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Analysis and simulation of a variational stabilization for the Helmholtz equation with noisy Cauchy data

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Abstract

This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied to appropriately perturb the underlying problem, which allows us to obtain a stable approximate solution. The corresponding approximate problem is of a hyperbolic equation, which is also a crucial aspect of this approach. Error estimates between the approximate and true solutions are derived with respect to the noise level. From this analysis, the Lipschitz stability with respect to the noise level follows. Some numerical examples are provided to see how our numerical algorithm works well.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

V.A.K. thanks Prof. Dr. Roselyn Williams and Prof. Dr. Desmond Stephens (Tallahassee, USA) for their support during the time V.A.K working at Florida A &M University. N.D.T. acknowledges Dr. Nguyen Thanh Long for the wholehearted guidance during his study at University of Science, and thanks Mr. Pham Truong Hoang Nhan for helpful discussions. The authors are thankful to Prof. Dr. Bruno Despres (Paris, France) for addressing the well-posedness of system (6).

Funding

This work was supported by the Faculty Research Awards Program (FRAP) at Florida A &M University, under the project “Approximation of a time-reversed reaction-diffusion system in cancer cell population dynamics” (007633).

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Authors and Affiliations

  1. Department of Mathematics, Florida A &M University, Tallahassee, FL, 32307, USA

    Vo Anh Khoa & Ajith Gunaratne

  2. Department of Mathematics and Computer Science, VNUHCM-University of Science, Ho Chi Minh City, Vietnam

    Nguyen Dat Thuc

  3. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Nguyen Dat Thuc

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  1. Vo Anh Khoa
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  2. Nguyen Dat Thuc
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  3. Ajith Gunaratne
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V.A.K. and N.D.T. organized and wrote this manuscript. V.A.K., N.D.T and A.G. contributed to all the steps of the proofs in this research together. All authors read and approved the final manuscript.

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Correspondence to Vo Anh Khoa.

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Communicated by Gunnar Martinsson.

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Khoa, V.A., Thuc, N.D. & Gunaratne, A. Analysis and simulation of a variational stabilization for the Helmholtz equation with noisy Cauchy data. Bit Numer Math 63, 37 (2023). https://doi.org/10.1007/s10543-023-00978-8

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