# Questions tagged [inverse-problem]

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

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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 ...
35 views

### 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 ...
333 views

### 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 ...
89 views

### 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 ...
129 views

### 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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### 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 ...
386 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. ...
246 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 ... 102 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 ...
514 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 $h$...
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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 ...
372 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 ...
277 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 ...
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1 vote
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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 $i = 1$, ...
236 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 vote
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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 ...
137 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 ...
306 views

1 vote