[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82568-en":3,"doc-seo-82568-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82568,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","A Superfast Direct Solver for 2D Type-II Inverse Nonuniform Discrete Fourier Transform Based on Hierarchically Semiseparable Matrix","A superfast direct inversion method addresses the 2D type-II nonuniform discrete Fourier transform (NUDFT) by factoring the NUDFT matrix A as A = G F, where F is the 2D DFT matrix and G is represented through a kernel matrix. The work approximates G with a hierarchically semiseparable (HSS) matrix and estimates its HSS rank. Combining an HSS least-squares solver with a 2D inverse fast Fourier transform yields efficient offline and online solvers, and the method is designed to serve as a robust preconditioner for ill-conditioned inverse NUDFT problems with grid-like sample layouts.","arXiv :2607 .00928v1 [math .NA] 1 Jul 2026  \nA Superfast Direct Solver for 2D Type-II Inverse Nonuniform Discrete Fourier Transform Based on Hierarchically Semiseparable Matrix  \nYingzhou Li∗, Jingyu Liu†  \nJuly 2, 2026  \nAbstract  \nThis paper proposes a direct inversion method for the 2D type-II nonuniform discrete Fourier transform (NUDFT) . The NUDFT matrix A is factored as A = GF , where G can be expressed as a kernel matrix and F is the 2D DFT matrix. We show that G can be approximated by a hierarchically semiseparable (HSS) matrix and give an estimate of the HSS rank. Then, using the least-squares solver for HSS matrix and the two-dimensional inverse fast Fourier transform, the inverse NUDFT problem can be solved efficiently. Our algorithm has an offline complexity of O 􀀀 M + N3/2 log3 N􀀁 where M and N are the size of rows and columns of the NUDFT matrix, respectively. Once the direct solver is built, it can be applied to a vector with an online complexity of O􀀀 M +N log3 N􀀁 . The proposed method can be used as a preconditioner for iterative methods, especially when the sample points are distributed on a grid such that A is ill-conditioned. Numerical results are provided to show the scaling performance of the inversion method and demonstrate the efficiency and robustness of it as apreconditioner.  \nKeywords nonuniform discrete Fourier transform, hierarchically semi-separable matrix  \n1 Introduction  \nThis paper considers the two-dimensional type-II nonuniform discrete Fourier transform (NUDFT) of the following form:  \nn [x] −1 n[y] −1  \nfj = X X ck [x], k [y]e−2πi(k[x]xj+k[y]yj) , 0 ≤ j ≤ M − 1 , (1.1) k [x]=0 k [y]=0  \nwhere the sample points { (xj , yj)} are distributed arbitrarily in [0 , 1)2 and the frequencies { (k[x], k [y])} are distributed on a Cartesian grid of contiguous integers. Throughout the paper, the number of samples points M is assumed to be larger than or equal to the number of frequencies, i.e. , N = n [x]n [y] . When the coefficients {ck [x], k [y] } are given and one aims to compute the target values {fj }, the problem is called the forward  \n∗ School of Mathematical Sciences, Fudan University; Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, [yingzhouli@fudan.edu.cn](yingzhouli@fudan.edu.cn)  \n†School of Mathematical Sciences, Fudan University, [jyliu22@m.fudan.edu.cn](jyliu22@m.fudan.edu.cn)  \nNUDFT. When the target values {fj } are given and one aims to determine the coefficients {ck [x], k [y] }, the problem is called the inverse NUDFT. Both the forward and inverse NUDFT have wide applications in scientific computing, such as signal processing [2], image reconstruction [7], and fast convolution [19] etc. Let the NUDFT matrix A be defined as 1  \nA 􀀀j,(k[x], k [y])􀀁 = e−2πi(k[x]xj+k[y]yj) , 0 ≤ j ≤ M − 1 , 0 ≤ k [x] ≤ n [x] − 1 , 0 ≤ k [y] ≤ n [y] − 1 , (1 .2)  \nwhere the column index (k[x], k [y]) is ordered in a lexicographical manner. That is, we have an underlying 1-to-1 map (k[x], k [y]) ↔ k [y] + k[x]n [y] . The forward NUDFT can be expressed as a matrix-vector multiplication f = Ac and the inverse NUDFT can be formulated as a linear least-squares problem minc ∥Ac−f∥2 , where c ∈ CN is the vectorized form of the coefficients {ck [x], k [y] } and f ∈ CM is the vector corresponding to the target values {fj } . Typically, the forward and inverse NUDFT have a complexity of O (MN) and O (MN2 ) respectively.  \n1.1 Related Work  \nIf the sample points are distributed on a uniformly Cartesian grid, i.e., (xj [x] , yj [y] ) = (j[x]/n [x], j [y]/n [y]) for 0 ≤ j [x] ≤ n [x] − 1 and 0 ≤ j [y] ≤ n [y] − 1, the NUDFT reduces to the discrete Fourier transform (DFT) . Consequently, the NUDFT matrix A has a Kronecker product structure A = A [x] ⊗ A [y], where A[x](j[x], k [x]) = e −2πk[x]j [x]/n [x] and A[y](j[y], k [y]) = e −2πik[y]j [y]/n [y] are the DFT matrices in the x and y directions, respectively. In this case, the forward NUDFT can be efficiently computed by the fast F","cbCaihcLf2om6Q6F","https://ap.wps.com/l/cbCaihcLf2om6Q6F","pdf",1153311,1,22,"English","en",105,"# Abstract\n# Introduction\n## Related Work","[{\"question\":\"What problem does the paper solve?\",\"answer\":\"The paper solves the inverse 2D type-II nonuniform discrete Fourier transform (NUDFT), formulated as a linear least-squares problem for recovering coefficients from target samples.\"},{\"question\":\"How is the NUDFT matrix factored in the proposed method?\",\"answer\":\"The NUDFT matrix A is factored as A = G F, where F is the 2D DFT matrix and G is expressed via a kernel matrix that is approximated using a hierarchically semiseparable (HSS) structure.\"},{\"question\":\"Why can the method be useful for iterative solvers?\",\"answer\":\"After building a direct solver, the approach can act as a preconditioner, especially when sample points are distributed so the system becomes ill-conditioned, improving efficiency and robustness of iterative methods.\"}]",1784181571,55,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"a-superfast-direct-solver-for-2d-type-ii-inverse-nonuniform-discrete-fourier-transform-based-on-hierarchically-semiseparable-matrix","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-superfast-direct-solver-for-2d-type-ii-inverse-nonuniform-discrete-fourier-transform-based-on-hierarchically-semiseparable-matrix/82568/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper solve?","Question",{"text":74,"@type":75},"The paper solves the inverse 2D type-II nonuniform discrete Fourier transform (NUDFT), formulated as a linear least-squares problem for recovering coefficients from target samples.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How is the NUDFT matrix factored in the proposed method?",{"text":79,"@type":75},"The NUDFT matrix A is factored as A = G F, where F is the 2D DFT matrix and G is expressed via a kernel matrix that is approximated using a hierarchically semiseparable (HSS) structure.",{"name":81,"@type":72,"acceptedAnswer":82},"Why can the method be useful for iterative solvers?",{"text":83,"@type":75},"After building a direct solver, the approach can act as a preconditioner, especially when sample points are distributed so the system becomes ill-conditioned, improving efficiency and robustness of iterative 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