Questions tagged [inverse-problem]
For questions pertaining to methods to estimate input parameters based upon output data.
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Overlap matrix and its inverse matrix
Now, we consider a non-orthonormal basis:
$$\mathcal{S}_N=\{|\alpha\rangle,a^\dagger|\alpha\rangle,a^{\dagger 2}|\alpha\rangle,\ldots,a^{\dagger N}|\alpha\rangle\},$$
where $|\alpha\rangle$ is the ...
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Estimating forces on a model from the displacements of nodes
In any FEM problem involving mechanics, we try to solve the differential equation for the displacement field, $u$ given the force vector in the nodes, $F$. In industry, we often see our automobiles ...
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adjoint method for degenerate problems
For the sake of the rest of the question, I'm interested in the porous medium equation
$$S\frac{\partial\phi}{\partial t} = \nabla\cdot K\phi\,\nabla\phi$$
where $S$ and $K$ are spatially-variable ...
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Converting distance matrix back into original data
Suppose that we have $N$ points, and a distance matrix $D \in \mathbb{R}^{N \times N}$ describing the Euclidean distance among those points. For now, assume that we do not necessarily know how many ...
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Identifying an unknown P.D.E. from solution data
I have a black-box simulation that produces the time evolution of a probability density function p(x, t) in 1 dimension from arbitrary initial conditions p(x, 0). The underlying simulation occurs on a ...
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Invert a huge sparse operator;
please help me with this question, I want to invert a huge sparse (non-circulant) this below in a $Ax=y$ equation:
$$(\lambda I+ \beta D+ \sigma C)x=y$$
where
I is an Identity Matrix,D is a Diagonal ...
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3
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A notion of resolution in inverse problems
Suppose I have a linear inverse problem of the form:
\begin{align}
Ax=b
\end{align}
I would like to reconstruct $x$ from the measurement $b$ via the objective
$$\min_x\{\vert\vert Ax-b\vert\vert^2_2+\...
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3
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Advantage of diagonal "jitter" for numerical stability?
In a machine learning code, that computes optimum parameters $\theta _{MLE}$ of a linear regression model, by maximum likelihood estimation:
$$ \boldsymbol \theta^\text{ML} = (\boldsymbol\Phi^T\...
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201
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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 ...
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2
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558
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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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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 ...
user14831
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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 ...
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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 $h$...
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2
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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 = \...
- 342
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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 ...
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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 ...
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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 $\...
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1
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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}...
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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 $i = 1$, ...
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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 ...
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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 ...
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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 ...
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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
$$
\hat{x}_{est}=(A^TA+\lambda^2 \Gamma^T\...
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How to invert a lagrangian polynomial
I'm reading the following paper (Grezlak and Oosterlee) and I have a specific question to a sentence on page 5. I quote:
"Since the mapping $y=g(x)$ is bijective and $g(x)$ is strictly
increasing,...
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Solving two inverse problems with same solution
I've got two inverse problems,
$$A_1 ~ x = b_1 \qquad A_2 ~ x = b_2$$
So far I've been solving them independently using Tikhonov Regularization and getting two estimates for $x$. However in my case $...
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What are the most popular wavelet or tight frame regularizers for image reconstruction problems?
A common approach to image reconstruction is to solve the convex optimization problem
\begin{equation}
\text{minimize} \quad \frac12 \| Ax - b \|^2 + \gamma \| Dx \|_1
\end{equation}
where $b$ is a ...
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SPECT reconstrction using MLEM
In Single-Photon Emission Computerized Tomography (SPECT) parallel beam reconstruction using Maximum-Likelihood Expectation–Maximization(MLEM), is it sufficient to scan the object around 180 degree? ...
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pointwise vs. continuous observations in PDE inverse problem
I work on an inverse problem for my Ph.D. research, which for simplicity's sake we'll say is determining $\beta$ in
$L(\beta)u \equiv -\nabla\cdot(k_0e^\beta\nabla u) = f$
from some observations $u^...
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Linear vs Non Linear inverse problems: Does non-linearity help?
This is not a typical question with a deterministic answer. If this is not the right place, feel free to close it.
For the past one year I have been working on various kinds of inverse problem. Most ...
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Inverted value is not consistent with expectation
We have a group of observations
$$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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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{\...
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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}...
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vote
1
answer
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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
$$\ J(\bf{x}) = f(\bf{x}) + \lambda |\bf{x}|_0 $$
$$\ \bigtriangledown ...
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Numerical methods for inverting integral transforms?
I'm trying to numerically invert the following integral transform:
$$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 $F(y)$ ...
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