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 ...
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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 $\mathcal{X}=[0,10]$...
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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 ...
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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 ...
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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 $h$ ...
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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 $p(w|a)$ ...
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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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
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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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
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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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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 ...
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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 ...
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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(\...
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