arXiv subjects

Bosu Choi

Publications and source records attributed to Bosu Choi.

6 recordsLinked to original sources

arXiv2024

Sampling error mitigation through spectrum smoothing: first experiments with ensemble transform Kalman filters and Lorenz models

In data assimilation, an ensemble provides a way to propagate the probability density of a system described by a nonlinear prediction model. Although a large ensemble size is required for statistical accuracy, the ensemble size is typically limited to a small number due to the computational cost of running the prediction model, which leads to a sampling error. Several methods, such as localization and inflation, exist to mitigate the sampling error, often requiring problem-dependent fine-tuning and design. This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum. It turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations. The efficacy of the new idea is validated through a suite of stringent test regimes of the Lorenz 96 turbulent model.

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arXiv2023

Weighted inhomogeneous regularization for inverse problems with indirect and incomplete measurement data

Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's characteristics, such as sparsity or smoothness. Inhomogeneous regularization, which incorporates a spatially varying exponent $p$ in the standard $\ell_p$-norm-based framework, has been used to recover signals with spatially varying features. This study introduces weighted inhomogeneous regularization, an extension of the standard approach incorporating a novel exponent design and spatially varying weights. The proposed exponent design mitigates misclassification when distinct characteristics are spatially close, while the weights address challenges in recovering regions with small-scale features that are inadequately captured by traditional $\ell_p$-norm regularization. Numerical experiments, including synthetic image reconstruction and the recovery of sea ice data from incomplete wave measurements, demonstrate the effectiveness of the proposed method.

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arXiv2020

Sparse Harmonic Transforms: A New Class of Sublinear-time Algorithms for Learning Functions of Many Variables

We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification(UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let $B$ be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality $|B|=N$. We will develop methods that approximate any function $f$ that is sparse in the BOPB, that is, $f:\mathcal{D}\subset R^D\rightarrow C$ of the form $f(\mathbf{x})=\sum_{b\in S}c_b\cdot b(\mathbf{x})$ with $S\subset B$ of cardinality $|S| =s\ll N$. Our method has a runtime of just $(s\log N)^{O(1)}$, uses only $(s\log N)^{O(1)}$ function evaluations on a fixed and nonadaptive grid, and not more than $(s\log N)^{O(1)}$ bits of memory. For $s\ll N$, the runtime $(s\log N)^{O(1)}$ will be less than what is required to simply enumerate the elements of the basis $B$; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as "sublinear-time". This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size $Ω(N)$.

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arXiv2019

Sparse Harmonic Transforms II: Best $s$-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

In this paper, we develop a sublinear-time compressive sensing algorithm for approximating functions of many variables which are compressible in a given Bounded Orthonormal Product Basis (BOPB). The resulting algorithm is shown to both have an associated best $s$-term recovery guarantee in the given BOPB, and also to work well numerically for solving sparse approximation problems involving functions contained in the span of fairly general sets of as many as $\sim10^{230}$ orthonormal basis functions. All code is made publicly available. As part of the proof of the main recovery guarantee new variants of the well known CoSaMP algorithm are proposed which can utilize any sufficiently accurate support identification procedure satisfying a {Support Identification Property (SIP)} in order to obtain strong sparse approximation guarantees. These new CoSaMP variants are then proven to have both runtime and recovery error behavior which are largely determined by the associated runtime and error behavior of the chosen support identification method. The main theoretical results of the paper are then shown by developing a sublinear-time support identification algorithm for general BOPB sets which is robust to arbitrary additive errors. Using this new support identification method to create a new CoSaMP variant then results in a new robust sublinear-time compressive sensing algorithm for BOPB-compressible functions of many variables.

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arXiv2019

Multiscale High-Dimensional Sparse Fourier Algorithms for Noisy Data

We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $Θ(ds \log s)$ average-case runtime and $Θ(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best $s$ Fourier modes. In this paper, we propose a multiscale sparse Fourier algorithm for noisy samples that proves to be both robust against noise and efficient.

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arXiv2019

High-Dimensional Sparse Fourier Algorithms

In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In ``Adaptive Sublinear Time Fourier Algorithm" by D. Lawlor, Y. Wang and A. Christlieb (2013), an efficient algorithm with $Θ(k\log k)$ average-case runtime and $Θ(k)$ average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth $N$, where $k$ is the number of significant modes such that $k\ll N$. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in the paper mentioned above. Note a higher dimensional signal can always be unwrapped into a one dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts: ``partial unwrapping'' and ``tilting''. These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.

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