Optimal Krylov On Average
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909566927634432 |
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| author | Luo, Qi Schäfer, Florian |
| author_facet | Luo, Qi Schäfer, Florian |
| contents | We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost, while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly, with minimal computational overhead. The problem has a closed-form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in GP hyperparameter training and competitive physics-informed neural networks problem to demonstrate the effectiveness of our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Krylov On Average Luo, Qi Schäfer, Florian Numerical Analysis 65F10, 68W20, 65B99, 65C05 We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost, while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly, with minimal computational overhead. The problem has a closed-form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in GP hyperparameter training and competitive physics-informed neural networks problem to demonstrate the effectiveness of our approach. |
| title | Optimal Krylov On Average |
| topic | Numerical Analysis 65F10, 68W20, 65B99, 65C05 |
| url | https://arxiv.org/abs/2504.03914 |