Daniel Shapero
• Member for 9 years, 1 month
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• Seattle, WA

TL,DR: either the good old Galerkin finite element method, or mesh-free / particle methods. There are a few things to unpack here. First, the simulation you show includes contact between an elastic ...

As the commenters have already said, for your specific problem, QR factorization might be a better approach not because of speed but because of numerical stability. It's still a good question to ask ...

Yes, but you have to mean symplectic on a higher-dimensional phase space than your original problem that includes previous steps too. As I understand there are also some subtle stability issues too. ...

First off, the PDE can be rewritten instead as $$\frac{\partial C}{\partial t} = \frac{\partial}{\partial x}C\frac{\partial C}{\partial x}$$ or, by applying the product rule in reverse again, as $$\... View answer 2 votes You can think of the ODE as a constraint between the parameters V and the observed variable y. To enforce constraints in optimization problems, you can introduce a Lagrange multiplier, which we'll ... View answer 7 votes Using an Eigen matrix type where the number of rows and columns is encoded into the type at compile time gives you an edge over LAPACK, where the matrix size is known only at runtime. This extra ... View answer Accepted answer 2 votes TL;DR: Let k be the diffusion coefficient, \theta the minimum angle between any two edges of the mesh, d the space dimension, and p the polynomial degree of the finite element basis you're ... View answer Accepted answer 6 votes If the only non-zero entries of A_{ij} have j in \{i - 1, i, i + 1\}, then A is a banded matrix with bandwidth 1. More generally, you can talk about matrices of bandwidth k where k is any ... View answer Accepted answer 10 votes If the operation is as trivial as you say, and each node has all the information necessary to carry out the operation, then the communication will be substantially more expensive than recomputing ... View answer 5 votes In addition to the voxel-based approach that rchilton suggests, you could also look at Delaunay-type algorithms. For example, the Computational Geometry Algorithms Library (CGAL) has some built-in ... View answer 2 votes I looked around a bit and found this paper (featuring no less than Shing-Tung Yau!) about the problem of generating meshes for the unit tangent bundle UT(S^2) on the 2-sphere S^2. As you point out,... View answer 2 votes I'm a little late to reply but I found this review to be really informative. One particularly nice feature of Hamiltonian flows is that they preserve volume in phase space. If we take an infinitesimal ... View answer 8 votes That depends on how well you know the coordinates and velocities. If you have exact values, you can get a reasonable answer using Hermite interpolation. This will give you a degree-3 polynomial in ... View answer Accepted answer 5 votes As always with numerical problems, it's a good idea to simplify things as much as possible to isolate what's really going wrong. You can simplify the problem by considering the evolution of a pure ... View answer 4 votes A quick and dirty approach to quantify the nonlinearity could be to evaluate$$\frac{1}{k}\frac{dk}{dT}\frac{\partial T}{\partial t}\delta t. It's dimensionless and quantifies how much relative ...

The number of subsets of size $m$ of a set of size $n$ is $\binom{n}{m}$, which is bounded by $\left(\frac{n}{m}\right)^m \le \binom{n}{m} \le \left(\frac{en}{m}\right)^m$. So in the limit of $m \ll ... View answer Accepted answer 6 votes As Brian Borchers pointed out, the question only makes sense up to factors of$2\pi$. I'll assume that you want an answer in$[0, 2\pi)$then. If there are really big contrasts in the magnitudes$|...

As Christian Clason alluded to in his comment, you have reinvented the ensemble Kalman filter! The Kalman filter is an algorithm for estimating, from imperfect measurements, the true state of a system ...

Your intuition is correct -- a bisection method cuts the (hyper)graph in two, and recursive bisection repeatedly applies this strategy until the desired number of cuts have been made. Direct ...

If you're interested in surface meshing, the book Delaunay Mesh Generation by Cheng, Dey, and Shewchuk is very good. Surface meshing is a pretty involved topic and I couldn't attempt to summarize it ...

I found the following books pretty helpful: Vogel - Computational Methods for Inverse Problems Parker - Geophysical Inverse Theory Aster, Borchers, Thurber - Parameter Estimation and Inverse Problems ...

As Bruno Levy pointed out, certain classes of physical systems have well-known structural properties. For example, consider a Hamiltonian system $\dot z = J\nabla H(z)$ where $J$ is a symplectic ...

In addition to Kirill's excellent answer, one thing I think work mentioning is Smale's $\alpha$-theory. One of the hard parts about applying Newton's method is that, if you don't make a really good ...

Suppose that the solution $u$ of the PDE lives in some function space $X$. We'll write the PDE as a bilinear form $A(u, v) = f(v)$ for all $v$ in $X$, where $f$ is some element of the dual space $X^*... View answer 1 votes If the index for a bin is some$d$-tuple of integers$\{k_1, \ldots, k_d\}$, one approach that might work is to come up with a hash function$h$for bin indices and use an unordered_map from the C++ ... View answer Accepted answer 3 votes Are you applying MINRES to the operator$B^{-1}A$? Unless$B$and$A$commute, which is unlikely, the product$B^{-1}A$is not symmetric. I'm guessing that this is what happened in your case because ... View answer 1 votes Three things immediately come to mind: R might not take advantage of sparsity when using the solve command to compute the inverse of a matrix. Usually, the inverse of a sparse matrix is dense, so the ... View answer 3 votes A good resource if you're interested in this sort of thing would be What is a Good Linear Finite Element? by Johnathan Richard Shewchuk. Large angles tend to degrade the quality of finite element ... View answer 5 votes The "100,000 unknowns" rule-of-thumb applies to sparse matrices rather than dense ones. A naive direct solver, which doesn't take advantage of sparsity at all, could in principle have$O(n^3)\$ ...