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 ...

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1answer
41 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
1
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0answers
7 views

Taking integration of a forward problem in a region of interest which is disretized-Microwave Imaging

I am trying to implemented the below formula in order to find the integration in matlab. However, I do not know how to do change of variables. The formula is ...
0
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0answers
9 views

Reconstruct a family of probability distributions having certain generalized hypergeometric moments

Reconstruct and/or otherwise characterize any/or all members of a certain one-parameter ($\alpha =\frac{1}{2}, 1, \frac{3}{2}, 2,\ldots$) family of univariate probability distributions (of ...
2
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0answers
41 views

An inverse problem for a parabolic equation and fixed-point theorem

I'm reading the paper "The determination of a parabolic equation from initial and final data" http://www.ams.org/journals/proc/1987-099-04/S0002-9939-1987-0877031-4/S0002-9939-1987-0877031-4.pdf. A ...
0
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0answers
42 views

Inverse spectrum problem - showing the existence of a 2x2 doubly stochastic matrix,

I am working through a couple of problems in Henryk Minc's book, Nonnegative Matrices, as a warm-up to understanding the inverse spectrum problem. This is Exercise 18 of Chapter VII of his book: ...
3
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1answer
79 views

An inverse spectrum problem in linear algebra,

I am reading the book, Nonnegative Matrices, by Henryk Minc, and came across an exercise that I would like to solve: Let $$\bar \sigma = (\bar\lambda_1, ... , \bar \lambda_n)=(\lambda_1, ... ...
1
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1answer
25 views

Inverse problems with Graphical Approximation and Graphs

Suppose an inverse problem with graphical approximation for the system where only a small subset of system features are known hence undetermined scenario. The system can be represented by a graph. ...
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0answers
23 views

References on Inverse Problems, Approximation theory and Algebraic geometry

For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined ...
1
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1answer
43 views

Direct numerical solutions for first kind Volterra integral equations

For clearly deliver my purpose, I rewrite this question. Consider first kind Volterra integral equations $$ \int_0^t k(t,s)f(s)ds=g(t) \quad 0\leq t\leq T $$ where $k(t,s)$ is continuous but not ...
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1answer
48 views

Cartesian to Spherical coordinate conversion specific case when Φ is zero and θ is indeterminant

Following is the conversion for spherical to cartesian coordinate \begin{align} x &= r \cos\theta \sin\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\varphi \end{align} and we are ...
1
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0answers
45 views

Can one hear the *material* of a drumhead?

"Can one hear the shape of a drum?" is a well known problem, originating from Kac, 1966, that questions whether an (idealized) drum head is completely specified by its spectrum. That is: is the ...
0
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1answer
38 views

how can we solve this equation without mellin transform?

given the equation (functional equation) $$ f(x)+f(2x)+f(3x)+.... =g(x) $$ we can use the Mobius tranform to obtain $$ f(x)=\sum_{n=1}^{\infty}g(nx)\mu(n) $$ however, what can we do with the ...
0
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0answers
22 views

Reference request: Inverse problem with stochastic error term

In many inverse problems there is an an error term resp. disturbance like $\|{y_\delta} - y \| \le \delta$ with noise level $\delta$, because only noisy data $y_\delta$ are known. Now I'm interested ...
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0answers
85 views

Relationship b/w PDE periodic boundary values

Consider the following homogeneous boundary value problem for function/potential $u(x,y)$ on the infinite strip $[-\infty,\infty]\times[0,\pi/4]$ w/positive periodic coefficient/conductivity ...
1
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1answer
37 views

Can I invert this simple nonlinear equation?

Suppose $y = x * B(x)$, where $x$ is a 2d array of positive nonzero real numbers, $*$ denotes pointwise multiplication, and $B(x)$ is a blurred version of $x$, that is, $B(x)=x \otimes p$, where $p$ ...
2
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3answers
93 views

How to solve for angle for simultaneous “additive trigonometric” equation

I've been trying to study concepts from the field of inverse-kinematics, but have run into a mathematical roadblock. To solve for an angle given a number is quite simple in itself $$ \sin(\theta) = ...
2
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0answers
29 views

Why does this algorithm converge?

Consider the following problem. Let $p_1, \dots, p_n \in (0,1)$ such that $\sum p_i = 1$. Let $m > 0$ such that $$ q_i := p_i + m \frac{p_i \log(p_i)}{\sum p_k \log(p_k)} < 1 $$ Suppose ...
1
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1answer
32 views

Jacobian of inverse of matrix $A(x) \in \mathcal{M}_{7\times7}$?

I have a matrix $A(x)$ where $x\in \mathbf{R}^{7}$. I have to calculate $\frac{\partial}{\partial x}A(x)^{-1}$ and then I will evaluate it at some $x_{0}$. Now this matrix is very dense so its not ...
5
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1answer
167 views

Please, help to identify this numerical constant

I'm trying to find an answer to this question. Let $K(k)$ be the elliptic integral of the first kind and $K'=K(\sqrt{1-k^2})$. According to Abel's theorem (see this link) we know that if ...
0
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0answers
4 views

General solution for $\lambda F - D_m D_p^+F = f$ is $F(x) = F_0(x) + c_1 u_0(x) + c_2 u_1(x)$. What happens when $u_i \notin C_I$

In the book of Petr Mandl Analytical treatment of one dimensional markov processes in page 34 one reads: The general form of solution $$F(x) = F_0(x) + c_1 u_0(x) + c_2 u_1(x)$$ in the case ...
1
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1answer
59 views

An asymptotic estimate for density of eigenvalues

this is the screenshot of the useful part of the cited book Let $\{\lambda_n\}$ be constants such that ${\lambda_n}=n^2\pi^2+\int_{0}^{1} q(t)\,dt +c_n \qquad \text{for} \quad n\rightarrow \infty$ ...
0
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1answer
29 views

Scattering data and IST

I'm looking into the IST (inverse scattering transform), in particular, its application to the kdv eq. I've picked a particular i.c. (though I don't think it's relevant to my question) for the ...
0
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0answers
12 views

Real ordinary differential operator and lenght element (Sturm-liouville operator)

The following passages are taken from the book Inverse Boundary Spectral problems - Alexander Katchalov, Yaroslav Kurylev, Matti Lassas pages 16-18 As we look at the vibrating string problem, We are ...
6
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2answers
120 views

Which theory is used to calculate the position and energy of a point source?

Consider an empty room with one point source that emits a stationary signal (constant sound, radioactive radiation, ...). The energy nor the position of the point source is known. We send someone in ...
4
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0answers
122 views

Given $g$ find an $f$ which is solution for $L f = g$. How do I do this?

I am learning about Stochastic processes. To characterize uniqueness of solutions to a given Stochastic differential equation, I need to find for each continuous function $g :\Bbb{R}^2_+ \to \Bbb{R}$ ...
2
votes
4answers
201 views

Prove that $\cos \arctan 1/2 = 2/\sqrt{5}$

How can we prove the following? $$\cos \left( \arctan \left( \frac{1}{2}\right) \right) =\frac{2}{\sqrt{5}}$$
1
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1answer
29 views

Methodologies for non-continuum inverse problem

I am solving an ill-posed inverse problem and having a difficult time researching related methodologies because I don't know the appropriate jargon/nomenclature. I have a system with several ...
2
votes
1answer
77 views

What is an Inverse problem in Mathematics?

I have come accross a lot of articles that talk about inverse problems. However, I dont really appreciate the uses due to my poor understanding of the notion. From the mathematics point of view, when ...
0
votes
0answers
46 views

Approximate inverse (or fast optimization) of non-linear least squares problem

Problem Statement Let ${\bf x}\in\mathbb{R}^N$ and ${\bf W}\in\mathbb{R}^{K\times N}$, ${\bf V}\in\mathbb{R}^{N\times K}$. We define $${\bf y} = f({\bf x}) = [{\bf V}[{\bf Wx}]_+]_+$$ where $[.]_+ = ...
3
votes
1answer
118 views

Find desired root (based on certain constraints) while using Newton-Raphson method

My function is a vector of dimension 6, and on using Newton Raphson method, the solution usually converges to the nearest root. However, I know that my function has multiple roots (it's an Inverse ...
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0answers
67 views

$L^{\infty}$ estimate on the boundary by means of Sobolev norms in the interior

I am studying this paper http://143.248.27.21/mathnet/paper_file/washington/gunther/cpde.pdf and I have a doubt I have not been able to solve since yesterday. In page 14, just before the last bunch of ...
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0answers
44 views

Explicit solution for conductivity equation with discontinuity

Let $B$ the unit disk in $\mathbb{R}^{2}.$ Let $B_{\epsilon}$ be the disk centered at the origin at of radius $\epsilon.$ Let me define $\displaystyle ...
1
vote
1answer
57 views

What is a good example of an algorithm that is hard to parallelise?

When I have 10 computers, the factorization of a number doesn't scale along. I am not sure how much faster it would go compared to a single computer, but not 10 times faster like one would expect. ...
1
vote
1answer
238 views

How to calculate the abduction and flexion angle from 3D co-ordinates of shoulder and elbow

I am using a "Microsoft Kinect for Windows" to track the $x$, $y$, and $z$ positions of my Shoulder, Elbow, Wrist, and Hand. Each position is normalised to my shoulder so that my shoulder is always at ...
1
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0answers
29 views

About consistency in an inverse problem formulation

I'm a beginner with inverse problems and I was reading about regularization techniques. Consider the problem: $$d=Kf_{\text{true}}$$ $d$ is a data vector, $K$ is an linear operador, $d=\hat{d}+\eta$ ...
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0answers
274 views

How can I invert/reverse a curve/ease function?

I have a range of values that represents a curve. This in turn is applied in programming to an interface - rotatable knobs to be precise. Let's say you have a knob that represents a value from 1-20. ...
-3
votes
1answer
24 views

If $z$ is inversely proportional to $x$ and $z=5$ when $x=7$, find the value of $x$ when $z=70$ [closed]

If $z$ is inversely proportional to $x$ and $z=5$ when $x=7$, find the value of $x$ when $z=70$.
3
votes
0answers
125 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
2
votes
1answer
53 views

How do I accurately find a point within a grid of 4 based on varying values?

I am specifically trying to find a source of heat, with heat sensors giving readings from 4 different points eg below: The diagram above represents a 4 x 4 grid where the numbers are heat sensors ...
3
votes
1answer
92 views

Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
2
votes
0answers
160 views

Phase retrieval via SDP (semidefinite program) of 2D test image (Matrix completion)

In SDP based phase retrieval we have intensity measurement (abs(FFT2(x))^2) of the form A(xo) = |ak,x|^2 =b^2, phase retrieval is then find x that obeys A(xo)=b The quadratic measurements can be ...
0
votes
1answer
121 views

How to calculate this matrix in component-form? (Undergrad)

If ${A}_{ab} = \delta_{ab} + \varepsilon_{abc}n^c$ and $B^{ab} = \frac{1}{1+n^2}(\delta^{ab} + n^an^b - \varepsilon^{abc}n_c)$ what is the correct way to evaluate $$C^{ab} = (AB)^{ab} $$ Here, ...
2
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2answers
46 views

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator ...
0
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1answer
96 views

Inverse Matrix Methods to find Nash Equilibriums

The board of directors of two companies determines the salary of its CEO according to the following reaction functions: S1 = 100,000 + (1/2) S2 S2 = 70,000 + (2/3) S1 Where Si is salary of company ...
7
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0answers
358 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
1
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0answers
35 views

Least squares problem equivalent to solving Poisson problem for graph embedding given edge lengths

Suppose we are given a set of edge lengths $\{e_j\}$ and want to recover vertex positions $\{x_i\}$ of a valid graph embedding that realizes the given edge lengths as best as possible. More precisely, ...
5
votes
2answers
43 views

Find $J: S^2\rightarrow \mathbb{R}$, if given $I:S^2\rightarrow \mathbb{R}$, s.t. $I(\vec{a})=\int_{S^2}\vec{n}\cdot\vec{a} J(\vec{n}) ds$.

I thought of the following problem, when we were discussing radiation intensity in an astrophysics lecture. Suppose $\mathbb{R}^3$ is filled with uniform radiation, i.e. there is a function ...
3
votes
2answers
100 views

Volterra equation for a Bessel type IVP that appears in inverse scattering

I have the following i.v.p. (Colton Kress-Inverse acoustic and electromagnetic scattering theory, Springer) $$y''(r)+(k^2n(r)-\frac{l(l+1)}{r^2})y(r)=0$$ $$y(0)=0, y'(0)=1$$ using the Liouville ...
1
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1answer
68 views

About Radon Transform

Recently I got to Know that, Radon Transformation has huge contribution to Computer Tomography. So I would like to know about Radon transformation in Mathematical point of view. Can any one suggest ...
1
vote
1answer
83 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...