@Article{icces.2023.09272,
AUTHOR = {Yanchuang Cao, Jun Liu, Dawei Chen},
TITLE = {A	Directional	Fast	Algorithm	for	Oscillatory	Kernels	with	Curvelet-Like	 Functions},
JOURNAL = {The International Conference on Computational \& Experimental Engineering and Sciences},
VOLUME = {25},
YEAR = {2023},
NUMBER = {4},
PAGES = {1--1},
URL = {http://www.techscience.com/icces/v25n4/53870},
ISSN = {1933-2815},
ABSTRACT = {Interactions	of	multiple	points	with	oscillatory	kernels	are	widely	encountered	in	wave	analysis.	For	large	
scale	problems,	its	direct	evaluation	is	prohibitive	since	the	computational	cost	increases	quadratically	with	
the	number	of	points.<br/>
Various	fast	algorithms	have	been	constructed	by	exploiting	specific	properties	of	the	kernel	function.	Early	
fast	 algorithms,	 such	 as	 the	 fast	 multipole	 method	 (FMM)	 and	 its	 variants,	 H2-matrix,	 adaptive	 cross	
approximation	 (ACA),	 wavelet-based	 method,	 etc.,	 are	 generally	 developed	 for	 kernels	 that	 are	
asymptotically	 smooth	when	 source	points	and	 target	points	are	well	 separated.	 For	 oscillatory	 kernels,	
however,	the	asymptotic	smoothness	criteria	is	only	satisfied	when	the	oscillation	is	insignificant,	thus	these	
fast	 algorithms	 are	 only	 suitable	 for	 low	 frequency	 cases.	 	 Fortunately,	 later	 the	 directional	 low	 rank	
property	of	highly	oscillatory	kernels	is	found,	based	on	which	various	directional	fast	algorithms that	are	
suitable	 for	 wideband	 problems are	 proposed,	 include	 the	 directional	 FMM,	 dirH2-ACA,	 and	 directional	
algebraic	FMM,	etc.	They	may	be	viewed	as	the	generalization	of	early	low-frequency fast	algorithms	to	high	
frequency	cases.<br/>
In	this	work,	a	curvelet-based	method	is	developed	by	generalizing	wavelet-based	method	to	high	frequency	
cases.	Multilevel	curvelet-like	functions	are	constructed	as	the	transform	of	the	original	nodal	basis.	Then
the	system	matrix	in	a	new	non-standard	form	is	derived	under	the	curvelet	basis,	which	would	be	nearly	
optimally	sparse	due	to	the	directional	low	rank	property	of	the	oscillatory	kernel. Its	sparsity	is	 further	
enhanced	via	a-posteriori	compression.	Finally, its	log-linear	computational	complexity	with	controllable
accuracy	is	demonstrated	by	numerical	results.<br/>
This	work	provides	another	highly	efficient	wideband	fast	algorithm	for	wave	problems.	Moreover,	it	gives	
an	 explicitly-sparse	 representation	 for the	 system	 matrix,	 which	 is	 expected	 to	 be	 beneficial	 for	 the	
development	of	fast	direct	solvers	and	efficient	preconditioners.},
DOI = {10.32604/icces.2023.09272}
}