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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28 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 }} ...
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17 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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24 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
35 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 ...
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43 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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47 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, ...
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98 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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56 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} ...
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34 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 ...
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132 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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77 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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223 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 ...
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556 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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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 ...
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1answer
33 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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65 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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46 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 ...
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330 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 ...
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3answers
108 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 ...
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2answers
211 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 ...
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33 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 ...
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1answer
106 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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23 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 ...
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547 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; ...
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1answer
344 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
414 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 ...
4
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4answers
319 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 ...