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Ryan Kurniawan - One of the best experts on this subject based on the ideXlab platform.
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Foundations of Computational Mathematics, 2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract: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.
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weak Convergence rates of spectral galerkin approximations for spdes with nonlinear diffusion coefficients
arXiv: Probability, 2014Co-Authors: Daniel Conus, Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak Convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak Convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak Convergence Problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak Convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak Convergence Problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak Convergence Problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak Convergence Problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kind of spatial, temporal, and noise numerical approximations for semilinear SEEs.
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weak Convergence rates of spectral galerkin approximations for spdes with nonlinear diffusion coefficients
The annual research report, 2014Co-Authors: Daniel Conus, Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak Convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp weak Convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [Math. Comp. 80 (2011) 89–117] for details. In this article, we solve the weak Convergence Problem emerged from Debussche’s article in the case of spectral Galerkin approximations and establish essentially sharp weak Convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak Convergence Problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak Convergence Problem emerged from Debussche’s article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito-type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak Convergence Problem emerged from Debussche’s article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs.
Muhammad Zia - One of the best experts on this subject based on the ideXlab platform.
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a convex relaxation approach to higher order statistical approaches to signal recovery
IEEE Transactions on Vehicular Technology, 2017Co-Authors: Huydung Han, Zhi Ding, Muhammad ZiaAbstract:In this paper, we investigate an efficient numerical approach for solving higher order statistical methods for blind and semiblind signal recovery from nonideal channels. We develop numerical algorithms based on convex optimization relaxation for the minimization of higher order statistical cost functions. The new formulation through convex relaxation overcomes the local Convergence Problem of existing gradient-descent-based algorithms and applies to several well-known cost functions for effective blind signal recovery, including blind equalization and blind source separation in both single-input–single-output (SISO) and multiple-input–multiple-output (MIMO) systems. We also propose a fourth-order pilot-based cost function that benefits from this approach. The simulation results demonstrate that our approach is suitable for short-length packet data transmission using only a few pilot symbols.
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a convex relaxation approach to higher order statistical approaches to signal recovery
arXiv: Information Theory, 2016Co-Authors: Huydung Han, Zhi Ding, Muhammad ZiaAbstract:In this work, we investigate an efficient numerical approach for solving higher order statistical methods for blind and semi-blind signal recovery from non-ideal channels. We develop numerical algorithms based on convex optimization relaxation for minimization of higher order statistical cost functions. The new formulation through convex relaxation overcomes the local Convergence Problem of existing gradient descent based algorithms and applies to several well-known cost functions for effective blind signal recovery including blind equalization and blind source separation in both single-input-single-output (SISO) and multi-input-multi-output (MIMO) systems. We also propose a fourth order pilot based cost function that benefits from this approach. The simulation results demonstrate that our approach is suitable for short-length packet data transmission using only a few pilot symbols.
Arnulf Jentzen - One of the best experts on this subject based on the ideXlab platform.
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Foundations of Computational Mathematics, 2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract: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.
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weak Convergence rates of spectral galerkin approximations for spdes with nonlinear diffusion coefficients
arXiv: Probability, 2014Co-Authors: Daniel Conus, Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak Convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak Convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak Convergence Problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak Convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak Convergence Problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak Convergence Problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak Convergence Problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kind of spatial, temporal, and noise numerical approximations for semilinear SEEs.
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weak Convergence rates of spectral galerkin approximations for spdes with nonlinear diffusion coefficients
The annual research report, 2014Co-Authors: Daniel Conus, Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak Convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp weak Convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [Math. Comp. 80 (2011) 89–117] for details. In this article, we solve the weak Convergence Problem emerged from Debussche’s article in the case of spectral Galerkin approximations and establish essentially sharp weak Convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak Convergence Problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak Convergence Problem emerged from Debussche’s article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito-type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak Convergence Problem emerged from Debussche’s article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs.
Stefan Gerhold - One of the best experts on this subject based on the ideXlab platform.
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the longstaff schwartz algorithm for levy models results on fast and slow Convergence
Annals of Applied Probability, 2011Co-Authors: Stefan GerholdAbstract:We investigate the Longstaff-Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090-2119] and Stentoft [Manag. Sei. 50 (2004) 1193-1203] to several Levy models, in particular the geometric Meixner model. A convenient setting to analyze this Convergence Problem is provided by the Levy-Sheffer systems introduced by Schoutens and Teugels.
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the longstaff schwartz algorithm for l e vy models results on fast and slow Convergence
arXiv: Probability, 2008Co-Authors: Stefan GerholdAbstract:We investigate the Longstaff--Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090--2119] and Stentoft [Manag. Sci. 50 (2004) 1193--1203] to several L\'{e}vy models, in particular the geometric Meixner model. A convenient setting to analyze this Convergence Problem is provided by the L\'{e}vy--Sheffer systems introduced by Schoutens and Teugels.
Hiroshi Saruwatari - One of the best experts on this subject based on the ideXlab platform.
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blind source separation based on a fast Convergence algorithm combining ica and beamforming
IEEE Transactions on Audio Speech and Language Processing, 2006Co-Authors: Hiroshi Saruwatari, Toshiya Kawamura, Tsuyoki Nishikawa, Akinobu Lee, Kiyohiro ShikanoAbstract:We propose a new algorithm for blind source separation (BSS), in which independent component analysis (ICA) and beamforming are combined to resolve the slow-Convergence Problem through optimization in ICA. The proposed method consists of the following three parts: (a) frequency-domain ICA with direction-of-arrival (DOA) estimation, (b) null beamforming based on the estimated DOA, and (c) integration of (a) and (b) based on the algorithm diversity in both iteration and frequency domain. The unmixing matrix obtained by ICA is temporally substituted by the matrix based on null beamforming through iterative optimization, and the temporal alternation between ICA and beamforming can realize fast- and high-Convergence optimization. The results of the signal separation experiments reveal that the signal separation performance of the proposed algorithm is superior to that of the conventional ICA-based BSS method, even under reverberant conditions.
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fast Convergence algorithm for ica based blind source separation using array signal processing
IEEE Signal Processing Workshop on Statistical Signal Processing, 2001Co-Authors: Hiroshi Saruwatari, Toshiya Kawamura, Kiyohiro ShikanoAbstract:We propose a new algorithm for blind source separation (BSS), in which independent component analysis (ICA) and beamforming are combined to resolve the low-Convergence Problem through optimization in ICA. The proposed method consists of the following two parts: frequency-domain ICA with direction-of-arrival (DOA) estimation, and null beamforming based on the estimated DOA. The alternation of learning between ICA and beamforming can realize fast- and high-Convergence optimization. The results of the signal separation experiments reveal that the signal separation performance of the proposed algorithm is superior to that of the conventional ICA-based BSS method.
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blind source separation combining frequency domain ica and beamforming
International Conference on Acoustics Speech and Signal Processing, 2001Co-Authors: Hiroshi Saruwatari, Satoshi Kurita, Kazuya TakedaAbstract:We describe a new method of blind source separation (BSS) on a microphone array combining subband independent component analysis (ICA) and beamforming. The proposed array system consists of the following three sections: (1) subband-ICA-based BSS section with direction-of-arrival (DOA) estimation; (2) null beamforming section based on the estimated DOA information; and (3) integration of (1) and (2) based on the algorithm diversity. Using this technique, we can resolve the low-Convergence Problem through optimization in ICA. The results of the signal separation experiments reveal that a noise reduction rate (NRR) of about 18 dB is obtained under the nonreverberant condition, and NRR of 8 dB and 6 dB are obtained in the case that the reverberation times are 150 msec and 300 msec. These performances are superior to those of both simple ICA-based BSS and simple beamforming method.