Transforming Design Spaces Using Pareto-Laplace Filters
Abstract: Optimization is a critical tool for addressing a broad range of human and technical problems. However, the paradox of advanced optimization techniques is that they have maximum utility for problems in which the relationship between the structure of the problem and the ultimate solution is the most obscure. The existence of solution with limited insight contrasts with techniques that have been developed for a broad range of engineering problems where integral transform techniques yield solutions and insight in tandem. Here, we present a ``Pareto-Laplace'' integral transform framework that can be applied to problems typically studied via optimization. We show that the framework admits related geometric, statistical, and physical representations that provide new forms of insight into relationships between objectives and outcomes. We argue that some known approaches are special cases of this framework, and point to a broad range of problems for further application.
- G. James, ed., Advanced Modern Engineering Mathematics, fifth edition ed. (Pearson Education, Harlow, United Kingdom, 2018).
- K. Ogata, Modern Control Engineering, 5th ed., Prentice-Hall Electrical Engineering Series. Instrumentation and Controls Series (Prentice-Hall, Boston, 2010).
- L. Debnath and D. Bhatta, Integral Transforms and Their Applications, 2nd ed. (Chapman & Hall/CRC, Boca Raton, Fla., 2007).
- C. Moore and S. Mertens, The Nature of Computation (Oxford University Press, Oxford, 2011).
- J. R. R. A. Martins and A. B. Lambe, Multidisciplinary Design Optimization: A Survey of Architectures, AIAA Journal 51, 2049 (2013).
- C. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27, 379 (1948).
- S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing, Science 220, 671 (1983).
- H. Gould, J. Tobochnik, and W. Christian, An Introduction to Computer Simulation Methods: Applications to Physical Systems, revised third edition ed. (Amazon Fulfillment, Wrocław, 2023).
- E. G. Teich, G. van Anders, and S. C. Glotzer, Identity crisis in alchemical space drives the entropic colloidal glass transition, Nat. Commun. 10, 64 (2019).
- E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).
- D. Frenkel and B. Smit, Understanding Molecular Simulation; from Algorithms to Applications (Academic Press, 2002).
- L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Butterworth-Heinemann, Oxford, 1980).
- N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading MA, 1992).
- J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Oxford University Press, Oxford, 2002).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.