Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials

Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials Lutz Kämmerer June 23, 2015 We present a new sampling metho that allows the unique reconstruction of (sparse) multivariate trigonometric polynomials. The crucial iea is to use several rank-1 lattices as spatial iscretization in orer to overcome limitations of a single rank-1 lattice sampling metho. The structure of the corresponing sampling scheme allows for the fast computation of the evaluation an the reconstruction of multivariate trigonometric polynomials, i.e., a fast Fourier transform. Moreover, we present a first algorithm that constructs a reconstructing sampling scheme consisting of several rank-1 lattices for a given frequency inex set. Various numerical tests inicate the avantages of the constructe sampling schemes. Keywors an phrases : sparse multivariate trigonometric polynomials, lattice rule, multiple rank-1 lattice, fast Fourier transform 2010 AMS Mathematics Subject Classification : 65T40, 65T50 1 Introuction During the last years, sparse mathematical moels has become very popular. Particularly, the numerical treatment of high imensional problems ask for suitable sparsity. However, even sparse problems may eal with huge amounts of ata such that efficient numerical methos are on the main focus in various fiels of research. In this paper, we consier multivariate trigonometric polynomials an introuce a new suitable spatial iscretization scheme that allows for fast evaluation an reconstruction algorithms. Previous papers on multivariate trigonometric polynomials that eal with single rank-1 lattices as sampling schemes, cf. [16, 9], or ranomly chosen sampling sets [5] le to ifferent limitations in the number of use sampling noes an computational expense, respectively. On the other han, wiely use concepts in higher imensions are so-calle function spaces of ominating mixe smoothness an efficient approximation methos that use hyperbolic cross trigonometric polynomials that are compute from sampling values along sparse gris, cf. [6, 17, 14, 3, 4],. However, research results on hyperbolic cross trigonometric Technische Universität Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany lutz.kaemmerer@mathematik.tu-chemnitz.e 1

polynomials [10] point out that even sparse gri iscretizations suffer from limitations, particularly, the conition number of the corresponing Fourier matrix increases with growing problem sizes. Anyhow, our new sampling metho is motivate by the structure of sparse gris. A sparse gri is a composition of specific rank-1 up to rank- lattices, cf. [1, 18, 15], that are wellsuite in orer to sample trigonometric polynomials with frequencies supporte on hyperbolic crosses. In the present paper, we introuce sampling schemes that are compositions of multiple rank-1 lattices. We investigate necessary an sufficient conitions on the sampling scheme in orer to guarantee a unique reconstruction of multivariate trigonometric polynomials. We emphasize that we are intereste in arbitrary multivariate trigonometric polynomials, i.e., we o not require any structure of the frequency support. Furthermore, we present fast algorithms, i.e., fast Fourier transforms (FFTs), that computes the evaluation of multivariate trigonometric polynomials at all sampling noes, cf. Algorithm 3, an the reconstruction of multivariate trigonometric polynomials from the function values at the sampling noes, cf. Section 4. As a matter of course, we are intereste in sampling sets that allow for the unique reconstruction of multivariate trigonometric polynomials supporte on specific frequency inex set. For a given frequency inex set, we present a metho, cf. Algorithm 5, that etermines a set of rank-1 lattices that guarantees the unique reconstruction of multivariate trigonometric polynomials supporte on this frequency inex set. The constructe sampling sets allow for a specific fast algorithm that computes the reconstruction in a irect way, cf. Algorithm 6. The present paper eals only with multivariate trigonometric polynomials. We stress on the fact that one may use the new sampling metho in an aaptive way similar to the ieas from [13]. The main ingreient therein is to use reconstructing sampling schemes for low-imensional frequency inex sets in orer to approximate functions using multivariate trigonometric polynomials in a imension incremental way. 2 Prerequisites In this paper, we eal with multivariate trigonometric polynomials p: T C, x k I ˆp k e 2πik x, where T = [0, 1) is the -imensional torus, the complex numbers (ˆp k ) k I C I are calle Fourier coefficients of p, an the frequency inex set I Z is of finite carinality. The term k x = j=1 k jx j is the usual scalar prouct of two -imensional vectors. Furthermore, the space of all trigonometric polynomials supporte on the frequency inex set I is enote by Π I := span{e 2πik x : k I}. The evaluation of the multivariate trigonometric polynomial p at a finite set X T of sampling noes is specifie by the matrix-vector prouct A(X, I)ˆp = p, where ˆp = (ˆp k ) k I C I is the vector of the Fourier coefficients of the trigonometric polynomial p, the right han sie p = (p(x)) x X contains the function values of p at all noes that 2

belong to the sampling set X, an the Fourier matrix A(X, I) is given by A(X, I) = ( e 2πik x) x X,k I. As usual, we assume a fixe orer of the elements of the sampling set X an the frequency inex set I in orer to use the matrix-vector notation. A very specific kin of sampling schemes are so-calle rank-1 lattices X = Λ(z, M) := { } j z mo 1: j = 0,..., M 1 T, M which are well-investigate in the fiel of numerical integration, cf. [15, 2]. The -imensional integer vector z Z is calle generating vector an the positive integer M N the lattice size of the rank-1 lattice Λ(z, M). The author stuie such sampling schemes for the reconstruction of trigonometric polynomials in [8]. Various specific examples illustrate the avantages of rank-1 lattices for the unique reconstruction of trigonometric polynomials. The main isavantage of a single rank-1 lattice sampling is the necessarily growing oversampling, i.e., M/ I necessarily increases for growing I in orer to ensure a unique reconstruction of trigonometric polynomials supporte on specifically structure frequency inex sets I, e.g., hyperbolic cross frequency inex sets I, cf. also [11, 7] for more etails on this specific topic. In orer to overcome this problem, we woul like to exten the metho of rank-1 lattice sampling in a more or less usual manner by sampling along multiple rank-1 lattices. The corresponing sampling sets are joine rank-1 lattices X = Λ(z 1, M 1, z 2, M 2,..., z s, M s ) := s Λ(z r, M r ) an calle multiple rank-1 lattice. The 0 is containe in each of the rank-1 lattices Λ(z r, M r ), r = 1,..., s. Consequently, the number of istinct elements in Λ(z 1, M 1, z 2, M 2,..., z s, M s ) is boune from above by Λ(z 1, M 1, z 2, M 2,..., z s, M s ) 1 s+ s r=1 M r. Due to the fact that there may be 1 r 1 < r 2 s such that Λ(z r1, M r1 ) Λ(z r2, M r2 ) > 1 or 1 r 3 s such that Λ(z r3, M r3 ) < M r3, we have to expect a lower number of elements within Λ(z 1, M 1, z 2, M 2,..., z s, M s ) than the upper boun promises. However, we o not care about uplicate rows within the Fourier matrix with the exception of the uplicates that arises from x = 0. We efine the Fourier matrix ( A := A(Λ(z 1, M 1, z 2, M 2,..., z s, M s ), I) := r=1 ) e 2πi j k z M 1 1 ( e 2πi j ) j=0,...,m 1 1, k I k z M 2 2 (e 2πi j Ms k zs ) where we assume that the frequency inices k I are in a fixe orer. In the following, we prove some basics about multiple rank-1 lattices. j=1,...,m 2 1, k I. j=1,...,m s 1, k I, 3

Lemma 2.1. Let Λ(z 1, M 1 ) an Λ(z 2, M 2 ) be two rank-1 lattices with relative prim lattice sizes M 1 an M 2. Then, the rank-1 lattice Λ(M 2 z 1 + M 1 z 2, M 1 M 2 ) is a supset of Λ(z 1, M 1 ) an Λ(z 2, M 2 ), respectively. Thus, the multiple rank-1 lattice Λ(z 1, M 1, z 2, M 2 ) Λ(M 2 z 1 + M 1 z 2, M 1 M 2 ) is a subset of the rank-1 lattice Λ(M 2 z 1 + M 1 z 2, M 1 M 2 ). Furthermore, the carinality of Λ(z 1, M 1, z 2, M 2 ) is given by Λ(z 1, M 1 ) + Λ(z 2, M 2 ) 1. Proof. Due to the coprimality of the numbers M 1 an M 2, the chinese reminer theorem implies that there exists one l {0, M 1 M 2 1} such that Consequently, we obtain l(m 2 z 1 + M 1 z 2 ) M 1 M 2 mo 1 = This yiels an l j 1 (mo M 1 ) an l j 2 (mo M 2 ). ( lz1 + lz ) ( 2 j1 z 1 mo 1 = mo 1 + j ) 2z 2 mo 1 mo 1. M 1 M 2 M 1 M 2 l(m 2 z 1 + M 1 z 2 ) M 1 M 2 mo 1 = j 1z 1 M 1 mo 1 l(m 2 z 1 + M 1 z 2 ) M 1 M 2 mo 1 = j 2z 2 M 2 mo 1 for j 2 = 0 an j 1 = 0, respectively. The coprimality of M 1 an M 2 irectly implies that Λ(z 1, M 1 ) Λ(z 2, M 2 ) = {0} an the assertion follows. Corollary 2.2. Let the multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ) with pairwise coprime lattice sizes M 1,..., M s be given. Then, the carinality of Λ(z 1, M 1,..., z s, M s ) is given by an the embeing Λ(z 1, M 1,..., z s, M s ) = 1 s + hols, where the generating vector z = s are explicitely given. s Λ(z r, M r ) r=1 Λ(z 1, M 1,..., z s, M s ) Λ(z, M) r=1 ( s l=1 l r Proof. An iterative application of Lemma 2.1 yiels the assertion. M l )z r an the lattice size M = s r=1 M r As a consequence of Corollary 2.2, a multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ) with pairwise coprime M 1,..., M s an a full rank matrix A(Λ(z 1, M 1, z 2, M 2,..., z s, M s ), I) is a subset of the rank-1 lattice Λ(z, M), z = s r=1 ( s l=1 M l )z r an M = s r=1 M r. The matrix A(Λ(z, M), I) has full column rank an, in particular, pairwise orthogonal columns, cf. [8, Lemma 3.1]. Further restrictions on the rank-1 lattices Λ(z 1, M 1 ),..., Λ(z s, M s ) allow for an easy etermination of the number of istinct sampling values that are containe in Λ(z 1, M 1,..., z s, M s ). l r 4

Corollary 2.3. Uner the assumptions of Corollary 2.2 an the aitional requirements M r is prim for all r = 1,..., s an 0 z r [0, M r 1] Z, we conclue s Λ(z 1, M 1,..., z s, M s ) = 1 s + M r. r=1 Proof. The aitional requirements of the corollary guarantee Λ(z r, M r ) = M r, r = 1,..., s. Hence, the statement follows irectly from Corollary 2.2. 3 Evaluation of Trigonometric Polynomials The evaluation of a trigonometric polynomial at all noes of a multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ) is simply the evaluation at the s ifferent rank-1 lattices Λ(z 1, M 1 ),..., Λ(z s, M s ). A corresponing fast Fourier transform is given by using s-times the evaluation along a single rank-1 lattice, cf. Algorithm 1, which uses only a single one-imensional fast Fourier transform. This yiels to a complexity of the fast evaluation at all noes of the whole multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ), cf. Algorithm 3, that is boune by terms containe in O ( s r=1 M r log M r + s I ). For etails on the single rank-1 lattice Fourier transform confer [12, 8]. Algorithm 1 Single Lattice Base FFT (LFFT) Input: M N lattice size of rank-1 lattice Λ(z, M) z Z generating vector of Λ(z, M) I Z frequency inex set ˆp = (ˆp k ) k I Fourier coefficients of p Π I ĝ = (0) Ms 1 l=0 for each k I o ĝ k zl mo M l = ĝ k zl mo M l + ˆp k en for p = ifft 1D(ĝ) p = Mp Output: p = Aˆp function values of p Π I Complexity: O (M log M + I ) The evaluation of multivariate trigonometric polynomials along multiple rank-1 lattices is guarantee by the inicate algorithm. Hence, we shift our attention to the reconstruction problem, i.e., to necessary an sufficient conitions on the sampling set Λ(z 1, M 1,..., z s, M s ) such that each trigonometric polynomial p Π I, that belongs to a given space of trigonometric polynomials, is uniquely specifie by its sampling values p(x), x Λ(z 1, M 1,..., z s, M s ) along the multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ). Moreover, we are intereste in a fast an unique reconstruction of each trigonometric polynomial p Π I from its values along the sampling set Λ(z 1, M 1,..., z s, M s ). 5

Algorithm 2 Ajoint Single Lattice Base FFT (alfft) Input: M N lattice sizes of rank-1 lattice Λ(z, M) z Z generating vector of Λ(z, M) I Z ( ( )) frequency inex set p = p j M z function values of p Π I j=0,...,m 1 ĝ = FFT 1D(p) â = (0) k I for each k I o â k = â k + ĝ k zl mo M l en for Output: â = A p ajoint evaluation of the Fourier matrix A Complexity: O (M log M + I ) Algorithm 3 Evaluation at multiple rank-1 lattices Input: M 1,..., M s N lattice sizes of rank-1 lattices Λ(z l, M l ) z 1,... z s Z generating vectors of Λ(z l, M l ) I Z frequency inex set ˆp = (ˆp k ) k I Fourier coefficients of p Π I 1: for l = 1,..., s o 2: p l =LFFT(M l, z l, I, ˆp) 3: en for 4: p = (p 1 [1],..., p 1 [M 1 ], p 2 [2],..., p 2 [M 2 ],..., p s [2],... p s [M s ]) Output: p = Aˆp function values of p Π I Complexity: O ( s l=1 M l log M l + s I ) 4 Reconstruction Properties of Multiple Rank-1 Lattices In orer to investigate the reconstruction properties of a sampling set X T, X <, with respect to a given frequency inex set I, we have to consier the corresponing Fourier matrix A(X, I). A unique reconstruction of all trigonometric polynomials p Π I from the sampling values (p(x)) x X implies necessarily a full column rank of the matrix A(X, I). We may use single rank-1 lattices Λ(z, M) as sampling set X, i.e., X = Λ(z, M). In [8], we investigate necessary an sufficient conitions on the reconstruction property of single rank-1 lattices Λ(z, M) as spatial iscretizations. Roughly speaking, we may have to expect oversampling that increase with growing carinality of the frequency inex set I, ue to the group structure of the lattice noes. Motivate by the construction iea of sparse gris, which are in general a union of ifferent lattices of several ranks, we will join a few rank-1 lattices as spatial iscretization in orer to construct sampling schemes Λ(z 1, M 1,..., z s, M s ) that guarantee full column ranks of the Fourier matrices A(Λ(z 1, M 1,..., z s, M s ), I) for given frequency inex sets I. In general, we are intereste in practically suitable construction strategies of such multiple rank-1 lattices 6

Algorithm 4 Ajoint evaluation at multiple rank-1 lattices Input: M 1,..., M s N lattice sizes of rank-1 lattices Λ(z l, M l ) z 1,... z s Z generating vectors of Λ(z l, M l ) I Z frequency inex set p 1 sampling ( values ) of p Π I, p =. p l = p( j M l z l ) p s j=1 δ 1,l,..., M l â = (0) k I for l = 1,..., s o g = (δ 1,l p[1], p[m 1 + + M l 1 l + 3],..., p[m 1 + + M l l + 1]) â = â + alfft(m l, z l, I, g) en for Output: â = A f result of ajoint matrix times vector prouct Complexity: O ( s l=1 M l log M l + s I ) Λ(z 1, M 1,..., z s, M s ). We call a sampling set Λ(z 1, M 1,..., z s, M s ) with et (A (Λ(z 1, M 1,..., z s, M s ), I)A(Λ(z 1, M 1,..., z s, M s ), I)) > 0 a reconstructing multiple rank-1 lattice for the frequency inex set I. For a given frequency inex set I an given sampling set Λ(z 1, M 1,..., z s, M s ), Λ(z 1, M 1,..., z s, M s ) I, we can check the reconstruction property in ifferent ways. For instance, one can compute the echelon form or (lower bouns on) the smallest singular value of the matrices A or A A in orer to check whether the rank of the matrix is full or not. We emphasize, that the complexity of the test methos is at least Ω ( I 2) an in the case of the computation of lower bouns on the smallest singular values, cf. [19], the test may fail. However, if the matrix A is of full column rank, one can reconstruct the Fourier coefficients of a multivariate trigonometric polynomial p Π I by solving the normal equation A Aˆp = A p. Usually, one approximates the inverse of the matrix A A using a conjugate graient metho an fast algorithms that compute the matrix-vector proucts associate with A an A, cf. Algorithm 3 an Algorithm 4 associate with Algorithm 2. Thus, the fast reconstruction of a trigonometric polynomial p Π I using the samples along a reconstructing multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ) for I is guarantee. Anyway, we are still intereste in a practically suitable construction strategy in orer to etermine reconstructing multiple rank-1 lattices Λ(z 1, M 1,..., z s, M s ) for given frequency inex sets I. In the following subsection, we present one possibility of such a construction. 4.1 Construction of reconstructing multiple rank-1 lattices In this subsection we characterize an algorithm that etermines reconstructing multiple rank- 1 lattice for given frequency inex sets I Z, I <. Accoringly, we assume the frequency inex set I of finite carinality being given an fixe. We suggest to reconstruct a 7

Algorithm 5 Determining reconstructing multiple rank-1 lattices Input: I Z frequency inex set σ R oversampling factor σ 1 n N number of ranom test vectors l = 0 while I > 1 o l = l + 1 M l = nextprime(σ I ) while M l {M 1,..., M l 1 } o M l = nextprime(m l ) en while for j = 1,..., n o choose ranom integer vector v (l) j [1, M l 1] compute K (l) j = I \ {k I : h I \ {k} with k v (l) j h v (l) j en for (mo M l )} etermine z l = v (l) j 0 such that K (l) j 0 = max j {1,...,n} K (l) j I = {k I : h I \ {k} with k z l h z l (mo M l )} en while Output: M 1,..., M l lattice sizes of rank-1 lattices an z 1,..., z l generating vectors of rank-1 lattices such that Λ(z 1, M 1,..., z l, M l ) is a reconstructing multiple rank-1 lattice trigonometric polynomial p Π I step by step. Inepenent of the structure of the frequency inex set I, we fix a lattice size M 1 I an a generating vector z 1 [1, M 1 1] an reconstruct only these frequencies ˆp k that can be uniquely reconstructe by the means of the use rank-1 lattice. The inices k of these frequencies ˆp k are simply given by I 1 = {k I : k z 1 h z 1 mo M 1 for all h I \ {k}}. Assuming that the frequencies (ˆp k ) k I1 are alreay etermine, we have to reconstruct a trigonometric polynomial p 1 Π I\I1 supporte by the frequency inex set I \ I 1 now. We etermine the sampling values of this trigonometric polynomial by p 1 (x) = p(x) k I 1 ˆp k e 2πik x. We apply this strategy successively as long as there are frequencies that have to be reconstructe. Algorithm 5 inicates one possibility of an algorithm that etermines a set of rank-1 lattices that allows for the application of the mentione reconstruction strategy. Obviously, in each step we are intereste in a rank-1 lattice that allows for the reconstruction of as many as possible frequencies. For that reason, we check a few generating vectors for its number of reconstructable frequency inices in Algorithm 5. We show that Algorithm 5 etermines a sampling set such that Λ(z 1, M 1,..., z s, M s ) implies a full column rank matrix A(Λ(z 1, M 1, z 2, M 2,..., z s, M s ), I). 8

Lemma 4.1. Let the matrix B C n m of the following form ( ) B1 B B = 2 B 3 B 4 be given. The matrices B 1 C n 1 m 1,..., B 4 C n 2 m 2 are submatrices of B, i.e., n = n 1 + n 2 an m = m 1 + m 2. In aition, we assume that B 1 has full column rank, i.e., the columns of B 1 are linear inepenent, B 4 has full column rank, i.e., the columns of B 4 are linear inepenent, an the columns of B 2 are not in the span of the columns of B 1. Then the matrix B has full column rank. Proof. The matrix B C n m, n m, has full column rank iff the columns of the matrix B are all linear inepenent, i.e. the formula m λ j b j = 0 (4.1) j=1 has the unique solution λ = 0. We will exploit the full column rank of the matrix B 1 C n 1 m 1, n 1 m 1, an, thus, we consier the sum m λ j b j = 0, j=1 where b j = (b j,l ) n 1 l=1 are vectors consisting of the first n 1 elements of the vectors b j. Due to the fact that the columns of B 2 C n 1 m 2 are not in the span of the columns of B 1 an the columns of B 1 are linear inepenent, we achieve λ j = 0 for all j = 1,..., m 1. Accoringly, (4.1) simplifies to m λ j b j = 0. j=m 1 +1 For the remaining vectors b j, j = m 1 + 1,..., m, we know that the vectors of the last m 2 components of b j are linear inepenent an, consequently, we obtain λ j = 0 for all j = 1,..., m. Theorem 4.2. Algorithm 5 etermines sampling sets such that the corresponing Fourier matrix A(Λ(z 1, M 1, z 2, M 2,..., z s, M s ), I) is a full column rank matrix. Proof. In orer to exploit Lemma 4.1 we will nee to rearrange the orer of the columns of A(Λ(z 1, M 1, z 2, M 2,..., z s, M s ), I) in a suitable way. We assume that M 1,..., M s an z 1,..., z s is the output of Algorithm 5. Consequently, we can etermine the following frequency inex sets I 1 = {k I : h I \ {k} with k z 1 h z 1 (mo M 1 )} an I 1 = I \ I 1, I 2 = {k I 1 : h I 1 \ {k} with k z 2 h z 2 (mo M 2 )} an I 2 = I 1 \ I 2,.. Is = {k Is 1 : h I s 1 \ {k} with k z s h z s (mo M s )} = an I s = Is 1 \ I s. 9

The resulting frequency inex sets I j, j = 1,..., s, yiel a isjoint partition of I. We rearrange the columns of the matrix A(Λ(z 1, M 1, z 2, M 2,..., z s, M s ), I) an we achieve a matrix K 1,1... K 1,s à :=....., K s,1... K s,s where the submatrices K r,l are given by K r,l := [1, s] 2 N 2. We efine the matrices à l := ( K r,t) r=1,...,l, t=1,...,l = (e 2πi j Mr k zr )j=1 δ 1,r,...,M r 1, k I l, (r, l) à l 1. K 1,l K l 1,l K l,1... K l,l 1 K l,l which are in fact submatrices of Ã. In particular, we obtain à 1 = K 1,1 an à s = Ã. In the following, we conclue the full column rank of à l from the full column rank of à l 1, the full column rank of K l,l, an the linear inepenence of each column of the matrix K 1,l. K l 1,l an all columns of the matrix à l 1, cf. Lemma 4.1. We start with l = 1, i.e., à 1 = K 1,1. Since Λ(z 1, M 1 ) is a reconstructing rank-1 lattice for I 1, the matrix à 1 has linear inepenent columns. Due to the construction of I 1 an I1, each column of one of the matrices K 1,r, r = 2,..., s, is not in the span of the columns of K 1,1. We prove the full rank of à l, l = 2,..., s, inuctively. For that, we assume à l 1 to be of full column rank. Aitionally, we know that K l,l is of full column rank, since Λ(z l, M l ) is a reconstructing rank-1 lattice for I l. In orer to apply Lemma 4.1, we have to show, that no column of the matrix K 1,l. K l 1,l, (4.2) is a linear combination of the columns of à l 1. We assume the contrary, i.e., let k I l be a frequency inex, a k the corresponing column of the matrix in (4.2), such that a k = à l 1 λ = K 1,1... K 1,l 1..... K l 1,1... K l 1,l 1 λ 1. λ l 1, (4.3) where we look for the vectors λ r C Ir, r = 1,..., l 1. We solve this linear equation recursively. The first M 1 rows of a k are not in the span of the pairwise orthogonal columns 10

of K 1,1 an the columns of the first M 1 rows of K 1,r, r = 2,..., l 1 are also not in the span of the columns of K 1,1. Consequently, we observe that λ 1 = 0 C I1 hols. Accoringly, the solution of (4.3) is given by λ := (λ 1,..., λ l 1 ) = (0,..., 0, λ }{{} 2,..., λ l 1 ) I 1 times an we search for the solution of K 1,2... K 1,l 1 λ 2 a k =....... K l 1,2... K l 1,l 1 λ l 1 Next, we consier the rows numbere by M 1 + 1,..., M 1 + M 2 1 an obtain (a k,j ) 1+M 1+M2 j=1+m 1 = (K 2,2,..., K 2,l 1) Since Λ(z 2, M 2 ) is a reconstructing rank-1 lattice for I 2 an there is no k l j=3 I j that aliases to a frequency inex k I 2 with respect to Λ(z 2, M 2 ), we obtain the result λ 2 = (0,..., 0) C I 2. These consierations lea inuctively to the formulas (a k,j ) 1 t+ t r=1 Mr j=3 t+ t 1 r=1 Mr = (K t,t,..., K t,l 1) λ 2. λ l 1 λ t. λ l 1., t = 1,..., l 1 an the result λ t = 0 C It, t = 1,..., l 1, in (4.3), which implies a k = 0 an is in contraiction to a k 1 = 1 (l 1) + l 1 r=1 M r > 0. Consequently, we apply Lemma 4.1 on the matrix à l an observe the full column rank of à l, in particular for l = s. Algorithm 5 etermines a reconstructing sampling scheme for all multivariate trigonometric polynomials with frequencies supporte on the inex set I. Moreover, the iea behin Algorithm 5 is the stepwise reconstruction of trigonometric polynomials as mentione above. Accoringly, we inicate the corresponing reconstruction strategy in Algorithm 6, where we use Algorithms 1 an 2 in orer to compute the require single lattice base iscrete Fourier transforms. As a consequence, we achieve a computational complexity of this algorithm which is boune by C ( s l=1 M l log M l + s( + log I ) I ), where the term C oes not epen on the multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ), the frequency inex set I, or the spatial imension. 5 Numerics Basically, we use Algorithm 5 with oversampling parameter σ = 1 an we choose n = 10 for our numerical examples. There is only a slight moification of the implemente algorithm: If there is an l such that max j {1,...n} K (l) j = 0, then we o not use the corresponing sampling values, since the sampling values o not ensure aitional information for the reconstruction. 11

Algorithm 6 Direct reconstruction of trigonometric polynomials p Π I from samples along reconstructing multiple rank-1 lattices that are etermine by Algorithm 5 Input: M 1,..., M s N lattice sizes of rank-1 lattices Λ(z l, M l ) z 1,... z s Z generating vectors of Λ(z l, M l ) I Z frequency inex set p 1 sampling ( values ) of p Π I, p =. p l = p( j M l z l ) p s j=1 δ 1,l,..., M l I R = {} ˆp = (0) k I for l = 1,..., s o I l = {k I \ I R : k z l h z l (mo M l ) for all h I \ (I R {k})} g l = (p[1], p[3 l + l 1 r=1 M r],..., p[1 l + l r=1 M r]) LFFT(M l, z l, I R, (ˆp k ) k IR ) (ˆp k ) k Il = alfft(m l, z l, I l, g l ) I R = I R I l en for Output: ˆp = (A A) 1 A p Fourier coefficients of p Π I Complexity: O ( s l=1 M l log M l + s( + log I ) I ) 5.1 Axis Crosses We consier so-calle axis crosses of a specific with N = 2 n N efine by I = I ac,n := {k Z : k = k 1 2 n } as frequency inex sets. One can use sampling values along single rank-1 lattices Λ(z, M) for the reconstruction of multivariate trigonometric polynomials supporte on axis crosses, cf. [8, Ex. 3.27]. The main isavantage of this approach is the necessary oversampling, i.e., one nees at least Λ(z, M) (N + 1) 2 sampling values in orer to ensure a unique reconstruction of trigonometric polynomials p Π I ac,n. We compare this lower boun on the number of sampling values to the number of frequency inices within Iac,N. This yiels oversampling factors of Λ(z, M) (N + 1)2 Iac,N 2N + 1 N 2 an thus the oversampling increases linearly in N. On the other han, we want to use multiple rank-1 lattices as spatial iscretizations. Our numerical tests, illustrate in Figure 5.1, allows for the conjecture that the oversampling factors stagnate for growing with N of the consiere axis crosses of fixe imension. Moreover, we observe small oversampling factors which are always below three. For comparison, sampling along a reconstructing single rank-1 lattice necessarily implies oversampling factors greater than 400 for N 2 14 an all imensions 2 20. 12

number of sampling noes number of frequency inices 3 2 1 2 2 2 5 2 8 2 11 2 14 2 17 2 20 2 23 refinement N =2 =3 =5 =10 =20 Figure 5.1: Oversampling factors of reconstructing multiple rank-1 lattices for axis crosses I ac,n. 5.2 Dyaic Hyperbolic Crosses In this secion, we consier so-calle yaic hyperbolic crosses efine by I = I hc,n := l N 0 l 1 =n Ĝ l, Ĝ l = Z ( 2 ls 1, 2 ls 1 ], s=1 where the number N = 2 n, n N 0, is a power of two. We applie Algorithm 5 to yaic hyperbolic crosses of ifferent refinements an imensions in orer to etermine reconstructing multiple rank-1 lattices. The resulting oversampling factors slightly grows with respect to the refinement an imension, cf. Figure 5.2. We observe that the oversampling factors seem to stagnate. In particular, Euler s number e bouns them from above in all our numerical tests. number of sampling noes number of frequency inices 3 2 1 2 2 2 5 2 8 2 11 2 14 2 17 2 20 2 23 refinement N =2 =3 =5 =10 Figure 5.2: Oversampling factors of reconstructing multiple rank-1 lattices for yaic hyperbolic crosses I hc,n. In the following, we compare three ifferent sampling methos in orer to reconstruct 13

Computational Time in Secons 10 4 10 2 HCFFT LFFT MLFFT 10 0 10 2 10 4 I 6 hc,n log 2 N 7 1 34 2 501 4 5 336 6 47 264 370 6882 664 192 8 10 12 Oversampling Factor X / I 6 hc,n Conition Number of A(X, I 6 hc,n ) 10 2 Sparse Gri X = S 6 N (HCFFT) Rank-1 Lattice X = Λ(z, M) (LFFT) Multiple Rank-1 Lattice (MLFFT) 10 5 10 4 HCFFT LFFT MLFFT 10 3 10 1 10 2 10 1 10 0 10 0 I 6 hc,n log 2 N 7 1 34 2 3 501 4 5 5 336 47 264 370 688 6 7 8 9 10 I 6 hc,n log 2 N 7 1 34 2 3 501 4 5 5 336 47 264 370 688 6 7 8 9 10 Figure 5.3: Six-imensional hyperbolic cross fast Fourier transforms for comparison. multivariate trigonometric polynomials with frequencies supporte on yaic hyperbolic cross inex sets: Sampling along sparse gris (HCFFT, cf. [6]), sampling along a reconstructing single rank-1 lattices (LFFT, cf. [7]), sampling along multiple rank-1 lattices (MLFFT). Our numerical tests illustrate the characteristics of the iscrete Fourier transforms (conition number of A(X, I), spent oversampling X / I ) an the corresponing fast algorithms (computational times). 14

Fixe Dimension = 6 Since we are intereste in numerical tests that visualize in some sense the asymptotic behavior of the three ifferent sampling methos, we fix the moerate imension = 6. Thus, we can compute conition numbers even for moerate refinements N = 2 n up to N = 1024, cf. Figure 5.3. We observe, that the iscrete Fourier transform base on multiple rank-1 lattice sampling nee only low overampling factors an that the corresponing Fourier matrices are well-conitione. Moreover, the computational complexity of the fast Algorithm of the Fourier transform (MLFFT) is illustrate an at least almost as goo as the computational complexity of the HCFFT with respect to N. Accoringly, the MLFFT seems to avoi both the isavantage of the HCFFT (growing conition numbers, cf. [10]) an the isavantage of the LFFT (growing oversampling factors an the associate fast growing computational times, cf. [11]). Fixe Refinement N = 2 2,..., 2 5 In [11, Fig. 4.2] we compare the computational times of sampling along sparse gris to sampling along reconstructing single rank-1 lattices. The lesson of this figure is clear: Unique sampling along reconstructing single rank-1 lattices may be not optimal ue to the fact that the number of necessarily use sampling values, cf. [7], may be not optimal, in general. Similar to this, we illustrate the computational times of ifferent fast algorithms in Figure 5.4. In particular, we mappe the computational times of the hyperbolic cross fast Fourier transform (HCFFT), i.e., the fast algorithm for the evaluation of hyperbolic cross trigonometric polynomials at all noes of a sparse gri an the fast algorithm for the reconstruction of hyperbolic cross trigonometric polynomials from the sampling values at sparse gri noes, multiple lattice fast Fourier transform (MLFFT) applie to hyperbolic cross trigonometric polynomials, i.e., the fast algorithm for the evaluation of multivariate trigonometric polynomials at all noes of a reconstructing multiple rank-1 lattice an ifferent fast algorithms for the reconstruction of multivariate trigonometric polynomials from the sampling values at a reconstructing multiple rank-1 lattice. In aition to the irect reconstruction, cf. Algorithm 6, we applie a CG metho with starting vector ˆp = 0 an a CG metho with starting vector ˆp that is the result of the irect reconstruction. We are intereste in the application of such a CG metho since the irect reconstruction suffers from growing relative errors err 2 := ˆp ˆp 2 ˆp 2, (5.1) where ˆp is the result of the reconstruction from sampling values of the hyperbolic cross trigonometric polynomial with frequencies ˆp, cf. Figure 5.5. This figure also contains the relative errors of the fast reconstruction metho from sampling values at sparse gris (HCFFT). We point out that this fast algorithm also suffers from growing relative errors. We i not apply a conjugate graient metho on the HCFFT since we expect huge computational costs ue to the expecte number of iterations of the CG metho which is inicate by the growing conition numbers of the corresponing Fourier matrices. 15

10 2 10 1 N = 4 10 3 10 2 N = 8 10 0 10 1 10 1 10 2 10 0 10 1 10 2 I hc,4 13 3 34 6 89 11 229 19 593 32 1 538 53 carinality of hyperbolic cross I hc,4 imension I hc,8 38 3 138 6 518 11 1 958 19 7 665 32 30 688 53 carinality of hyperbolic cross I hc,8 imension 10 3 10 2 10 1 10 0 10 1 10 2 Ihc,16 104 3 501 6 N = 16 2 586 11 13 852 19 63 671 30 carinality of hyperbolic cross I hc,16 imension 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Ihc,32 272 3 1 683 6 N = 32 11 584 11 69 460 347 392 18 27 carinality of hyperbolic cross I hc,32 imension HCFFT evaluation MLFFT irect reconstruction & CG HCFFT irect reconstruction MLFFT reconstruction only CG MLFFT evaluation multiple rank-1 lattice construction (Alg. 5) MLFFT irect reconstruction Ihc,N /2000 Figure 5.4: Computational times in secons of the fast algorithms computing the yaic hyperbolic cross iscrete Fourier transforms with respect to the problem size I hc,n. However, the runtime of the new fast algorithms (MLFFT) behave similar to the runtime of the HCFFT with respect to the imension. Even the CG methos have a similar runtime 16

N = 16 N = 32 10 9 10 11 10 11 err2 10 13 err2 10 13 10 15 10 15 I hc,16 104 3 501 6 2 586 11 13 852 63 671 19 30 I hc,32 272 3 1 683 6 11 584 11 69 460347 392 18 27 carinality of hyperbolic cross I hc,16 imension carinality of hyperbolic cross I hc,32 imension HCFFT irect reconstruction MLFFT irect reconstruction MLFFT irect reconstruction & CG MLFFT reconstruction only CG Figure 5.5: Relative errors, cf. (5.1), of the fast algorithms computing the yaic hyperbolic cross iscrete Fourier transforms with respect to the problem size I hc,n. behavior. This inicates that the number of iterations that are use uring the CG metho only slightly increases with growing imension which may be cause by boune conition numbers of the Fourier matrices, cf. Figure 5.6. Conition Number of A(X, I hc,32 ) 10 7 10 5 Sparse Gri X = S32 (HCFFT) Multiple Rank-1 Lattice (MLFFT) Ihc,32 Ihc,32 2 /1000 10 3 10 1 2 4 6 8 10 12 14 16 18 20 imension Figure 5.6: Conition numbers of the Fourier matrices using corresponing sparse gris an reconstructing multiple rank-1 lattices as sampling schemes. 17

inex s 24 20 16 12 8 4 maximal inex (max(s)) average inex minimal inex (min(s)) 2 1 2 4 2 8 2 12 2 16 2 20 carinality I of frequency inex set I Figure 5.7: Inices s of reconstructing multiple rank-1 lattices for ranomly chosen tenimensional frequency inex sets I. 5.3 Ranom frequency inex sets The complexity of the new fast Fourier transform algorithms, cf. Algorithms 3, 4, an 6, mainly epens on the number s of rank-1 lattices that are joine in orer to buil the multiple rank-1 lattice Λ(z 1, M 1,..., z s, M s ). We emonstrate the behavior of s for ten-imensional ranomly chosen frequency inex sets. Each component of the inices k I Z 10 are roune values that are chosen from a normal istribution with mean zero an variance 10 000. We consiere frequency inex sets I of ifferent carinalities which are powers of two, i.e. I {2, 4, 8,..., 2 20 }. We prouce 1 000 ifferent frequency inex sets for each of the carinalities an constructe one reconstructing multiple rank-1 lattice for each of the frequency inex sets. Figure 5.7 plots the carinality of the frequency inex set I against the maximal number s, the minimal number s an the average of the numbers s that occurre in our numerical tests. We observe that the number s behave logarithmically with respect to the carinality of I. We woul like to point out that we observe a similar behavior in all numerical tests that we treate before. Furthermore, the oversampling factors s r=1 M r/ I are less than Euler s number e. In aition, we compute the conition numbers of the Fourier matrices A(Λ(z 1, M 1,..., z s, M s ), I) for the cases I 2 12 an obtaine low conition numbers less than 16. The average of the conition numbers were less than seven in each of the cases I = 2 n, n = 1,..., 12. 6 Conclusion The paper presents a new sampling metho that allows for the unique sampling of sparse multivariate trigonometric polynomials. The main iea is to sample along more than one lattice gri similar to the sparse gris. In contrast to sparse gris we restrict the use lattices to rank-1 lattices. The constructive iea that etermines reconstructing multiple rank-1 lattices, cf. Algorithm 5, allows for another point of view. The resulting multiple rank-1 lattices Λ(z 1, M 1,..., z s, M s ) are subsampling schemes of huge single reconstructing rank-1 lattices Λ(z, M) for the frequency inex set I, where z an M are specifie in Corollary 2.2. The Fourier matrix A(Λ(z, M), I) that belongs to this huge rank-1 lattice Λ(z, M) has 18

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