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13

A fairly simple method would be to choose a basis in function space and convert the integral transformation to a matrix. Then you can just invert the matrix. Mathematically, here's how that works: you need some set of orthonormal basis functions $T_i(x)$. (You can get away without them being normalized too, but it's easier to explain this way.) Orthonormal ...

13

If the solution of $Ax=b$ is unstable, the matrix is very ill-conditioned (i.e., has a very large condition number), and (paraphrasing Lanczos) no amount of mathematical trickery can make it stable. The best you can hope for is to solve a different problem that is a) stable and b) gives you a solution that is sufficiently close; this is called regularization....

9

You can write your problem as $\min \| Fm - g \|_{2}^{2}$ where $F=\left[ \begin{array}{c} A_{1} \\ A_{2} \\ \alpha I \\ \end{array} \right]$ and $g=\left[ \begin{array}{c} b_{1} \\ b_{2} \\ 0 \\ \end{array} \right].$ You can use whatever linear least squares solver you want to solve this problem.

9

Your problem sounds like Independent Component Analysis. Where $x_i$ are the measurements in which the source signals have got mixed and $b_i$ are the values emitted by the sources. The $A$ in your equations is called the unmixing matrix. There is an iterative procedure to estimate $A$ and hence, the $b_i$s, based on the maximum likelihood principle. Refer ...

9

The term inverse crime for a numerical test of a parameter identification method that uses data contained in the range of the discrete(!) forward operator used for the inversion (thus essentially reducing the problem to a well-posed finite-dimensional one that behaves fundamentally different from the original infinite-dimensional one -- it is important to ...

9

You want to minimize $\min \| Ax -y \|_{2}^{2} + x^{T}B^{T}Bx=\| Ax -y \|_{2}^{2} + \| Bx \|_{2}^{2}$ Recall that $\| u \|_{2}^{2} + \| v \|_{2}^{2}= \left\| \left[ \begin{array}{c} u \\ v \end{array} \right] \right\|_{2}^{2}$. Thus your problem can be written as $\min \| Hx - g \|_{2}^{2}$ where $H=\left[ \begin{array}{c} A \\ B \end{array} \... 7 There is another approach called the adjoint method, which is commonly used in inverse problems for PDE and which is quite easy to generalize to other problems. This is going to be long. Your observations give you a field$u_{\text{exp}}$; you'd like to find a value of$\mu$for which$\frac{1}{2}\iint(u - u_\text{exp})^2dx\hspace{2pt}dt$is a minimum, ... 6 It's a question of how you choose$\lambda$and$\Gamma$(of course). Think for a moment about what happens if you choose$\Gamma=I$and make$\lambda$large: in that case you say that it is more important to you to minimize the regularization term$\|Ix\|^2=\|x\|^2$than to minimize the misfit$\|Ax-b\|^2$. Obviously, making the term$\|x\|^2$small means ... 6 The measurements of this field are often spotty and missing chunks; why interpolate to get a continuous field of dubious fidelity if that can be avoided? You're perfectly right - most of the time, interpolation to a continuous field covering the entire domain is not an option. Think about weather prediction problems, where measurements (point-sources) are ... 6 I have an inverse problem in which the optimal positions for a variable number of injections needs to be determined. Are there any thoughts on what I could do? Or any sources you can point me to (searching "optimising a variable number of variables" unsurprisingly isn't very fruitful)? One potential strategy is using a mixed-integer formulation, ... 6 I think your premise is flawed: If you wish to do research on a topic, you first need to learn the fundamentals -- no matter whether it's "pure" or "applied" math (a distinction which is not really useful in general) -- and the best place is from a good textbook (or, even better, attending a good lecture). Once you get to the cutting edge, you need to switch ... 5 I think you're looking for the paper by Calderon on electrical impedance tomography: @InCollection{Cal80, author = {A. Calder{\'o}n}, title = {On an inverse boundary value problem}, booktitle = {Seminar on Numerical Analysis and its Applications to Continuum Physics}, pages = {65--73}, publisher = {Soc. Brasileira ... 5 This problem is actually more of an optimal control problem for a partial differential equation. As a starting point, I would recommend the following books: F. Tröltzsch, Optimal control of partial differential equations, AMS, 2010 M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE constraints, Springer The first one in particular is an ... 5 To expand on @GoHokies's answer: If you're interested in regularity questions, you can also ask what "point measurements" really are. In physical practice, you cannot measure anything at a "point". Rather, you are always going to get some kind of average over some kind of space-time chunk: a thermometer is not a point but an extended object, and it takes ... 5 First, a disclaimer: I'll answer specifically within the context of Bayesian inverse problems, not the wider statistical theory of Bayesian inference (which tends to devolve into philosophy at some point...) Second, a general point: If you are only computing a MAP estimate and are not trying to extract higher order moments from the posterior distribution, ... 5 Consider the vector$y_i$for a given time$t_i$. We are looking for the linear operator$\mathcal{B}$that maps$y_i$to$y_{i+1}by noting that \begin{align} \dot{y}&= \mathcal{A} y\\ \frac{y_{i+1}-y_i}{\Delta t}&=\mathcal{A} y_i\\ \Rightarrow y_{i+1}&=(\mathcal{A}\Delta t+I) y_i\\ y_{i+1}&=\mathcal{B} y_i \end{align} For a set of vectors...

5

It's a well-known result that the 2d tomography problem is weakly ill-posed (singular values decay as $O(1/\sqrt{n})$ even with full data and strongly ill-posed (the singular values decay exponentially) if you don't have complete angular coverage. See for example: M. E. Davison. The Ill-Conditioned Nature of the Limited Angle Tomography Problem. SIAM ...

5

So, you want to invert your matrix $A=\Phi^T\Phi$. For $A$ to be invertible it must not have zero eigenvalues. We can show that $A$ is positive semi-definite as follows. Positive semi-definite means that the eigenvalues of $A$ are $\geq 0$. This is equivalent to showing $y^TAy \geq 0, \forall y \neq 0$. $$y^TAy = y^T\Phi^T\Phi{y}=(\Phi{y})^T(\Phi{y}) \geq ... 4 Changing the relative weight of the two terms is equivalent to changing your prior. The fundamental issue here is that you've got a problem with far more parameters to estimate than data points. You will need to use a fairly strong prior to get tight bounds on the fitted parameters, and the choice of that prior will have a strong influence on your ... 4 No. Multi-objective optimization is concerned with the simultaneous optimization of two (or more) competing objectives where you do not wish to decide on the trade-off between the two beforehand. However, this is not the situation you have: Only the exact minimizer of the Dirichlet energy D is a solution to your forward mapping; the actual value of the ... 4 Some remarks: Notation: [t_1,t_2]: the time interval; \Omega: the spatial domain; \bar{u}^i(x): the known tumour profile at t_i; \left\| \cdot \right\|_\Omega: a suitably chosen norm on \Omega. I assume that the design variables D and k do not depend on t. The cost functional should be a scalar (real-valued), something like:$${\cal G}(u;...

4

There are infinite functions that have the same integral over a given domain, so you would need to make assumptions on the type of functions that you want to allow. The easiest approach that comes to my mind is assuming a constant function in each cell, then $$p_i \equiv p(x) = \frac{P(V_i)}{V_i} \quad \forall x \in V_i\, .$$ Then you can use this ...

4

As I understand, your ultimate goal is to solve an inverse problem (i.e., infer some parameters from given data / observations). To this end, you want to apply Bayesian Inference, which relates the posterior (i.e., the probability distribution of the unknown parameters) to the likelihood (i.e., the probability model of observing some values given the ...

4

Think of the simplest case when $\Phi$ is a scalar value. Not well defined: $$\boldsymbol \theta^\text{ML} = (0^T 0)^{-1}0^T ~ y = \frac{1}{0} 0~y= \frac{0}{0}$$ Well defined: $$\boldsymbol \theta^\text{ML} = (0^T 0 + \kappa)^{-1}0^T~y =\frac{1}{\kappa} 0 ~y= 0$$

4

Look up something on Tikhonov regularization, also known as ridge regression in machine learning. This is a standard technique (but I agree that the explanation in that notebook is somewhat poor). Technically speaking, it does not affect the numerical stability of that algorithm, but it modifies the problem to a more well-conditioned one, from $\min \|\Phi \... 3 The paper: I. Anjam, J. Valdman, "Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements", Applied Mathematics and Computation, 267, (2015), 252–263; states the following. http://www.sciencedirect.com/science/article/pii/S0096300315004191 "A finite element discretization is done in terms of edge elements, typically Raviart–Thomas elements [12] for ... 3 As a sidenote, the physical dimensions of your scaling parameters$\sigma,\alpha$are different. So a statement of the form$\sigma\ll\alpha$as given in your question doesn't make any sense: are 3kg of apples much less or much more than 5 micrometers of yarn? In addition to the references others have given before, let me also recommend the book by ... 3 The short answer is that in general this problem is intractable (NP-Hard) with the$\| x \|_{0}$regularization. Are you willing to consider minimizing$ f(x) + \lambda \| x \|_{1}\$ instead? There are many methods for the 1-norm regularized problem. There are also lots of theoretical results that give conditions under which 1-norm minimization ...

3

To add my own experience to ChristianClason's post: it's not uncommon to for graduate students in computational science to read pure mathematics textbooks and take pure mathematics classes. I graduated with an engineering degree and have read large portions of several pure math books. Part of the reason I did so is because my adviser required me to take a ...

3

In addition to Brian's reference, there was also a review article in SIAM Review a few years ago that summarized the state of research at the time: http://epubs.siam.org/doi/abs/10.1137/S0036144598342032

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