
$(T_2f)(x) = \int_0^1 K(x,...
- 165
2
votes
0
answers
56
views
How does one invert $e^{(k+i)\theta}+e^{(k-i)\theta}=e^{k\theta}+c$ to obtain $\theta$, given known $k,c$?
I need to find a real solution for $\theta$ in the following complex equation:
$e^{(k+i)\theta} + e^{(k-i)\theta} = e^{k\theta} + c$
Here $k$ and $c$ are positive real numbers.
The problem is related ...
- 51
-1
votes
1
answer
73
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Find an ODE with given functions as solutions
I am teaching a class on differential equations lately, and in the course of trying to invent problems for a midterm, I got to wondering how to generate an ordinary differential equation having ...
- 14.9k