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
23 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
36 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 ...
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0answers
16 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
80 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
35 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
64 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
27 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
30 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 ...
6
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1answer
158 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 ...
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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
58 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
27 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
10 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
116 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
120 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
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4answers
142 views

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

How can we prove the following? $$\cos \left( \arctan \left( \frac{1}{2}\right) \right) =\frac{2}{\sqrt{5}}$$
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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
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1answer
71 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
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0answers
45 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
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1answer
104 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
60 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
43 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
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1answer
56 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
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1answer
177 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 ...
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0answers
27 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
193 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. ...
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votes
1answer
23 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
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0answers
104 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
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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
85 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
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0answers
139 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
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1answer
98 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
90 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
353 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 ...
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0answers
34 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
99 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
61 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
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1answer
81 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, ...
0
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0answers
49 views

alternating direction method of multipliers for nonlinear inverse problems?

I have a standard inverse problem with L1 regularization: $\|F(\mathbf{x})-\mathbf{y}\|^2_2+\alpha\|\mathbf{x}\|_1$, where $F(\mathbf{x})$ is nonlinear. I am wondering if this is a good problem to use ...
1
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1answer
148 views

Inverse problem with 4D and 2D matrix

I was trying to solve an inverse problem in mechanics and computing it in Matlab, when i found something unknown for me, and so I haven't any idea on how to compute it. Basically, after ...
4
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0answers
63 views

Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
1
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1answer
61 views

Determining whether an uncountable set of integral equations yield a unique solution

I am interested in the set of numbers $\alpha>0$ for which there exists a function $g:\mathbb{R}\to[0,1]$ satisfying $$ \forall r\in \mathbb{R} \qquad f(r) = \int\limits_\mathbb{R}\! g(\alpha ...
5
votes
1answer
390 views

When is $R \, A^{-1} \, R^t$ invertible?

In the context of a Gaussian model, I came across a matrix product $R \, A^{-1} \, R^t$ where $R$ is a $m \times n$ rectangular matrix and as implied $A$ is $n \times n$ and invertible. On which ...
3
votes
1answer
201 views

Inverse LaPlace Transform of the square root of Rational, Monic 1st Degree Polynomials

I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the ...
1
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1answer
33 views

nonnegative inverse eigenvalue problem

Is it true that for any set of $n$ distinct positive real numbers $$\lambda_1 < \cdots < \lambda_n$$ there is a real symmetric matrix $A_{n \times n}$ with eigenvalues $\lambda_1, \ldots, ...
1
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1answer
101 views

pseudo-inverse by SVD decomposition has not accurate results?

The goal is finding $\frac{{\partial f}}{{\partial {\bf{A}}}} = 0$ where $ f\left( {{\bf{A}},{\boldsymbol{\alpha }}} \right) = {\left( {{{\bf{p}}^{\bf{T}}}{{\bf{A}}^{\bf{T}}}{\boldsymbol{\alpha }} ...
1
vote
1answer
118 views

How to approximate a smooth function

Now I have a target smooth function f which is infinitely differentiable over $R^d$, $f \in C^{\inf}(R^d)$. $f = \Sigma c_ig_i(x)$, where $c_i$s are unknown coefficients and $g_i(x)$ is a smooth ...