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Reviewer: Goran Trajkovski

We have learned to expect the cutting edge in numerical mathematics and related areas from Acta Numerica , and this, its 14th volume, is no disappointment. About 570 pages are devoted to seven invited surveys at the forefront of current research. The contributors are some of the best known among their peers, the topics are carefully selected, and the material is elaborated on in a form suitable for researchers as well as for practitioners in the field. The topics are drawn from the whole spectrum of numerical math problems, and, as such, it is hard to identify a common topical niche, apart from the fact that they reflect the state of the art of mainstream research in the domain. The first paper, "Numerical solution of saddle point problems," by Benzi, Golub, and Liesen, focuses on the problem of solving large linear systems in a saddle point form. These problems emerge frequently out of technical and scientific applications. The paper surveys the properties of saddle point matrices, solution algorithms, null space methods, coupled direct solvers, preconditioning, and multilevel methods, as well as the available software for solving these omnipresent problems. Sparse matrices and large problems dominate the chapter, as one would expect, due to their presence in a plethora of engineering problems, especially for solving saddle point problems from partial differential equations (PDEs) and PDE-constrained optimization. This chapter also provides a survey of available software packages on the Web. Despite the ambitious number of topics this chapter brushes upon, it gives well-balanced coverage of everything important for following the main ideas. Deckelnick, Dziuk, and Elliott's paper, "Computation of geometric partial differential equations and mean curvature," surveys the computation of curvature-dependent interface motion governed by geometric PDEs, and, in particular, parametric mean curvature flow, flow of graphs, flow of level sets, the phase field approach, the anisotropic mean curvature flow, and the four order flows. The ideas in this chapter are covered with strong mathematical rigor, but with scarce illustration when conveying some major points. In "Random matrix theory," Edelman and Rao give an overview of accomplishments in stochastic linear algebra and operators from a perspective relevant to numerical analysis. Different from standard numerical analysis, where the term random is used more or less only in the context of random number generators, the equations themselves are random, and emerge from a variety of engineering and financial applications. This area has emerged from the needs of heavy atom physics and multivariate statistics, and is expected to become a valuable engineering tool, as it has picked up speed in wireless communications, combinatorial math, and financial analysis. As the results thus far have been promising-to say the least-we expect to see a large wave of engineering research using random matrices theory and its results when attempting practical problems. This is probably the most comprehensive coverage of this topic to date. In "Numerical methods for large-scale nonlinear optimization," Gould, Orban, and Toint review the latest developments in optimization for solving unconstrained as well as differently constrained problems, including nonlinearly constrained ones. Historically speaking, this paper gives an overview of the reigning mainstream methods in the domain in a manner easily understandable for both academic and industrial purposes. With "Computational chemistry from the perspective of numerical analysis," by Le Bris, the book slowly transitions into more directly applicable numerical analysis. The marriage of two disciplines, such as chemistry and numerical analysis, is always a positive event, especially when one discovers the extent to which one discipline can be applied to the other. This chapter, which is mathematically strict and well written, adds to the applicability of most problems covered throughout the whole volume. It is a motivational read in that it initiates inquiry into new areas for implementing known numerical methods and modifying them to fit real-life purposes. This work is followed by "Steady-state convection-diffusion problems," by Stynes. It treats problems abundant in physical processes, known from the Navier-Stokes equations, or from the Black-Scholes equation in the domain of financial modeling. The last paper, "Total variation and level set methods in image science," by Tsai and Osher, provides a substantial overview of the use of level sets with a variety of examples from the earth sciences and medical applications. The algorithms given can be implemented easily. In conclusion, this volume is a very useful one. It provides an overview of current developments in numerical mathematics and scientific computing, with practical considerations. Is it a textbook, a research companion, or a journal__?__ Well, it is a little bit of each. Online Computing Reviews Service