From SIAM Review: Survey and Review
By J. M. Sanz-Serna
The Survey and Review section of the latest issue of SIAM Review contains two papers. “Cubature, Approximation, and Isotropy in the Hypercube,” by L. N. Trefethen, is a thought-provoking article, with implications on many different areas of applied mathematics, including functional approximation, numerical linear algebra, data compression, cubature, optimization, and others. When Richard Bellman coined the expression “the curse of dimensionality” in 1961, he thought that the curse was “a malediction that has plagued the scientist from earliest days.” In fact, it is plaguing us more and more because the dimensionality of the problems being tackled increases continually. For problems posed on a hypercube in \(\mathbb{R}^N\), a tensor-product grid with \(n+1\) points along each coordinate axis has a total of \((n+1)^N\) points and is linked in a natural way to the anisotropic, \((n+1)^N\)-dimensional space of polynomials of degree \(\le n\) in each of the \(N\) coordinate variables. For large \(N\), the number \((n+1)^N\) is huge, even if \(n\) is small: this is the curse of dimensionality. The author argues that many algorithms that dispel the malediction and get by with substantially fewer grid points/degrees of freedom can only be successful in special cases, for instance, for anisotropic functions well aligned with the coordinate axes. Unfortunately little has been done to characterize mathematically such special cases. The use of the isotropic space of polynomials of total degree \(\le n\) introduces difficulties of its own, because, for \(N\) large, the hypercube \([-1,1]^N\) is very anisotropic: “most” of it is well outside the unit ball of \(\mathbb{R}^N\).
Our second paper is “On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows,” by V. John, A. Linke, C. Merdon, M. Neilan, and L. G. Rebholz. Most readers will be aware of the crucial role played by the Navier-Stokes equations in engineering and applied mathematics, and many will know that the corresponding numerical solution may be far from trivial, with difficulties stemming from the nonlinearity of the inertial force, the smallness of the viscous term, etc. This paper concentrates on a relatively unknown source of trouble. In the Navier--Stokes equations, perturbing the external force \(\bf f\) by the addition of a conservative force \(-\nabla V\) changes the pressure field \(p\) into \(p-V\) but has no effect on the velocity field \(\bf u\). Unfortunately many stable, convergent mixed finite element discretizations used in practice do not inherit this property, due to the way they handle the incompressibility constraint (this constraint is of course “dual” to the pressure force \(-\nabla p\)). Such discretizations are said to be nonrobust (with respect to pressure). Nonrobustness has a negative impact not only on accuracy, but also on conservation of mass and on the structure of error bounds. The paper clearly shows the disadvantages of nonrobustness and reviews the available remedies for partially fixing this problem or even suppressing it completely.
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J. M. Sanz-Serna is a full professor at Universidad Carlos III de Madrid. He is the section of editor of the Survey and Review section of SIAM Review.