Questions tagged [inverse-problem]

For questions pertaining to methods to estimate input parameters based upon output data.

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67 views

Inverse problem with uncertain forward operator

Suppose I want to solve a linear inverse problem. In this example we take a convolution with the kernel: $$\frac{1}{(y^2+z^2)^{3/2}}$$ We only take a fixed $z$ for the computation and convolve with ...
3
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2answers
105 views

Optimization of expensive model with many parameters

I have a physical model which takes $\sim50$ parameters and gives $\sim2000$ outputs taking tens of minutes to run. I need to optimize these parameters to give outputs as close as possible to data. ...
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2answers
185 views

Different questions about “Inverse Physics problems”

I am in a context of forecasts in astrophysics. Don't be too rude if questions seem to you stupid or naive but rather indulgent, I am just looking for better undertsand all these numerical methods of ...
6
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1answer
90 views

Is there any theory of the minimum amount of data for tomographic reconstruction?

I'm doing an experiment on synthetic data and I want to generate enough data but not too much. So I wonder if there is any rule for the minimum number of projection angles and detector count. For ...
4
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2answers
187 views

Compute point-spread-function between original and blurred image

Take an image $f$ with some characters on it (below, hjFu3). Let's apply a filter $h$ on it to obtain a second image $g$ where the text is not visible. Is there a way to compute what kind of filter $h$...
3
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2answers
101 views

How to reconstruct a 2D field from its integral?

General question I work on the plane where I have a two-dimensional shape $V$ that is cut in a collection of parts $\{V_i\}$ that do not overlap $ V_i ~~\text{s.t.}~~ \bigcup_i \overline{V}_i = \...
3
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1answer
72 views

Smoothness regularisation of a 2D field on a triangular mesh?

I'm working on an inverse problem where the solution is the values of a 2D scalar field at the vertices of a 2D triangular mesh, such that the field can be defined continuously inside the mesh via ...
6
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1answer
113 views

numerical solution of an under-determined linear equation in high dimensions

I need to solve a linear regression problem $$Ax=y$$ which is hugely underdetermined. I have around $10^6$ features but only $10^3$ equations. So $A$ is a $1,000\times 1,000,000$ matrix and $y$ a ...
2
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1answer
329 views

Finding probability vectors from an implicit equation

I have $q$ $n$-dimensional vectors $\vec y_i$ and a matrix $\hat B$ of shape $n\times m$. I'm looking for $q$ $m$-dimensional vectors $\vec x_i$ such that: $\vec y_i=\hat B \vec x_i$ each vector $\...
5
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1answer
105 views

Numerical error of a spectral-domain Poisson solver

In $\mathbb{R}^n$, I would like to solve a Poisson equation (given $f$, solve for $u$): $$\nabla^2 u = f$$ assuming Neumann boundary condition (i.e. $\partial u = 0$ at boundaries). I solved it in ...
4
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1answer
289 views

adjoint method for reaction-diffusion problem

I'm trying to code a parameter estimation for a reaction-diffusion problem. Namely, knowing the distribution of tumor density $u$ at time $0$ and $T_f$ ($u^0$ and $u^f$), what are the best ...
5
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2answers
177 views

Algorithm for finding initial conditions of differential equations given trajectory

Let's say I'm given a system of three first-order differential equations in three variables, where all of the equations are known, and we additionally know the trajectory of two of the variables at a ...
12
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1answer
362 views

Inverse problem in linear ODE

I have a linear ordinary differential equation (ODE) with a system matrix with constant coefficients: $$\dot{y}(t) = \mathcal{A}\; y(t), \quad y(0) = y_0$$ with $y(t) \in \mathbb{R}^{n \times 1}$ and $...
11
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1answer
722 views

First appearance of the phrase “inverse crime”

In research on inverse problems, it's common to construct a synthetic data set from a known set of parameters and then test whether the inversion technique can reconstruct those parameters. In doing ...
1
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0answers
80 views

Can x-ray back-projection be converted to hard-field magnetic induction tomography?

This is a question about hard-field back-projection as used in x-ray tomography, applied magnetic induction tomography. Al-Zeibak and Saunders have shown that x-ray filtered backprojection can be ...
4
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0answers
102 views

Inverse problems with a discrete set of known parameters

What are the techniques on inverse problems to discover the distribution of parameters from a discrete set of values? For instance, I know that my domain where the PDE is defined is made up of ...
3
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1answer
686 views

Regularization vs constrained optimization of an ill posed tomography problem

I am trying to solve an ill-posed linear system of equations. The particular system has 160 equations and 400 variables. Moreover, the condition number of the left hand side matrix is of order $10^{16}...
1
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1answer
161 views

Computing preconditioner for a non-linear conjugate gradient implementation

Consider the following steps for the $i$-th non-linear conjugate gradient iteration, in the context of 3D electromagnetic inversion, and as discussed in (Newman and Boggs, 2004): (1) set $i = 1$, ...
5
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2answers
159 views

Computing inverse functions of functions of two variables

There are several functions of two or three variables that I am working with. For this question I have made a small set showing the resistivity, $\rho$, in n$\Omega$m, of copper as a function of its ...
1
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1answer
867 views

Edge and Nodal finite element methods in MATLAB for Magnetic induction tomography

What is the difference between edge finite elements and nodal finite elements? This for use in modeling the eddy current problem in classical electromagnetism. I am attempting to convert MATLAB code ...
7
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1answer
124 views

reformulating inverse problem as multi-objective optimization

I'm working on an inverse problem for my Ph.D. research, for which I'll write the objective functional as $J(\theta) = E(G(\theta) - u^o)$, where $\theta$ are the parameters, $G$ is the forward map ...
6
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1answer
181 views

Tikhonov (Ridge) Regression and Normalization

For a typical Ridge Regression method for solving an inverse problem $$ \min_x ||A~x - b||^2 + \lambda^2||\Gamma~x||^2 $$ Which has an analytical solution of $$ \hat{x}_{est}=(A^TA+\lambda^2 \Gamma^T\...
2
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1answer
48 views

How to invert a lagrangian polynomial

I'm reading the following paper (Grezlak and Oosterlee) and I have a specific question to a sentence on page 5. I quote: "Since the mapping $y=g(x)$ is bijective and $g(x)$ is strictly increasing,...
8
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1answer
197 views

Solving two inverse problems with same solution

I've got two inverse problems, $$A_1 ~ x = b_1 \qquad A_2 ~ x = b_2$$ So far I've been solving them independently using Tikhonov Regularization and getting two estimates for $x$. However in my case $...
2
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1answer
71 views

What are the most popular wavelet or tight frame regularizers for image reconstruction problems?

A common approach to image reconstruction is to solve the convex optimization problem \begin{equation} \text{minimize} \quad \frac12 \| Ax - b \|^2 + \gamma \| Dx \|_1 \end{equation} where $b$ is a ...
2
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0answers
52 views

SPECT reconstrction using MLEM

In Single-Photon Emission Computerized Tomography (SPECT) parallel beam reconstruction using Maximum-Likelihood Expectation–Maximization(MLEM), is it sufficient to scan the object around 180 degree? ...
12
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2answers
224 views

pointwise vs. continuous observations in PDE inverse problem

I work on an inverse problem for my Ph.D. research, which for simplicity's sake we'll say is determining $\beta$ in $L(\beta)u \equiv -\nabla\cdot(k_0e^\beta\nabla u) = f$ from some observations $u^...
6
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0answers
239 views

Linear vs Non Linear inverse problems: Does non-linearity help?

This is not a typical question with a deterministic answer. If this is not the right place, feel free to close it. For the past one year I have been working on various kinds of inverse problem. Most ...
2
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0answers
50 views

Inverted value is not consistent with expectation

We have a group of observations $$y = f(x_1, x_2, x_3) \enspace .$$ We have also a forward model $y = f(x_1, x_2)$. The forward model does not include $x_3$ because $x_3$ might include dozens of ...
4
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3answers
213 views

Objective function scaling in an Inverse Problem

I am trying to solve a large scale inverse problem using the Bayesian formulation. To estimate the Maximum a Posteriori Estimation (MAP) solution I will have to minimize the following objective ...
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0answers
158 views

Simple MCMC Algorithm in Matlab

I would be really glad to get some specific advise on how to implement a simple MCMC algorithm (in Matlab, if possible). I'm not yet too familiar with optimization methods. My problem goes as follows: ...
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0answers
291 views

How can I efficiently solve $Ax$=$b$ given $A$ is symmetric and contains very small (even negative) eigenvalues using EIGEN

Currently I am using the EIGEN C++ library to try to solve $x$ from the equation $Ax$ = $b$. One problem I encountered is that the matrix $A$ is a correlation matrix with size > 5000 and can ...
5
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2answers
141 views

Inverse problem with changing number of variables

I have an inverse problem in which the optimal positions for a variable number of injections needs to be determined. If the number of injections was fixed, I could easily imagine implementing ...
4
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1answer
639 views

Numerically stable approach for calculating x in Ax=b

I have an equation $Ax=b$ for which I need to solve for numerous $x$ matrices given $b$. Both $x$ and $b$ are nx1 matrices. Unfortunately, $A$ is a 32x32 matrix and inversion gives highly unstable ...
6
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2answers
969 views

Algorithm for optimizing Ax = b with unknown A and known x values

I would like to solve for the optimum $A$ values for a series of matrix equations $Ax_{1} = b_{1}, Ax_{2} = b_{2} ... Ax_{n} = b_{n}$ where only the $x$ values are known and when I start with an ...
3
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0answers
114 views

Creating FEM mesh for image region — what is the most suitable shape function?

I wish to create a FEM mesh to solve an inverse elasticity problem, for an irregular domain. This domain is given by a medical image, so it is discretised and each square on the grid has one scalar ...
2
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2answers
404 views

Do you spend time reading pure math books as a graduate student on computational math

I am going to start my first year of a research-oriented master program on inverse problem. From what I know, unlike pure math students, applied math students usually don't spend the first year ...
5
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3answers
179 views

Algorithms for radiation treatment planning

I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, ...
6
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1answer
486 views

Inverse advection-diffusion problem, solving for a drift coefficient with experimental data?

I am investigating a physical process where I believe the 1-D advection-diffusion equation: \begin{equation} \frac{\partial u}{\partial t} = -\frac{\partial}{\partial x}[\mu(x,t) u(x,t)] + \frac{\...
2
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0answers
131 views

Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system $$ x=A^{-1}b $$ This inverse problem is basically a least square problem with a rank-1 update. $$ x=[uv^{T}...
1
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1answer
55 views

an over view of sparsity promoting inversion techniques [closed]

I have a function called f(x) which is convex and I can have access to its first order derivative , my objective function is $$\ J(\bf{x}) = f(\bf{x}) + \lambda |\bf{x}|_0 $$ $$\ \bigtriangledown ...
11
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1answer
941 views

Numerical methods for inverting integral transforms?

I'm trying to numerically invert the following integral transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$ So for a given $F(y)$ ...
2
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2answers
294 views

Research in Inverse Problem and Numerical PDE

I am taking a Thesis-based Master degree now and I am going to choose my supervisor soon. I plan to take a PHD degree after graduation, so if possible, I wish my PHD research area could be an ...
2
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1answer
172 views

How to solve this numerical technique problem?

Well, in a numerical technique test we were given the following problem: A physical phenomenon is modeled such that, $F(f,d) = A(f)/d^2 + L$; Where, $F$ is a function of frequency $f$ and distance $...
4
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1answer
140 views

Early work on inverse problems

Long time ago I came across with a paper that covered early theoretical work (first half of 20th century) in the field of inverse problems. I remember there was a reference to a paper which proved ...
7
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5answers
594 views

Recommended Route for Mastering Inverse PDE Problems

I would like to master Inverse PDE Problems particularly with the use of Finite Elements. My problem is I don't know where to start. Should I begin by reading a book on Inverse Problems or on PDE-...
10
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1answer
922 views

How can I determine the initial values of pseudo-random number generator if the sequence is given?

Suppose I knew that a random number sequence was generated by a linear congruential generator. That is, $x_{n+1}=(aX_n+c) \bmod m$ If I am given the entire period (or at least a large contiguous ...
8
votes
1answer
227 views

Finding the fixed point of an operator

What numerical methods are available for finding the fixed point of an operator $A$ that is acting on functions $f : [a,b] \rightarrow [a,b]$? I am looking for the function $f$ for which $Af = f$. ...