# Tag Info

18

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. ...

5

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 $n$ ...

4

The meanings of those terms depend on context. Superconvergence is usually used to mean you are converging faster than the "optimal" rate, and occasionally this sort of weirdly fast convergence can be proven rigorously. One example in DG is that hyperbolic problems generally have an optimal estimate that decays like $h^{n+1/2}$ but some recent ...

4

Pre-asymptoticity in conjugate gradient method: Asymptotic behavior sometimes assumes certain conditions. For example, given $h$ the meshsize or similar, and $h\to 0$: $$\|f_h - g_h \| \leq c h.$$ but what if $$\|f_h - g_h \| \leq c k h^2$$ if we assume $kh = O(1)$ then $f_h$ is asymptotic to $g_h$, this is asympototic analysis. If we do not assume that ...

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

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 $c$": ...

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 ...

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 ...

3

Preasymptotic refers to the following concept: A priori error estimates only say that as the mesh size $h\rightarrow 0$, we have that (for example) $\|e\|\le Ch^2$. But this is an asymptotic statement: it is not an equality that holds for all $h$ but one typically only sees the quadratic decay whenever $h$ is small enough. In other words, in numerical ...

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. ...

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

Superconvergent refers to the concept that sometimes convergence is faster than one would usually expect. As an example, we know that for the Laplace equation, pointwise convergence happens as $\|u-u_h\|_{L_\infty} \le Ch^2 |\log h|$ when using linear elements. But, on sufficiently regular meshes, it can be shown that at certain points (on quadrilateral ...

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

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

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" ...

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 ...

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