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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18 views

'A' transpose inverse equals to 'B' transpose

I searched everywhere but I could not find a solution to this problem. Let $A$ and $B$ be invertible matrices with $AB = I$. Show that
5
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0answers
191 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
13 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
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2answers
39 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
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2answers
79 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 ...
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1answer
29 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 ...
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1answer
48 views

How to find the inverse Laplace transform of $s/(s^2+s+1)$? [closed]

How to find the inverse Laplace transform $\displaystyle L^{-1} \left\{ \frac s{s^2+s+1}\right\} $ ? Can someone explain this question I don't really understand it.
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1answer
55 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, ...
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0answers
23 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
51 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 ...
3
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0answers
27 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 $g(x)$ ...
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1answer
56 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 ...
4
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1answer
125 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
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1answer
53 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 ...
0
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1answer
19 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, ...
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0answers
42 views

how this computes the eigenvalues?

I've read somewhere that the following iteration referred to as "inverse orthogonal iteration" (I don't know why?) can be used to compute the $p$ smallest eigenvalues of $A$ in absolute value. I ...
1
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1answer
50 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 }} ...
0
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0answers
24 views

Iteratively solve linear equations with rank-1 updates on LHS and RHS

What is the best way to iteratively solve updating equations of the form $$ Ax=b $$ $$ (A+c_1v_1^\intercal)x_1=b+ \alpha_1 d_1 $$ $$ (A+c_1v_1^\intercal+c_2v_2^\intercal)x_2=b+\alpha_1d_1+\alpha_2d_2 ...
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0answers
37 views

Total Variation minimization problem

Thanks for reading this thread. I have a object function, with constraints, I am trying to minimize. The object function is the Total Variation of an image. The Total Variation is defined as: ...
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1answer
54 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 ...
3
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0answers
53 views

Sources on Inverse Spectral Theory

I follow the book "Inverse Spectral Theory" by J. Pöschel and E. Trubowitz in which functional analysis is very much involved. I am curious about another approach using more real analysis and theory ...
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0answers
98 views

Second-Order Tikhonov Regularization

In the second-order Tikhonov regularization approach $\min\left\|Gm - d \right\|_2^2 + \alpha\ ^2 \left\|\Gamma x\right\|_2^2$ (1) given that $\Gamma\ $ contains second order derivatives, ...
2
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0answers
147 views

Inverse vs. adjoint operators

I hope this is enough of a question (and less of a start of a discussion) to be allowed here. I am trying to get my head around the ideas of inverse and adjoint operators. To keep it simple, let's ...
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0answers
64 views

How to solve for the matrix in a set of equations involving the matrix exponential?

I was wondering how to solve the following problem (in a least-squares sense): $$ \mathbf{y}_1 = e^{Ax_1} \mathbf{y}_0 \\ \mathbf{y}_2 = e^{Ax_2} \mathbf{y}_0 \\ \vdots\\ \mathbf{y}_n = e^{Ax_n} ...
2
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0answers
38 views

Inferring a probability distribution from another probability distribution

Let $A$ and $B$ be real-valued random variables, with $f_A$ and $f_B$ their probability density functions. Let's say we can observe the values of $A$ many times and estimate $f_A$ fairly precisely. We ...
2
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0answers
210 views

Iterative solver for the pseudo inverse

I have got the following equation: $\begin{bmatrix} \hat{c}\\ \hat{\lambda} \end{bmatrix}=\begin{bmatrix} B^T \cdot B & H^T\\ H & 0 \end{bmatrix}^+\begin{bmatrix} B^T \cdot Y\\ 0 ...
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1answer
96 views

Roughning matrix format for the first-order Tikhonov Regularization (inverse problem)

I have been trying to solve the regularized least square problem of min||Gm-d||^2 + a ||Lm||^2 using first order Tikhonov regularization method. the general ...
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1answer
361 views

How would you solve a Tikhonov Regularized Least Squares problem with nonnegative constraints?

For a Tikhonov Regularized Least Squares problem with nonnegative constraints, what are some methods that solve it? Are methods solving a Least Squares problem with nonnegative constraints and the ...
6
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1answer
969 views

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
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0answers
29 views

Inverse of $(U^H X U + D)$ where U is unitary, X and D diagonal

Given complex unitary matrix U, and full rank diagonal matrices X and D with positive entries. I'm looking for an efficient way to compute: $(U^HXU+D)^{-1}$ The matrix inversion identity doesn't ...
2
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1answer
44 views

Confusion related to inverse problems in statistics

I am getting started with inverse problems in statistics. However, I didn't something related to it. I was reading this paper http://math.uni-heidelberg.de/studinfo/reiss/CavalierInvProb.pdf. It ...
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1answer
74 views

A question on the equivalence of an inverse problem and a probabilistic model

Suppose a model, $$y = x + \eta$$ In the engineering terms, think of $y$ as the observed data, $x$ is the desired (unknown) object, and $\eta$ as the noise. So, we have a noisy observation. Suppose ...
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1answer
50 views

Existence of solution for degenerate equation

Suppose all arguments are in $\mathbb{R}^3$, $G(x):\mathbb{R}^3\rightarrow \mathbb{C}$ is a smooth function vanishes at point set $x_j\in\Omega$$j=1\cdots,m$,with multiplicity as 1, And the equation ...
4
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1answer
334 views

Existence of function

[also asked here:http://mathoverflow.net/questions/122191/existence-of-a-function] All arguments are in $\mathbb{R}^3$. Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a ...
1
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3answers
116 views

Recover filter coefficients from filtered noise

I have a digital signal which may be represented as noise filtered with an FIR (finite impulse response) filter. Let us suppose that the noise consists of pulses (nonzero samples on a zero ...
4
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2answers
239 views

My covariance matrix is computationally singular. Does it make sense to use the pseudoinverse instead?

I have a large covariance matrix, something like 1000 x 1000. The matrix is not singular, but rather computationally singular due to approximations taking in inversion algorithms. Does it make sense ...
4
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0answers
39 views

inverse spectral problem, how to recover the function $ q(x) $

given the problem of a second order sturm liouville operator $$ - \frac{d^{2}}{dx^{2}}y(x)+q(x)y(x)=zy(x) $$ with the boundary conditions $ y(0)=0=y(\infty) $ if i know the spectral meassure ...
0
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1answer
135 views

Gelfand-Levitan -Marchenko method and Sturm-Liouville operator

Given the Sturm-Liouville operator $$ - \frac{d^{2}}{dx^{2}}y(x)+y(x)q(x)=zy(x),$$ my question is how to use spectral data to obtain $ q(x) $ inside the last equation by the ...
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0answers
24 views

entire function and second Order Schroedinger (Stur Liouville) operator

from the physics and mathematics we now that $$ \frac{\sin(\sqrt u)}{\sqrt u} $$ $$ J_{l}(\sqrt u) $$ have only real zeros on the other hand these functions are entire, have no poles, only zeros ...
3
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2answers
912 views

Inverse problem from pdes

A linear inverse problem is given by: $\ \mathbf{d}=\mathbf{A}\mathbf{m}+\mathbf{e}$ where d: observed data, A: theory operator, m: unknown model and e: error. To minimize the effect of the noise; ...
4
votes
1answer
446 views

How do I numerically calculate a function from its noisy gradient using “global integration”?

I have the model $\ s(x,y)=x^2+y^2, 0 \leq x \leq 1, 0 \leq y \leq 1 $. Instead of observing the model directly I am observing the derivatives of the model + some noise (e): $\ p(x,y)=s_x+e, ...
3
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1answer
550 views

Unstable linear inverse problem: which “dampening” Tikhonov matrix should I use?

A linear inverse problem is given by: $\ \mathbf{d}=\mathbf{A}\mathbf{m}+\mathbf{e}$ where d: observed data, A: theory operator, m: unknown model and e: error. The Least Square Error (LSE) model ...
5
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4answers
381 views

a good book on inverse problems for engineers

I'm looking for a book on inverse problems which is suitable for engineers, both introduction and practical applications are required. Currently I'm looking to Parameter Estimation and Inverse ...