Foundations of Computational Mathematics

Journal: Foundations of Computational Mathematics

Abbreviation

Found. Comput. Math.

Publisher

Springer

ISSN

1615-3375
1615-3383

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Publications 1 - 10 of 24

Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the hp-Version
Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria (2016)
Foundations of Computational Mathematics
Improved Resolution Estimate for the Two-Dimensional Super-Resolution and a New Algorithm for Direction of Arrival Estimation with Uniform Rectangular Array
Liu, Ping; Ammari, Habib (2024)
Foundations of Computational Mathematics
In this paper, we develop a new technique to obtain improved estimates for the computational resolution limits in two-dimensional super-resolution problems and present a new idea for developing two-dimensional super-resolution algorithms. To be more specific, our main contributions are fourfold: (1) Our work improves the resolution estimates for number detection and location recovery in two-dimensional super-resolution problems; (2) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems [or direction of arrival Problems (DOA)]. The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (3) Inspired by the new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (4) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys some advantages compared to the conventional DOA algorithms.
Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws
Fjordholm, Ulrik S.; Käppeli, Roger; Mishra, Siddhartha; et al. (2017)
Foundations of Computational Mathematics
Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Jentzen, Arnulf; Kurniawan, Ryan (2021)
Foundations of Computational Mathematics
Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions. In the recent article (Conus et al. in Ann Appl Probab 29(2):653–716, 2019) this weak convergence problem has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome this weak convergence problem in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild Itô-type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
Stable Phase Retrieval in Infinite Dimensions
Alaifari, Rima; Daubechies, Ingrid; Grohs, Philipp; et al. (2019)
Foundations of Computational Mathematics
The problem of phase retrieval is to determine a signal f∈ H, with H a Hilbert space, from intensity measurements | F(ω) | , where F(ω) : = ⟨ f, φω⟩ are measurements of f with respect to a measurement system (φω)ω∈Ω⊂H. Although phase retrieval is always stable in the finite-dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if H is infinite-dimensional: in that case phase retrieval is never uniformly stable (Alaifari and Grohs in SIAM J Math Anal 49(3):1895–1911, 2017; Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016); moreover, the stability deteriorates severely in the dimension of the problem (Cahill et al. 2016). On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function |F| of intensity measurements is concentrated on disjoint sets Dj⊂ Ω , i.e. when F=∑j=1kFj where each Fj is concentrated on Dj (and k≥ 2). Motivated by these considerations, we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing F up to a phase factor that is not global, but that can be different for each of the subsets Dj, i.e. recovering F up to the equivalence F∼∑j=1keiαjFj.We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance, if the measurement system is a Gabor frame or a frame of Cauchy wavelets.
ENO Reconstruction and ENO Interpolation Are Stable
Fjordholm, Ulrik S.; Mishra, Siddhartha; Tadmor, Eitan (2013)
Foundations of Computational Mathematics
We prove that the ENO reconstruction and ENO interpolation procedures are stable in the sense that the jump of the reconstructed ENO point values at each cell interface has the same sign as the jump of the underlying cell averages across that interface. Moreover, we prove that the size of these jumps after reconstruction relative to the jump of the underlying cell averages is bounded. Similar sign properties and the boundedness of the jumps hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on nonuniform meshes, indicate a remarkable rigidity of the piecewise polynomial ENO procedure.
Subexponential-Time Algorithms for Sparse PCA
Ding, Yunzi; Kunisky, Dmitriy; Wein, Alexander S.; et al. (2024)
Foundations of Computational Mathematics
Covariance's Loss is Privacy's Gain: Computationally Efficient, Private and Accurate Synthetic Data
Boedihardjo, March; Strohmer, Thomas; Vershynin, Roman (2024)
Foundations of Computational Mathematics
The protection of private information is of vital importance in data-driven research, business and government. The conflict between privacy and utility has triggered intensive research in the computer science and statistics communities, who have developed a variety of methods for privacy-preserving data release. Among the main concepts that have emerged are anonymity and differential privacy. Today, another solution is gaining traction, synthetic data. However, the road to privacy is paved with NP-hard problems. In this paper, we focus on the NP-hard challenge to develop a synthetic data generation method that is computationally efficient, comes with provable privacy guarantees and rigorously quantifies data utility. We solve a relaxed version of this problem by studying a fundamental, but a first glance completely unrelated, problem in probability concerning the concept of covariance loss. Namely, we find a nearly optimal and constructive answer to the question how much information is lost when we take conditional expectation. Surprisingly, this excursion into theoretical probability produces mathematical techniques that allow us to derive constructive, approximately optimal solutions to difficult applied problems concerning microaggregation, privacy and synthetic data.
Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients
Kuo, Frances Y.; Schwab, Christoph; Sloan, Ian H. (2015)
Foundations of Computational Mathematics
Hypersurfaces and their singularities in partial correlation testing
Lin, Shaowei; Uhler, Caroline; Sturmfels, Bernd; et al. (2014)
Foundations of Computational Mathematics

Publications 1 - 10 of 24

Journal: Foundations of Computational Mathematics

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