# Tag Info

19

Wikipedia gives a good definition Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysts are typically interested in proving mathematical results about their algorithms, ...

10

if my years in the industry have taught me anything, it's this: everything depends on the grid. developing a robust solver that efficiently converges to machine zero might be the flashy rock star job, but the unsung heroes are the developers that improve our gridding algorithms. if you're looking for a really great way to lose the effects of a vortex, try ...

9

This is coordinate descent. I believe it's used on very large-scale problems when other methods like gradient descent might be too slow (e.g., http://epubs.siam.org/doi/abs/10.1137/100802001). It should converge to a local minimum, but it also would require more steps than something like gradient descent or Newton-type methods.

8

A phenomenological model is based on observations of a system rather than on physical theory. Other physically based models are based on fundamental physical principles such as Newton's laws of motion. Both kinds of models might end up being expressed in the form of mathematical equations and called mathematical models. In practice, models used in many ...

6

As someone who moved from Engineering to Scientific Computing during Grad school as an incidental need of the kind of work I was doing here are my two cents: Numerical analysis would focus on the math and algorithms side of things. Figuring out what techniques to use to solve a particular mathamatical problem that does not have an analytical solution e.g. ...

6

This seems to be called an exponential search, doubling search, or galloping search. I've also heard it called a geometric expansion search or something similar. In principal, it is similar to the geometric expansion strategy that is often used for resizing dynamic arrays in computer programming. Resizing dynamic arrays in this way ensures that adding $n$ ...

5

Traditionally, the convergence of optimization algorithms has been analyzed in terms of the asymptotic rate of convergence. A quadratically convergent algorithm has $x_{k} \rightarrow x^{*}$ and $\lim_{k \rightarrow \infty} \frac{\| x_{k+1}-x^{*} \|}{\| x_{k}-x^{*} \|^{2}}=L< \infty$ while a linearly convergent algorithm has $\lim_{k \rightarrow \infty} \... 4 The approach you described originally (only one iteration optimizing in each of the three variables x,y,z ) is not guaranteed to converge to the optimal solution unless F(x,y.z) is variable separable into univariate functions. Therefore, what you describe is not technically an optimization "method", but an optimization "heuristic", similar to an operator ... 4 The obvious answer is "it depends". However, it's not helpful. I would certainly separate the work in mathematical modeling and actual numerical simulation. Sometimes it might be a bit tough to draw the line in between, but I think it's usually possible. Thus, by using work in mathematical modeling and numerical simulation does not seem to be redundant and ... 4 I would call the constraint "upper- and lower-bounds on the maximum element." Note that you are actually dealing with two separate constraints. Define the max element function as follows $$\max:\mathbb{R}^{n}\to\mathbb{R}\qquad\max(x)\equiv\max_{i\in\{1,\ldots,n\}}x_{n}.$$ Your first constraint is "take the max element and ensure that it is less than$c$": ... 3$A$is a discretized version of your differential operator + enforced boundary conditions. The names for$A$can vary depending on the way the PDE is being discretized. For example, in FEM, it will be stiffness matrix. For integral equation methods (technically not a PDE) applied to Maxwell equations, such a matrix is usually called impedance matrix. ... 3 Lower level optimization problems being solved within a top or higher level algorithm are called subproblems. So the algorithm or routine to solve subproblems could be called a "subproblem solver". Googling "subproblem solver" shows that this term is not that uncommon. If there is a specific type of subproblem being solved, that can be incorporated, such as ... 3 For me, there is a clear hierarchy going from reality to a simulation. The first step is to understand reality as much as you can and propose a model for this reality, typically without formally writing down equations. You define what physical/chemical/biological/... processes are involved. Already in this step, you introduce an error: you can never model ... 2 You may want to check out Section 3 in chapter III of: E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, Second., vol. 31. Berlin: Springer-Verlag, 2006. and/or: A. Murua, “The Hopf Algebra of Rooted Trees, Free Lie Algebras, and Lie Series,” Foundations of ... 2 Here is some generic (application-independent) terminology I've seen in papers:$A$: The coefficient matrix''$b$: The right hand side''$x$: The unknown'' Also, sometimes$A$may be called the coefficient operator'' if it is considered as a linear operator rather than a matrix. 2 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 partitioning on the other hand tries to immediately divide up the graph. Part of the divide between the two is historical. Some of the earliest successful heuristics ... 2 To partially echo @aeroNotAuto, meshing algorithms are crucial. Here is a useful page listing important papers on meshing, from a Berkeley course taught by Jonathan Shewchuck. 2 In the US, the pronunciation of Eulerian is actually "You-leh-rian". But it's also ok (and understandable to everyone who is educated enough to understand the word) to say "Oy-leh-rian" (which is closer to the German original -- or at least about as close as American speakers typically get to a German-origin word). "Guassian" is actually spelled "Gaussian" ... 2 Like I mentioned in your other question already, don't worry about the language part too much if there is a way to express what you are trying to say in "natural" language (rather than trying to be overly formal). Here, I would say something like this: The solution of this dynamical system is periodic in the sense that the$y(t)$and$\theta(t)\$ ...

1

The term immersed boundary method has nothing to do with the term direct numerical simulation. The immersed boundary method is a numerical methodology that is often used to handle heterogeneous fluid regions in your simulation domain. For example: Fluid-structure interaction Multi-phase flows The term direct numerical simulation is often used in the LES ...

1

Yes, its commonly used to describe the magnitude of an outputs change with respect to an input. In CFD optimization we provide a sensitivity vector to the optimizer (or the derivatives of each output to each input). In this case the matrix inversion is sensitive to (for example) small differences on the right hand side or floating point error.

1

To my knowledge these are the same things. However, this type of thing is common. For example, the proper orthogonal decomposition also has field-specific names. Others call it principal component analysis, the Karhunen--Loeve expansion, or empirical orthogonal functions. It is also no different than an autoencoder with linear activation function. I'm sure ...

1

If I correctly understood your question, you wonder what exactly to call "a model for blood flow in the heart": the equations or their solution. I think part of the ambiguity comes from the fact that there are a lot of definitions of model. Even if we take only the mathematical models into considerations, there would be a lot of details and border cases to ...

1

I would suggest that a slightly modified version of @GoHokies suggestion array layout or even a more precise, but a bit wordish array memory layout should suffice and be unambiguous. In my opinion, this term is the clearest one, describing what row-major and column-major (and possibly some other variants) is. Google search (Apr 7, 2018, google.ca): "array ...

Only top voted, non community-wiki answers of a minimum length are eligible