Stable Identification of Sources Located on Interface of Nonhomogeneous Media

Mones, José Julio Conde; Aguayo, Emmanuel Roberto Estrada; Óliveros, J.; Hernández-Gracidas, Carlos Arturo; Morín-Castillo, M. M.

doi:10.3390/math9161932

Cited by 4 publications

(5 citation statements)

References 33 publications

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“…Our approach demonstrates that the algorithm can be parallelized to take advantage of the unique capabilities of FPGAs. This implementation demonstrates the feasibility and efficiency of utilizing FPGAs for inverse source problems, which is an important and growing field [2,3,9,20,21]. Hence, the implementation developed in this work provides a new and promising solution for future research in this field.…”

Section: Implementation Of Algorithms In Fpgasmentioning

confidence: 86%

“…The Tikhonov regularization method is the best known among them; it depends on a parameter that must be chosen in terms of the measurement error. The identification problem studied in this work only presents numerical instability since the problem has a unique solution [7,9]. The regularization in this work considers two parameters.…”

Section: Fpgas and Inverse Source Problemsmentioning

confidence: 99%

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FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

Oliveros-Oliveros,

Conde-Sánchez,

Hernández-Gracidas

et al. 2024

Applied Sciences

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Objective: This work presents an implementation of a stable algorithm that recovers sources located at the boundary separating two homogeneous media in field-programmable gate arrays. Two loop unrolling architectures were developed and analyzed for this purpose. This inverse source problem is ill-posed due to numerical instability, i.e., small errors in the measurement can produce significant changes in the source location. Methodology: To handle the numerical instability when recovering these sources, the Tikhonov regularization method in combination with the Fourier series truncation method are applied in the stable algorithm. This stable algorithm is implemented in two different architectures developed in this work: The first architecture (Mode 1) allows for different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (Mode 2) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. Results: The architectures were tested on four devices of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two hardware implementations of the stable algorithm were validated using synthetic examples implemented in MATLAB, which shows the advantages of each architecture. Contributions: We developed two efficient architectures based on a loop unrolling design for source identification problems. These are effective strategies to divide and assign tasks to the configurable hardware, and they appear as an appropriate technique for implementing the algorithm. The first one is simple and allows for different operating speeds. The second one uses a control system based on multiplexors that reduce resource consumption and complexity of the design and can be used for multichannel problems. From the numerical test, we found the regularization parameters. The synthetic examples developed here can be considered for similar problems and can be extended to concentric spheres.

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Section: Implementation Of Algorithms In Fpgasmentioning

confidence: 86%

Section: Fpgas and Inverse Source Problemsmentioning

confidence: 99%

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

Oliveros-Oliveros,

Conde-Sánchez,

Hernández-Gracidas

et al. 2024

Applied Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our approach shows that the algorithm can be parallelized, taking advantage of the unique capabilities of FPGAs. This implementation demonstrates the feasibility and efficiency of utilizing FPGAs for inverse source problems, which is an important and growing field [2,3,[12][13][14]. Hence, the implementation developed in this work provides a new and promising solution for future research in this field.…”

Section: Algorithm Implementation In Fpgamentioning

confidence: 89%

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-homogeneous Circular Region

Oliveros,

Sánchez,

Gracidas

et al. 2023

Preprint

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This work implements a stable algorithm in Field-Programmable Gate Arrays (FPGAs) for which two architectures (unrolling the loop) were developed and analyzed. The algorithm recovers sources located at the boundary separating two homogeneous media that make up a two-dimensional non-homogeneous region from measurements on the boundary of such region. The problem of recovering these sources is an ill-posed inverse problem as small errors in the measurement can produce important changes in the source location. Inverse source problems have many applications in different areas, such as engineering and medicine, making the proposed implementation important. The first architecture (mode one) allows considering different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (mode two) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. The architectures were tested on four FPGAs of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two FPGA implementations of the algorithm were validated using synthetic examples implemented in MATLAB. The results presented here can be extended to concentric spheres and complex geometries.

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“…When there are superficial sources, they are defined on S 1 and the boundary condition, corresponding to the flux on the boundary S 1 , changes. For example, in inverse electroencephalography, these superficial currents correspond to bioelectrical cortical sources [29,30].…”

Section: Mathematical Model For a Region With Two Conductive Layersmentioning

confidence: 99%

“…In the next Section, we show that this solution can be achieved using a conjugate gradient algorithm operating in space U . This methodology has been used in other works to solve inverse problems in elliptic differential equations in similar or different contexts (see, for example, [18,19,30,34]).…”

Section: Optimality Conditionsmentioning

confidence: 99%

Stable Numerical Identification of Sources in Non-Homogeneous Media

et al. 2022

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In this work, we present a numerical algorithm to solve the inverse problem of volumetric sources from measurements on the boundary of a non-homogeneous conductive medium, which is made of conductive layers with constant conductivity in each layer. This inverse problem is ill-posed since there is more than one source that can generate the same measurement. Furthermore, the ill-posedness is due to the fact that small variations (or errors) in the measurement (input data) can produce substantial variations in the identified source location. We propose two steps to solve this inverse problem in some classes of sources: we first recover the harmonic part of the volumetric source, and, in a second step, we compute the non-harmonic part of the source. For the reconstruction of the harmonic part of the source, we follow a variational approach based on the reformulation of the inverse problem as a distributed control problem, for which the cost function incorporates a penalized term with the input data on the boundary. This cost function is minimized by a conjugate gradient algorithm in combination with a finite element discretization. We recover the non-harmonic component of the source using a priori information and an iterative algorithm for some particular classes of sources. To validate the numerical methodology, we develop synthetic examples both in circular (simple) and irregular (complex) regions. The numerical results show that the proposed methodology allows to recover the complete source and produce stable and accurate numerical solutions.

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Stable Identification of Sources Located on Interface of Nonhomogeneous Media

Cited by 4 publications

References 33 publications

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-homogeneous Circular Region

Stable Numerical Identification of Sources in Non-Homogeneous Media

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