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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3
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1answer
64 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 ...
1
vote
0answers
33 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 ...
1
vote
0answers
31 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
43 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. ...
0
votes
0answers
13 views

Fredholm integral with functions constrained to [0;1]

I am trying to feed information about the solution when numerically solving an inverse problem given by a Fredholm integral of the form $$ g(t)=\int_{a}^{b}K(t,s)f(s)ds. $$ Say I know $g(t)$ and ...
1
vote
1answer
43 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
vote
0answers
18 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, $d=\hat{d}+\eta$ ...
1
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0answers
44 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
15 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 $z=70$.
0
votes
0answers
10 views

Basis functions which allow non-negativity to be enforced?

I'm working on an inverse problem where I am trying to estimate a probability distribution $f(x)$ as a sum of basis functions: $$ f(x) = \sum_{i}^{M} c_i \phi_i(x) $$ Where the $\mathbf{c}$ are ...
3
votes
0answers
59 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 ...
0
votes
0answers
20 views

Optimum sampling of equation

While solving a ill conditioned under determined linear system Ax = b, I have huge number of equations(lets say 10^4) or say relationship between unknowns. I don't want to use all relationships to ...
2
votes
1answer
49 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
63 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
76 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
41 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, ...
1
vote
2answers
34 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
votes
1answer
63 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 ...
6
votes
0answers
322 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
vote
0answers
27 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
42 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
89 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
vote
1answer
41 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
71 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
votes
0answers
35 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
vote
1answer
111 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
votes
0answers
58 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
vote
1answer
60 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
249 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
125 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
vote
1answer
27 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
vote
1answer
72 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
80 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
votes
0answers
59 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 ...
1
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0answers
142 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, ...
3
votes
0answers
216 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 ...
1
vote
0answers
73 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
votes
0answers
44 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 ...
3
votes
0answers
275 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 ...
0
votes
1answer
113 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 ...
1
vote
1answer
514 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 ...
10
votes
3answers
2k 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 ...
1
vote
0answers
33 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
votes
1answer
57 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 ...
1
vote
1answer
78 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 ...
1
vote
1answer
57 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
votes
1answer
339 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
vote
3answers
120 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
votes
2answers
282 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
votes
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 ...