Semantics of Integrating and Differentiating Singularities
summary
A formal semantics for programs that evaluate and differentiate singular integrals, giving a denotational model and an operational Monte Carlo estimator that is proven unbiased and consistent; a practical JAX implementation (SingularFlow, ~100 lines) that reuses samples and uses a symmetric-reflection sampler; benchmarks on the finite Hilbert transform and four aerodynamics/mechanical singular integral equations show numerically matching ground truth while naive Monte Carlo and straightforward AD produce orders-of-magnitude errors; JIT yields comparable performance.
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chapters (11)
- 0:00Singular function semantics
- 1:46Automatic differentiation
- 2:52Finite part integrals
- 4:38Aerodynamics modeling
- 6:26Semantics in SingularFlow
- 8:11Language semantics for integration
- 10:46Signal processing applications
- 13:15Crack propagation problems
- 14:38Related work and future
- 16:46Conclusion and open source
- 17:28Q&A: Monte Carlo variance
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