On Sat, Nov 11, 2017 at 1:33 PM, Jed Brown <[email protected]> wrote:
> Matthew Knepley <[email protected]> writes: > > > On Sat, Nov 11, 2017 at 1:12 PM, Jed Brown <[email protected]> wrote: > > > >> Matthew Knepley <[email protected]> writes: > >> > >> >> Matrix and graph are equivalent concepts. > >> > > >> > > >> > This is clearly wrong. A matrix is the coordinate representation of a > >> > linear operator, and thus has a specific > >> > behavior under coordinate transformations. A graph is just > connectivity, > >> > and really just a relation. I cannot > >> > count the number of times Barry has ranted about this on petsc-maint > >> > (usually about Vecs and arrays). The > >> > mathematical object is not its data structure. > >> > >> A graph Laplacian certainly does transform under coordinate > >> transformation and indeed, we use that property to design effective > >> coarsening strategies. That one basis strikes you as intrinsically > >> "more canonical" does not mean it isn't a linear operator. > >> > > > > That is one operator. This is argument by anecdote. An arbitrary graph > > is not a linear operator, but an arbitrary matrix definitely is (the > > coordinate representation of one). > > Dude, we solve linear systems and eigenproblems for arbitrary graphs. > It isn't an anecdote. > > Barry (rightly) objects to a 2D array representing a function on a grid > being considered a Matrix. We don't "apply" it as a linear operator. > There is no "vector" on which it operates. > > But we absolutely do with a graph. Our vectors are functions at the > vertices of the graph. Applying the graph Laplacian tells us about > local compatibility of the field over the vertices. It is entirely > analogous to fields over a grid. You don't need a concept of "grid > refinement" to have matrices. > I don't think makes sense. You are saying, because a linear operator (the graph Laplacian) can be defined using the graph, then the graph is identical with this operator. I do not agree. Whereas the matrix means nothing else but the linear operator which it represents. Matt -- What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead. -- Norbert Wiener https://www.cse.buffalo.edu/~knepley/ <http://www.caam.rice.edu/~mk51/>
