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.

Filter by
Sorted by
Tagged with
1
vote
0answers
15 views

Approach to finding a function for a binary pattern

Below is a binary matrix of shape [64,63]. I want to find an equation to compute the values of this matrix as a function of rows and column indices. Are there any common strategies to attack such ...
3
votes
0answers
38 views

$L^2$ norm of inverse differential operator

This has come up in Lemma 1 of Mandache's 2001 paper on exponential instability for the inverse problem of the Schrodinger operator. Let $\Omega = B(0,1)$ in $\mathbb{R}^d$. Suppose $r_0\in (0,1)$ and ...
3
votes
1answer
107 views

Determine the structure of all finite sets $A$ of integers such that $|A| = k$ and $|2A| = 2k + 1$.

An exercise in Nathanson's text: Additive Number Theory, Inverse problems and the geometry of sumsets is the following (Excercise 16, P.No.37): Determine the structure of all finite sets $A$ of ...
0
votes
1answer
8 views

Condition inequality in perturbed LS

I have two matrices $A \in \mathcal{M}_{n,d}(\mathbb{R})$ and $B \in \mathcal{M}_{d,d}(\mathbb{R})$ with $B$ being symmetric definite-positive. I am trying to find a condition on $A$ for which I have ...
1
vote
0answers
24 views

Computing $L^2$ norm on a curved domain

I am currently working in a field related to elastography which tries to solve an inverse problem (of elasticity) similar to the one described below. Given a measured displacement field $u_{measured}$...
1
vote
1answer
29 views

References on parameter identification for ODE models from time-series data

Say I've got a system of ODEs, e.g. a generalized Lotka-Volterra system of the form: $$\dot x_i = x_i \left( r_i + \sum_{j=1}^n b_{ij} x_j \right),$$ where I've got $n$ ODEs and a total of $n^2 + n$ ...
2
votes
1answer
34 views

Given the distance between more than 3 points and no prerequisite for coordinate values, can the distance formula be reversed?

Very similar to this, but the accepted solution given only works for 3 points, not n. I have a filled-out logic grid of distances between 16 points and I would like to plot a plausible configuration ...
0
votes
0answers
22 views

Equivalence of two Riemannian metrics

So I have this problem on a course on geometric inverse problems. I am working in $\mathbb{R}^n$. I have the metrics $$g_1=c(r)^{-2} dr^2+(r/c(r))^2 d\theta^2$$ and $$g_2=a(r)^2dr^2+b(r)^2d\theta^2.$$ ...
0
votes
0answers
18 views

constraint optimization solver

I have the equation bellow which I want to solve for $I$ with constraints on $I$. where $I$ is a vector and $ lb \leq I_{i} \leq ub $ . Note that $ size(I)=(N,1)$ , $ size(A_{m})=(N',m)$ , $ size(K)...
-3
votes
2answers
106 views

How can I solve $A\operatorname{diag}(x)=B\;?$

How can I solve $A\operatorname{diag}(x)=B\;?$ I am actually an engineer and recently involved in this field. Thanks.
2
votes
0answers
25 views

Radon transform maps a Schwartz function to a differentiable function

This isn't supposed to be hard, but I just can't seem to be able to write this down. My Radon transform is defined as follows: Let $f\in C_c^{\infty}(\mathbb{R}^2)$, then $$Rf(s,\omega)=\int_{\mathbb{...
0
votes
0answers
23 views

Conjugate search directions in BFGS algorithm - Convex Optimization

I have been studying optimization techniques specifically the BFGS algorithm and often in papers the result is given that the set of search direction vectors $p = \{ p_1, p_2, ... , p_n \}$ are ...
1
vote
0answers
61 views

Unboundedness of complex $R^{-1}$ implies unboundedness of real $R^{-1}$?

Define the Radon transformation as $Rf(\varphi,s) = \int_{x \in L(\varphi,s)} f(x) dx_L = \int_{t=-\infty}^{\infty} f(s\theta + t\theta^\bot)dt$ where $\theta = \theta(\varphi) = (\cos \varphi, \sin \...
1
vote
0answers
26 views

Find a point on function graph that has a fixed arc-length distance from another point [duplicate]

The problem (as often is in math) is finding $x$: \begin{equation} d = \int_{x_0}^x\sqrt{1+[f'(t)]^2} \ \text{dt} \end{equation} It's an inverse problem, and I'm looking for an approximation of the ...
0
votes
0answers
5 views

How can i find Jacobian $J_r$ in the integral?

I am studying for linear integral equation but i dont understand how to find jacobian $J_r$. Could you help me please? Any help appriciated. Thank you in advance. Single layer operator is defined as ...
0
votes
1answer
43 views

Inverse Laplace transform of a multiplication of a constant matrix and an exponential matrix

Let $\mathcal{L}^{−1}\{⋅\}(z)$ be an inverse Laplace transform. Let $A, B$ be square matrixes, $I$ an identity matrix, and $\hat{\Phi}_z = \mathcal{L}\{\Phi(t)\}(z)$. I have: $\hat{\Phi}_z = A[Iz - B]...
0
votes
0answers
20 views

Understanding Linear Transformations for Optimizing Stability of Inversion

In an engineering context, I'm working with a system of the form: $$f = Ax + \eta$$ where $f$ is the observed data, $A$ is a linear transformation that models a physical system, $x$ is some unknown ...
1
vote
0answers
20 views

Inverse eigenvalue problem for arrowhead matrix

Consider a diagonal $N\times N$ matrix $\boldsymbol{H}=\text{diag}(\omega_1,\omega_2, \dots,\omega_N)$ with ordering $\omega_1<\omega_2<\dots<\omega_N$, where the eigenvector $\boldsymbol{t}...
4
votes
1answer
33 views

Can we reconstruct a function $f$ by knowing its scalar product with its own shift, $\langle T_xf,f \rangle$?

Assume $f:\mathbb{R} \to \mathbb{C}$ is a function, let's say square integrable. Assume we know the values $$ a(x) = \langle T_xf,f\rangle = \int f(y-x)\overline{f(y)} \, dy, $$ for $x \in \mathbb{R}$...
1
vote
0answers
55 views

Radon transform inversion formula

I have trouble following Deans's derivation of the inverse Radon transform formula for $n=2$ on this page of his book "The Radon Transform and Some of its Applications" (see snapshot) Formulas (3.9) ...
2
votes
0answers
33 views

DFT and inverse problem

Let two functions $I,J:\{0,1,\ldots,n\}\times \{0,1,\ldots,m\} \mapsto \{0,1,\ldots,p\}$ and $h$ a $v\times v$ matrix ($v\ll n,m$) such that $$ J = I * \underbrace{h*\ldots *h}_{p \text{ times}} $$ ...
0
votes
1answer
46 views

Inverse function of function with 2 variables.

Maybe this question has possible duplicate or something, but i still don't get it. Suppose i have this function : $$f(x,y)=x+y$$ What is the inverse of this? Is it possible? Bcz, i'm not learning ...
0
votes
2answers
29 views

Can absolute max eigenvalue of A and B tell whether A+B is invertible?

The teaching assistant explained to me that if $A,B\in\mathbb R^{nxn}$ are symmetric, $|\,\text{max eigenvalue of }B\,|\le1\le2\le|\,\text{max eigenvalue of }A\,|$, and $|\,\text{min eigenvalue of }A\...
2
votes
0answers
73 views

Textbook recommendation on Inverse Problem Theory

I am looking for a textbook on inverse problem theory, which includes both Regularization and Bayesian approach for inverse problem solving, including data assimilation, preferably machine learning ...
1
vote
0answers
28 views

Order Reduction of a field

I have a problem with a static (time invariant, but spatially varying) field expressed only as a set of discrete set of values at some points of an arbitrarily shaped domain in 2 or 3 dimensions. For ...
2
votes
1answer
21 views

Estimate Signal by Its Convolution by 2 Different Kernels

I have a discrete Signal $s$ that has been convoluted with two functions $h_1$ and $h_2$. I measure the result of this convolution: $$y_1=s*h_1, \quad y_2=s * h_2.$$ I have a short time segment (for ...
0
votes
0answers
37 views

what property between two matrices affects on the trace of multiply of them

I have two matrices $A$ with dimension $n\times n$ and $B$ with dimension $n\times m$ that $m<n$. In my problem matrix, $A$ is fixed and given, but the elements of the $B$ are variables that can ...
1
vote
0answers
22 views

Why do we choose $\lambda,\mu \in C^1(\partial D)$ and $\partial D $ in the class of $C^4$ for this problem?

I am reading article with title " Integral equation methods in Inverse Obstacle scattering with a Generalized Impedance Boundary Condition" written by Rainer Kress. The problem is fomulated in the ...
2
votes
1answer
33 views

X=Y+Z, know the distribution of X, under which conditions can I recover(pin down) the distribution of Y and Z?

You may add any conditions you want if this is doable. Otherwise, it would be great if you could provide any impossibility result. Also greatly appreciate it if you could guide me to related problems,...
1
vote
0answers
91 views

Directional derivative of cost function under optimization. Finding bounds on decision variables.

Given the optimization problem \begin{align} \boldsymbol{ x}^*, \boldsymbol{y}^* &= \min_{ \boldsymbol{x}, \boldsymbol{y}}( V( \boldsymbol{x}, \boldsymbol{y}))\\ s.t: \quad & \...
1
vote
0answers
13 views

Accuracy in Bayesian updating

Hello I have a question about the accuracy in Bayesian updating. I use the following procedure to compute a posterior distribution: Generate synthetic measurement data, i.e. I know the true mean and ...
0
votes
1answer
55 views

Inverse Problem for positive definite matrices

Does there exists a positive definite matrix $A^H = A\in \mathbb C^{n \times n}$ under some condition such that for a given $x,y \in \mathbb C^n$, where $x \neq 0$. $Ax = y$ holds good. What I have ...
2
votes
1answer
75 views

Discretising the Fourier Integral gives a high condition number

I have the following integral equation $$ \int_0^1 e^{-2\pi i s t} f(t)\, \text{d} t = g(s), \hspace{3em} -1/2 \leq s \leq 1/2$$ where $f$ is to be found, and $g$ is known. I believe this problem is ...
5
votes
2answers
80 views

Can we determine summands from their partial sums?

Assume there are non-negative numbers $\lambda_1\le \ldots\le \lambda_n\in[0,\infty)$. You are given the (ordered) list $s_1\le\ldots\le s_{2^n}\in[0,\infty)$ of all partial sums, i.e. every $s_i$ is ...
2
votes
0answers
15 views

Finding the domain shape, where the fundamental Helmholtz Eigenmode has certain properties at a boundary

The following is a problem about finding the shape of a domain such that the fundamental Helmholtz eigenmode has certain properties at the boundary of the domain. I search for an analytical or ...
1
vote
0answers
35 views

Can anyone help with the inverse problem and tuning parameters

For my final year project i want to model the population of London using the Verhulst logistic model. However, to gain more marks i wish to use the inverse problem to tune the parameters to make the ...
3
votes
1answer
103 views

How to determine Tikhonov regularization parameter using standard deviation?

I have a linear ill posed problem with the form: $$Ax=b.$$ One approach to this problem is Tikhonov regularization, replacing it with $$\min_x( \|Ax-b\|^2+\alpha\|x\|^2 ).$$ In https://en.wikipedia....
0
votes
0answers
25 views

Finding conditions such that the following inequality holds.

Let $ \boldsymbol{J} \in \mathbb{R}^{m \times n} $, with $ n>m $ be an invertible full row-rank matrix. Further: \begin{equation}\label{key} f= |\boldsymbol{J}^{-1} \boldsymbol{1}_{\pm}|_1 \...
2
votes
1answer
60 views

Radon transform expressed via delta distribution

The Radon transform for $f\in C_c^\infty(\mathbb{R}^n)$ is defined by \begin{equation} Rf(\theta , s) := \int _{\{ x^T\theta =s \}} f(x)dx. \ \ \ (1) \end{equation} Now it says, that an equivalent ...
3
votes
0answers
84 views

Construct operator with given spectrum. [duplicate]

The Spectrum of a bounded operator on a Banach space $X$ is always a compact subset of $\mathbb{C}$. What about the converse? Given any compact subset $K \subset \mathbb{C}$ is it always possible to ...
1
vote
1answer
61 views

On a nonlinear regression problem

Consider the function $f\colon \mathbb{R}^2\to \mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data $\{(x_1^{(i)},...
0
votes
0answers
59 views

What is the compactness uniqueness argument

I have come across the term "compactness uniqueness argument" in many inverse problems papers and books but the proof are usually omitted. Could anyone give such an example and relevant references?
1
vote
1answer
44 views

Is there a reason for why some problems reduce to Caldéron's inverse conductivity equation?

Is there a reason for why some problems reduce to Caldéron's inverse conductivity equation? Or is it "coincindence"? Such problems include, optical tomography, inverse scattering. E.g. in https://...
2
votes
0answers
2k views

How is the Jacobian matrix computed in finite difference problems?

I have come across many papers which reference the Jacobian when solving certain finite difference inverse problems. And I have seen many articles and textbooks which discuss the mathematical ...
2
votes
0answers
34 views

Find the solution of an outer product induced system

Sorry if the question is lame, but I'm struggling to find the answer to the following problem: Given a matrix $A\in \mathbb{R}^{n,n}$ and a column vector $b\in \mathbb{R}^{n}$, how can one find the ...
3
votes
0answers
83 views

About $ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $

We know for $ u > 1 $ $$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$ Usually this is shown by using differentiation under the ...
-1
votes
1answer
52 views

Inverse function on matrices with help of Cayley-Hamilton theorem?

I have been thinking about inverse functions of matrices lately. (Yees yees, I know usually for anything more complicated than reals we need to define/select branch and for reals to select interval ...
3
votes
0answers
45 views

Integral equation in polar coodinate system

I need an inversion formula with the form $f(r)=\cdots$, from this integral relation: $$g(r)=\frac{1}{2\pi}\int_0^{2\pi}d\theta\,f\left(\sqrt{r^2+r_0^2-2rr_0\cos\theta}\right)$$ where $r_0\geq0$ is a ...
2
votes
0answers
92 views

Harmonic functions interpolation

Denote by $B = \{(x,y) \in \mathbb{R}^2|x^2 + y^2 < 1\}$ the open unit ball in $\mathbb{R}^2$, and by $S$ it's boundary, i.e. the unit sphere. For some $n > 0$ let $x_1,...,x_n \in B$, $y_1,...,...
2
votes
0answers
41 views

Uniqueness of charge distribution

Some exercises in general physics ask for students to find specific charge distributions on the boundaries of given conductors in various of situations. Usual answers goes like follows; firstly using ...

1
2 3 4 5