Journal of the Acoustical Society of America, Oct 1, 2020
Room impulse responses can be simulated using various methods that differ in their computational ... more Room impulse responses can be simulated using various methods that differ in their computational complexity and accuracy. While boundary element methods (BEM) potentially can provide accurate simulations in complex geometries including effects such as diffraction and absorption, their use is limited due to relatively large values of parameter kD (k is the wavenumber and D is the room diameter). The size of the mesh also grows as the square of this parameter. We show that the fast multipole (FMM) accelerated BEM realized on multicore workstations can handle problems for rooms of D ∼ 10 m for the frequency range up to ~5 kHz (kD ∼ 1000). For such problems, needing meshes with several million elements, stable and efficient methods for high frequency FMM are needed. We discuss these issues and demonstrate results of such computations both in frequency and time domains. In simple cases numerical solutions are compared with analytical solutions obtained by the method of images. For more complex cases including rooms of different shapes with baffles and openings two methods, the direct and indirect BEM, are studied and their performance discussed. [This work is supported by VisiSonics Corporation.]
This chapter considers various translations and rotation transforms, which enable expansion of so... more This chapter considers various translations and rotation transforms, which enable expansion of solutions of the Helmholtz equation in various domains, and express solutions in various reference frames. Some of the properties and methods for the computation of the structures of translation and rotation coefficients are described. The translation coefficients obey a number of symmetries, which are particularly important for their computations, since one can substantially reduce the amount of computational work. They also can be derived from symmetries of the Clebsch–Gordan coefficients or related symbols. Several theorems and derived recurrence relations for the translation and rotation coefficients are proved. Recurrence relations among the fundamental solutions of the Helmholtz equation produce recurrence relations for the reexpansion coefficients due to invariance of the differential operators ә z , ә x±iy with respect to translations. Their proofs are based on the theorems for differentiation of elementary solutions in spherical coordinates. These relations provide faster computation of these coefficients than methods based on expansions involving Clebsch-Gordan or similar coefficients.
Active Noise Cancellation (ANC) at a target area in an open space, as opposed to cancellation in ... more Active Noise Cancellation (ANC) at a target area in an open space, as opposed to cancellation in the ears through headphones, can lead to future applications. For instance, a personal acoustic environment in an airplane seat or inside a car, or a quiet zone in a noisy shared workspace can be possible using such open-space ANC without any uncomfortable on-body audio equipment. Recent advancements reinforce the practicality of such systems. However, regularization of the cancellation signal has been a crucial challenge in open-space ANC as it causes amplification of noise at locations away from the target area. This work presents a spherical harmonics-domain feed-forward spatial ANC method with a room-wide global cost function to address this issue. This room-wide global cost function is used for optimizing the set of regularization hyperparameters, while at run time only local information captured by a microphone array surrounding the target listening zone is required. Numerical experiments applying the proposed method in a simulated reverberant room show the effectiveness of the proposed method in creating a specific zone of silence with low to moderate noise amplification in the rest of the room.
2012 SC Companion: High Performance Computing, Networking Storage and Analysis, 2012
ABSTRACT We use a particle-based method to simulate incompressible flows, where the Fast Multipol... more ABSTRACT We use a particle-based method to simulate incompressible flows, where the Fast Multipole Method (FMM) is used to accelerate the calculation of particle interactions. The most time-consuming kernelsâ"the Biot-Savart equation and stretching term of the vorticity equationâ"are mathematically reformulated so that only two Laplace scalar potentials are used instead of six, while automatically ensuring divergence-free far-field computation. Based on this formulation, and on our previous work for a scalar heterogeneous FMM algorithm, we develop a new FMM-based vortex method capable of simulating general flows including turbulence on heterogeneous architectures, which distributes the work between multi-core CPUs and GPUs to best utilize the hardware resources and achieve excellent scalability. The algorithm also uses new data structures which can dynamically manage inter-node communication and load balance efficiently but with only a small parallel construction overhead. This algorithm can scale to large-sized clusters showing both strong and weak scalability. Careful error and timing trade-off analysis are also performed for the cutoff functions induced by the vortex particle method. Our implementation can perform one time step of the velocity+stretching for one billion particles on 32 nodes in 55.9 seconds, which yields 49.12 Tflop/s.
IEEE Antennas and Propagation Society International Symposium (IEEE Cat. No.02CH37313)
Abstract For geometries such as multiple spheres the scattering problem can be solved more effici... more Abstract For geometries such as multiple spheres the scattering problem can be solved more efficiently using semi-analytical (spectral) techniques. Here we develop such a method, which in some sense is analytical since it is based on solutions in the form of infinite series. At the same time the method is numerical, since in the simple form presented here, it requires the solution of a large linear system for determining coefficients in the series, to satisfy boundary conditions on multiple spheres. The solution is based on decomposing ...
2014 IEEE International Parallel & Distributed Processing Symposium Workshops, 2014
The fast multipole method (FMM) is often used to accelerate the calculation of particle interacti... more The fast multipole method (FMM) is often used to accelerate the calculation of particle interactions in particle-based methods to simulate incompressible flows. To evaluate the most time-consuming kernels -- the Biot-Savart equation and stretching term of the vorticity equation, we mathematically reformulated it so that only two Laplace scalar potentials are used instead of six. This automatically ensuring divergence-free far-field computation. Based on this formulation, we developed a new FMM-based vortex method on heterogeneous architectures, which distributed the work between multicore CPUs and GPUs to best utilize the hardware resources and achieve excellent scalability. The algorithm uses new data structures which can dynamically manage inter-node communication and load balance efficiently, with only a small parallel construction overhead. This algorithm can scale to large-sized clusters showing both strong and weak scalability. Careful error and timing trade-off analysis are also performed for the cutoff functions induced by the vortex particle method. Our implementation can perform one time step of the velocity+stretching calculation for one billion particles on 32 nodes in 55.9 seconds, which yields 49.12 Tflop/s.
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, 2004
This chapter presents various algorithms that can be used for reduction of complexity of translat... more This chapter presents various algorithms that can be used for reduction of complexity of translation. It shows some exact decompositions of the rotation, truncated coaxial translation, and truncated general translation matrices. Fast rotation algorithm has complexity 0(p 2 log p) and can be used in various methods of translations. The other two algorithms relies on the translation matrix diagonalization via fast Legendre and fast spherical transforms, which can be performed with complexity 0(p 2 log 2 p ). It is the fastest complexity that can be currently achieved for a single translation with truncated translation matrices. Low- and high-frequency asymptotic expansions are also reviewed, which shows that for very low frequencies, an arbitrary translation can be performed with complexity 0(p 2 log p) . For higher frequencies, complete asymptotic representations that enable any order asymptotic expansions for coaxial translation operators, are obtained. These expansions, however, are derived for fixed sizes of the truncated translation matrices, and further research is needed to obtain uniform asymptotic representation of translation operators. The diagonal forms of the translation operators interpreted as a method of signature function achieves complexity 0(p 2 ) for translations when implemented for the class of band-unlimited functions. Theoretically, this is the lowest complexity of translations and while the tests show that the method works and can be practical, its accuracy is limited at low frequencies and the algorithm is unstable for a sequence of translations. This instability can be damped by the application of the spherical filters, which control the function bandwidths.
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, 2004
This chapter focuses on O(p 3 ) translation methods, which have a low asymptotic constant and is ... more This chapter focuses on O(p 3 ) translation methods, which have a low asymptotic constant and is considered as the best available translation methods for low and moderate frequencies. O(p 3 ) matrix-based translation methods are stable to errors and do not require additional data filtering, which is an important stabilization step for high-frequency translation methods based on diagonal forms of translation operators. The chapter considers some of the basics of representation of translation operators, including matrix and integral representations, and shows that rotational–coaxial translation decomposition with recursive computation of the rotation and translation matrix elements results in O(p 3 ) translation algorithm with relatively small asymptotic constant. This method is faster than O(p 4 ) method based on direct product of the translation matrix with the vector of expansion coefficients. The chapter also introduces another 0(p 3 ) translation method based on sparse matrix decomposition of the translation operator, where the sparse matrices are differentiation matrices commutative with translation matrices.
Journal of the Acoustical Society of America, Oct 1, 2020
Room impulse responses can be simulated using various methods that differ in their computational ... more Room impulse responses can be simulated using various methods that differ in their computational complexity and accuracy. While boundary element methods (BEM) potentially can provide accurate simulations in complex geometries including effects such as diffraction and absorption, their use is limited due to relatively large values of parameter kD (k is the wavenumber and D is the room diameter). The size of the mesh also grows as the square of this parameter. We show that the fast multipole (FMM) accelerated BEM realized on multicore workstations can handle problems for rooms of D ∼ 10 m for the frequency range up to ~5 kHz (kD ∼ 1000). For such problems, needing meshes with several million elements, stable and efficient methods for high frequency FMM are needed. We discuss these issues and demonstrate results of such computations both in frequency and time domains. In simple cases numerical solutions are compared with analytical solutions obtained by the method of images. For more complex cases including rooms of different shapes with baffles and openings two methods, the direct and indirect BEM, are studied and their performance discussed. [This work is supported by VisiSonics Corporation.]
This chapter considers various translations and rotation transforms, which enable expansion of so... more This chapter considers various translations and rotation transforms, which enable expansion of solutions of the Helmholtz equation in various domains, and express solutions in various reference frames. Some of the properties and methods for the computation of the structures of translation and rotation coefficients are described. The translation coefficients obey a number of symmetries, which are particularly important for their computations, since one can substantially reduce the amount of computational work. They also can be derived from symmetries of the Clebsch–Gordan coefficients or related symbols. Several theorems and derived recurrence relations for the translation and rotation coefficients are proved. Recurrence relations among the fundamental solutions of the Helmholtz equation produce recurrence relations for the reexpansion coefficients due to invariance of the differential operators ә z , ә x±iy with respect to translations. Their proofs are based on the theorems for differentiation of elementary solutions in spherical coordinates. These relations provide faster computation of these coefficients than methods based on expansions involving Clebsch-Gordan or similar coefficients.
Active Noise Cancellation (ANC) at a target area in an open space, as opposed to cancellation in ... more Active Noise Cancellation (ANC) at a target area in an open space, as opposed to cancellation in the ears through headphones, can lead to future applications. For instance, a personal acoustic environment in an airplane seat or inside a car, or a quiet zone in a noisy shared workspace can be possible using such open-space ANC without any uncomfortable on-body audio equipment. Recent advancements reinforce the practicality of such systems. However, regularization of the cancellation signal has been a crucial challenge in open-space ANC as it causes amplification of noise at locations away from the target area. This work presents a spherical harmonics-domain feed-forward spatial ANC method with a room-wide global cost function to address this issue. This room-wide global cost function is used for optimizing the set of regularization hyperparameters, while at run time only local information captured by a microphone array surrounding the target listening zone is required. Numerical experiments applying the proposed method in a simulated reverberant room show the effectiveness of the proposed method in creating a specific zone of silence with low to moderate noise amplification in the rest of the room.
2012 SC Companion: High Performance Computing, Networking Storage and Analysis, 2012
ABSTRACT We use a particle-based method to simulate incompressible flows, where the Fast Multipol... more ABSTRACT We use a particle-based method to simulate incompressible flows, where the Fast Multipole Method (FMM) is used to accelerate the calculation of particle interactions. The most time-consuming kernelsâ"the Biot-Savart equation and stretching term of the vorticity equationâ"are mathematically reformulated so that only two Laplace scalar potentials are used instead of six, while automatically ensuring divergence-free far-field computation. Based on this formulation, and on our previous work for a scalar heterogeneous FMM algorithm, we develop a new FMM-based vortex method capable of simulating general flows including turbulence on heterogeneous architectures, which distributes the work between multi-core CPUs and GPUs to best utilize the hardware resources and achieve excellent scalability. The algorithm also uses new data structures which can dynamically manage inter-node communication and load balance efficiently but with only a small parallel construction overhead. This algorithm can scale to large-sized clusters showing both strong and weak scalability. Careful error and timing trade-off analysis are also performed for the cutoff functions induced by the vortex particle method. Our implementation can perform one time step of the velocity+stretching for one billion particles on 32 nodes in 55.9 seconds, which yields 49.12 Tflop/s.
IEEE Antennas and Propagation Society International Symposium (IEEE Cat. No.02CH37313)
Abstract For geometries such as multiple spheres the scattering problem can be solved more effici... more Abstract For geometries such as multiple spheres the scattering problem can be solved more efficiently using semi-analytical (spectral) techniques. Here we develop such a method, which in some sense is analytical since it is based on solutions in the form of infinite series. At the same time the method is numerical, since in the simple form presented here, it requires the solution of a large linear system for determining coefficients in the series, to satisfy boundary conditions on multiple spheres. The solution is based on decomposing ...
2014 IEEE International Parallel & Distributed Processing Symposium Workshops, 2014
The fast multipole method (FMM) is often used to accelerate the calculation of particle interacti... more The fast multipole method (FMM) is often used to accelerate the calculation of particle interactions in particle-based methods to simulate incompressible flows. To evaluate the most time-consuming kernels -- the Biot-Savart equation and stretching term of the vorticity equation, we mathematically reformulated it so that only two Laplace scalar potentials are used instead of six. This automatically ensuring divergence-free far-field computation. Based on this formulation, we developed a new FMM-based vortex method on heterogeneous architectures, which distributed the work between multicore CPUs and GPUs to best utilize the hardware resources and achieve excellent scalability. The algorithm uses new data structures which can dynamically manage inter-node communication and load balance efficiently, with only a small parallel construction overhead. This algorithm can scale to large-sized clusters showing both strong and weak scalability. Careful error and timing trade-off analysis are also performed for the cutoff functions induced by the vortex particle method. Our implementation can perform one time step of the velocity+stretching calculation for one billion particles on 32 nodes in 55.9 seconds, which yields 49.12 Tflop/s.
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, 2004
This chapter presents various algorithms that can be used for reduction of complexity of translat... more This chapter presents various algorithms that can be used for reduction of complexity of translation. It shows some exact decompositions of the rotation, truncated coaxial translation, and truncated general translation matrices. Fast rotation algorithm has complexity 0(p 2 log p) and can be used in various methods of translations. The other two algorithms relies on the translation matrix diagonalization via fast Legendre and fast spherical transforms, which can be performed with complexity 0(p 2 log 2 p ). It is the fastest complexity that can be currently achieved for a single translation with truncated translation matrices. Low- and high-frequency asymptotic expansions are also reviewed, which shows that for very low frequencies, an arbitrary translation can be performed with complexity 0(p 2 log p) . For higher frequencies, complete asymptotic representations that enable any order asymptotic expansions for coaxial translation operators, are obtained. These expansions, however, are derived for fixed sizes of the truncated translation matrices, and further research is needed to obtain uniform asymptotic representation of translation operators. The diagonal forms of the translation operators interpreted as a method of signature function achieves complexity 0(p 2 ) for translations when implemented for the class of band-unlimited functions. Theoretically, this is the lowest complexity of translations and while the tests show that the method works and can be practical, its accuracy is limited at low frequencies and the algorithm is unstable for a sequence of translations. This instability can be damped by the application of the spherical filters, which control the function bandwidths.
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, 2004
This chapter focuses on O(p 3 ) translation methods, which have a low asymptotic constant and is ... more This chapter focuses on O(p 3 ) translation methods, which have a low asymptotic constant and is considered as the best available translation methods for low and moderate frequencies. O(p 3 ) matrix-based translation methods are stable to errors and do not require additional data filtering, which is an important stabilization step for high-frequency translation methods based on diagonal forms of translation operators. The chapter considers some of the basics of representation of translation operators, including matrix and integral representations, and shows that rotational–coaxial translation decomposition with recursive computation of the rotation and translation matrix elements results in O(p 3 ) translation algorithm with relatively small asymptotic constant. This method is faster than O(p 4 ) method based on direct product of the translation matrix with the vector of expansion coefficients. The chapter also introduces another 0(p 3 ) translation method based on sparse matrix decomposition of the translation operator, where the sparse matrices are differentiation matrices commutative with translation matrices.
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