Periodic Reporting for period 4 - 321 (from Cubic To Linear complexity in computational electromagnetics)
Reporting period: 2022-03-01 to 2023-08-31
Computational Electromagnetics (CEM) is the scientific field at the origin of all new modeling and simulation tools required by the constantly arising design challenges of emerging and future technologies in applied electromagnetics. CEM is at the interface between electrical engineering, advanced computing, and applied mathematics and it focuses on developing fast and efficient solvers to characterize the electromagnetic interactions, radiation, and scattering of large, multiscale, and complex objects. As in many other technological fields, however, the trend in all emerging technologies in electromagnetic engineering is going towards miniaturized, higher density and multi-scale scenarios. Computationally speaking this translates into a steep increase of the number of degrees of freedom. Given that the design cost (the cost of a multi-right-hand side problem dominated by matrix inversion) can scale as badly as cubically with these degrees of freedom, this can sensibly compromise the practical impact of CEM on future and emerging technologies. The Grand Challenge of 321 is to tackle this Holy Grail in Computational Electromagnetics by investigating a dynamic Fast Direct Solver for Maxwell Problems that would run in a linear-instead-of-cubic complexity for an arbitrary number and configuration of degrees of freedom. To attain this, the project’s objectives include the investigation of new modelling and solution strategies for a large plethora of scenarios and frequencies, new fast solution strategies, and impacting applications in dosimetry, bioelectromagnetism, and neuroimaging. The action has achieved all foreseen objectives resulting in several impactful results, including new mathematical and computational tools, fast solvers, and their applications to important real case scenarios. The new paradigm in fast solution strategies and a new plethora of technologies to achieve them that the project has obtained will have a substantial impact on the discipline within and well beyond the scope of this action.
The first part of the project has been devoted to the investigation of transformations towards Hermitian positive definite (HPD) problems. We have in fact investigated and obtained the first integral formulation for electromagnetics which is HPD independently of the sign of the original problem. Standard strategies can achieve such a desirable outcome only at the cost of increasing the condition number and thus making the modelling slow and unstable. Instead we have obtained here an HPD formulation for which not only the condition number does not worsen but is even improved by several orders of magnitude.
A significant challenge arising when solving electric problems containing loops and handles (most notably in electronic circuits) with full wave high frequency solvers is to handle the breakdown present in the lower part of the frequency spectrum. One common option is the use of magnetic formulations that do not suffer from this breakdown. A stable discretization of these equations, however, is extremely expensive in the low frequency limit. This problem has been solved in this project, with the first formulation that decouples frequency and integration precision for magnetic type equations. This resulted in the magnetic complement of the Calderón equation quite popular for electric formulations. Moreover, the linear combination of the two formulations leads to a high frequency stable equation that can model electromagnetic problems from zero to high frequency.
As per the project’s plan we have also extended these schemes to penetrable cases. We have obtained a full wave high frequency solver that encompasses the Eddy Current regime without instabilities, this will be quite impacting in electromagnetic compatibility and several strategies to extend our results to scenarios containing inhomogeneities and anisotropies have been achieved. Among the new schemes we have originated, we have notably obtained the first full wave integral equation solver attaining stable forward results in electroencephalography-based (EEG) neuroimaging. This will have a substantial impact on several applications, including diagnostics and treatment in epilepsy as well as in brain stimulation and neural implants.
The project has then achieved the fast direct solution strategies. This has been obtained by introducing new mathematical tools that allowed to handle particularly efficiently and in linear complexity the spectral error that would have otherwise prevented the direct solutions to occur. The technology has been applied to both metallic and penetrable bodies, including the inhomogeneous case. This new framework has already received recognition from the community with several best paper and honorable mentions awards.
The new obtained solvers have been parallelized on high performance computing platforms and applied to real case scenarios including neuroimaging and terahertz modelling. Finally, the obtained technology has been distributed as an open source package.
The project’s results have been disseminated in 31 publications already published. These contributions have collectively reveiced 10 paper awards and recognitions, showing how the community has already well-received and recognized the value of the project contributions. The project has also been widely communicating in the form of activities towards schools, press releases and articles, and general public presentations.
A significant challenge arising when solving electric problems containing loops and handles (most notably in electronic circuits) with full wave high frequency solvers is to handle the breakdown present in the lower part of the frequency spectrum. One common option is the use of magnetic formulations that do not suffer from this breakdown. A stable discretization of these equations, however, is extremely expensive in the low frequency limit. This problem has been solved in this project, with the first formulation that decouples frequency and integration precision for magnetic type equations. This resulted in the magnetic complement of the Calderón equation quite popular for electric formulations. Moreover, the linear combination of the two formulations leads to a high frequency stable equation that can model electromagnetic problems from zero to high frequency.
As per the project’s plan we have also extended these schemes to penetrable cases. We have obtained a full wave high frequency solver that encompasses the Eddy Current regime without instabilities, this will be quite impacting in electromagnetic compatibility and several strategies to extend our results to scenarios containing inhomogeneities and anisotropies have been achieved. Among the new schemes we have originated, we have notably obtained the first full wave integral equation solver attaining stable forward results in electroencephalography-based (EEG) neuroimaging. This will have a substantial impact on several applications, including diagnostics and treatment in epilepsy as well as in brain stimulation and neural implants.
The project has then achieved the fast direct solution strategies. This has been obtained by introducing new mathematical tools that allowed to handle particularly efficiently and in linear complexity the spectral error that would have otherwise prevented the direct solutions to occur. The technology has been applied to both metallic and penetrable bodies, including the inhomogeneous case. This new framework has already received recognition from the community with several best paper and honorable mentions awards.
The new obtained solvers have been parallelized on high performance computing platforms and applied to real case scenarios including neuroimaging and terahertz modelling. Finally, the obtained technology has been distributed as an open source package.
The project’s results have been disseminated in 31 publications already published. These contributions have collectively reveiced 10 paper awards and recognitions, showing how the community has already well-received and recognized the value of the project contributions. The project has also been widely communicating in the form of activities towards schools, press releases and articles, and general public presentations.
The project has obtained the following substantial advances with respect to the state of the art
1) The first existing modelling tool (for now in the metallic case) that renders any initial electromagnetic problem well-conditioned, fast solvable and Hermitian Positive Definite. HPD systems have several favorable properties including sharp convergence bounds, simplified error analyses, and spectral bounds. The effects of these properties in other scenarios, such as high order methods, have also been investigated with very promising outcomes. This technique can also be applied to filtering via manipulation of integral transforms (including wavelets, frames, and related transforms) and has already found applications in neuroimaging.
2) The first magnetic modelling framework that requires computationally feasible and low order integration precisions for arbitrary frequencies. The approach we have found can be applied in several other scenarios.
3) A new family of integral models that handle anisotropies, inhomogeneities, and losses in an extremely stable and computationally efficient way. These technologies will play a game changing role in several application scenarios and we are currently exploiting, with substantial results already obtained, their impact in neuroimaging and bioelectromagnetic assessments.
4) As an additional outcome of the project it has introduced a new mathematical tool: the quasi-Helmholtz filter. This opens the possibility of manipulating electromagnetic operators in linear complexity, changing the spectrum in unwanted regions allows for a new paradigm in forward, fast direct, and inverse solution strategies of electromagnetic problems.
5) The project has originated a new family of fast matrix-vector multiplication schemes and fast direct solver strategies. These new schemes, enabled by operator filtering allows for compression of both static and dynamic kernels providing substantial advances with respect to the state of the art in the field.
1) The first existing modelling tool (for now in the metallic case) that renders any initial electromagnetic problem well-conditioned, fast solvable and Hermitian Positive Definite. HPD systems have several favorable properties including sharp convergence bounds, simplified error analyses, and spectral bounds. The effects of these properties in other scenarios, such as high order methods, have also been investigated with very promising outcomes. This technique can also be applied to filtering via manipulation of integral transforms (including wavelets, frames, and related transforms) and has already found applications in neuroimaging.
2) The first magnetic modelling framework that requires computationally feasible and low order integration precisions for arbitrary frequencies. The approach we have found can be applied in several other scenarios.
3) A new family of integral models that handle anisotropies, inhomogeneities, and losses in an extremely stable and computationally efficient way. These technologies will play a game changing role in several application scenarios and we are currently exploiting, with substantial results already obtained, their impact in neuroimaging and bioelectromagnetic assessments.
4) As an additional outcome of the project it has introduced a new mathematical tool: the quasi-Helmholtz filter. This opens the possibility of manipulating electromagnetic operators in linear complexity, changing the spectrum in unwanted regions allows for a new paradigm in forward, fast direct, and inverse solution strategies of electromagnetic problems.
5) The project has originated a new family of fast matrix-vector multiplication schemes and fast direct solver strategies. These new schemes, enabled by operator filtering allows for compression of both static and dynamic kernels providing substantial advances with respect to the state of the art in the field.