When is a finite matrix a "good" approximate representation of an operator? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T17:37:23Z http://mathoverflow.net/feeds/question/36409 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36409/when-is-a-finite-matrix-a-good-approximate-representation-of-an-operator When is a finite matrix a "good" approximate representation of an operator? Jiahao Chen 2010-08-22T22:08:38Z 2010-10-04T09:22:15Z <p>I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions</p> <p>$\rho(r) = \sum_{i=1}^N q_i \phi_i (r)$</p> <p>where $\phi_i (r)$ is normalized to $\int_{\mathbb{R}^3} \phi_i (r) dr = 1$ and has the interpretation of being the shape of some charge distribution (shape) of a unit charge. $\rho$ and the $\phi_i$s are real-valued but may be positive in some regions and negative in others. The basis functions are nonorthogonal and local in space but not strictly compact. Let's say for now that we use spherical Gaussians of the form $\phi_i (r) \propto \exp (-\alpha_i |r-R_i|^2)$, where $R_i$ is where the basis function is centered around. The number of basis functions chosen scales approximately as the number of atoms, as we expect charge to concentrate around atomic nuclei. (We may add additional basis function per atom of different shapes until we achieve a reasonable approximation to the desired shape of the charge distribution around an atom.)</p> <p>The energy of the system can then be given by</p> <p>$E = \frac 1 2 \sum_{i,j=1}^N q_i q_j J_{ij}$</p> <p>where the matrix $J$ has elements</p> <p>$J_{ij} = \int_{\mathbb{R}^{3\times 2}} \frac{\phi_i(r_1) \phi_j(r_2)}{|r_1 - r_2|} dr_1 dr_2$</p> <p>and represents the Coulomb interaction between the unit charge distributions $\phi_i$ and $\phi_j$.</p> <hr> <p>One way to look at the matrix $J$ is as a finite dimensional (approximate) representation of the Coulomb operator $\hat J = 1 / {|r_1 -r_2|}$. We know that $\hat J$ has certain nice properties such as positivity, so we expect a "good" representation of $\hat J$ should be a symmetric positive definite matrix.</p> <p>My question is this: are there conditions on the discrete representation (possibly expressible as conditions on the {$\phi_i$} basis) to detect whether or not a given claimed representation $J$ is "good" in that it preserves such properties? Or asked another way, if I have some matrix $J$ which is claimed to represent $\hat J$, what are necessary and sufficient conditions on its matrix elements for it to be a "good" representation of $\hat J$?</p> <p>I hope the question makes sense, and that I am not misusing too much terminology.</p> http://mathoverflow.net/questions/36409/when-is-a-finite-matrix-a-good-approximate-representation-of-an-operator/36429#36429 Answer by Jerry Gagelman for When is a finite matrix a "good" approximate representation of an operator? Jerry Gagelman 2010-08-23T07:43:20Z 2010-08-23T07:43:20Z <p>The answer to your question, "are there conditions on the discrete representation ... to detect whether or not a given claimed representation $J$ is 'good'...", is yes. One usually measures "good" in terms of a norm.</p> <p>I think your operator $\hat{J}$ is actually an integral operator, right?</p> <p>$$\hat{J}(u)(r) = \int \frac{u(s)}{|r-s|}~ds$$</p> <p>To keep things as general as possible, this is a map between two normed linear spaces $\hat{J}: V\to W$. Your finite matrix can be regarded as the <em>representation</em> of $\hat{J}$ on some finite dimensional subspace <code>$V_n\subseteq V$</code>; call it <code>$J_n:V_n \to W$</code>. (Here $n$ is a parameter, maybe the basis size, etc.) Writing <code>$P_n:V \to V_n$</code> for some appropriate projection operator, your measure of goodness will be</p> <p>$$\| \hat{J}u - J_nP_nu\|_W \leq O(n^{-\alpha}), \quad n\to \infty, \quad u\in V,$$</p> <p>where $\alpha > 0$ and where the asymptotic constant in the big-O notation invovles the quantity <code>$\|u\|_V$</code> in some way.</p> <p>This is more an attempt to formulate your question mathematically rather than to answer it. Does it look like something you can work with?</p>