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By Gilbert Strang (auth.), David Y. Gao, Hanif D. Sherali (eds.)

The articles that include this exotic annual quantity for the Advances in Mechanics and Mathematics sequence were written in honor of Gilbert Strang, a global well known mathematician and extraordinary individual. Written by way of prime specialists in complementarity, duality, worldwide optimization, and quantum computations, this assortment finds the great thing about those mathematical disciplines and investigates contemporary advancements in worldwide optimization, nonconvex and nonsmooth research, nonlinear programming, theoretical and engineering mechanics, huge scale computation, quantum algorithms and computation, and knowledge theory.

Much of the fabric, together with some of the methodologies, is written for nonexperts and is meant to stimulate graduate scholars and younger college to enterprise into this wealthy area of study; it is going to additionally gain researchers and practitioners in numerous components of utilized arithmetic, mechanics, and engineering.

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The original work of Monge and Kantorovich on continuous flows has been enormously extended by Evans [11], Gangbo and McCann [15], Rachev and R¨ uschendorf [25], and Villani [33]. Our challenge problem requires the movement of material F (x, y) from Ω to ∂Ω. The bottleneck is in moving from the interior of S to the minimal cut ∂S. The distribution of material is uniform in S, and its destination is uniform along ∂S, to use all the capacity allowed by |v| ≤ 1. How is the shortest path (Monge) flow from S to ∂S related to the maximum flow?

References [1] M. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, John Wiley & Sons, New York, 2000. 2 Variational Principles and Residual Bounds for Nonpotential Equations 23 [2] G. Auchmuty, Duality for Non-Convex Variational Principles, J. Diff. Eqns. 50 (1983), 80—145. [3] G. Auchmuty, Saddle Point Methods, and Algorithms, for Non-Symmetric Linear Equations, Numer. Funct. Anal. Optim. 16 (1995), 1127—1142. [4] G. Auchmuty, Min-Max Problems for Non-Potential Operator Equations, Optimization Methods in Partial Differential Equations (S.

11) a ∈Nv,0 The next result summarizes some basic estimates for Πh . Its proof can be found in [21]. 2. There exists an h-independent constant C > 0 such that for all v ∈ V and f ∈ L2 (Ω), 34 V. Bostan, W. 14) kh−1/2 (v − Πh v)k20;γ ≤ C|v|21;Ω . 5) is uh ∈ Vh , a(uh , vh −uh )+j(vh )−j(uh ) ≥ (vh −uh ) ∀ vh ∈ Vh . 16) The discrete problem has a unique solution uh ∈ Vh by the standard existence and uniqueness result on elliptic variational inequalities. We need the following characterization of the finite element solution, similar to that of the solution of the continuous problem.

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