Convex optimization and sum-of-squares programming, coming from the field of real algebraic geometry, have recently been applied in the analysis of PDEs and ODEs. When the underlying equations are formulated in terms of polynomials, one can formulate sum-of-squares auxiliary functions using computer-assisted methods, and compute sharp bounds on integral functions [1], study existence of traveling waves [2], nonlinear stability questions through constructing Lyapunov functions [3], as well as find basins of attraction and much more. Sum-of-squares problems can be solved numerically via semidefinite programming, a step up from linear programming where approximate solutions can be found using modern interior-point methods.

[1]: A. Chernyavsky, J.J. Bramburger, G. Fantuzzi, D. Goluskin, Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization, SIAM J. Optim. 33, 481–512, 2023.

[2]: J.J. Bramburger, D. Goluskin, Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization, Proc. R. Soc. A 476, 20200450, 2020.

[3]: G. Fantuzzi, D. Goluskin, Bounding extreme events in nonlinear dynamics using convex optimization, SIAM J. Appl. Dyn. Syst. 19, 1823-1864, 2020.