A Plane Wave Method Based on Approximate Wave Directions for Two

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(∆ + κ2(r))u(ω, r)=0, r = (x, y) Ω. (1.2) ∈ Recently, the Geometric Optics Ansatz-based plane wave discontinuous Galerkin (GOPWDG) method for (1.2) was proposed in [21], in which a high accuracy L2 error of approximate solution was established for the case with a variable wave number κ(r). It is known that L2 errors of the finite element solutions of a usual elliptic equation converge to zero when h 0. However, this property does not hold → for all the methods mentioned above: the L2 errors of the finite element solutions for Helmholtz equation (1.2) (with a suitable boundary condition) do not decrease when h decreases unless hω 0 or the number of basis functions on every element → also increases. Let the mesh sizes h satisfy a basic assumption hω = O(1), and let the number of basis functions on every element be fixed. A natural question is whether the previous property can be kept for Helmholtz equations with large wave numbers? The answer is positive definite, provided that basis functions are constructed carefully. In [3, 16, 10, 11], plane wave type methods based on the geometrical optics ansatz were proposed to remove the pollution effect in the sense that the L2 errors of the approximate finite element solutions decrease when ω increases or h decreases (assuming hω = const.). The key idea is to define each basis function as the form p(r)eiκd·r, where p(r) is a polynomial and d denotes a local approximation of a wave direction vector determined by the geometric optics ansatz. In [3], assuming that a good local approximation d of every wave direction is known (for example, it has been computed by the ray tracing technique), the best L2 approximation of the resulting finite element space was derived. An important observation made in [10, 11] is that the wave directions are independent of ω, so every local approximation d can be preliminarily computed by the numerical 3 micro-local technique (NMLA) [2] based on approximate solutions of an auxiliary low-frequency problem. The approximate wave directions are incorporated as plane wave basis functions of a finite element space that is applied to the discretization of the considered high-frequency problem [10, 11]. It has been shown in [10, 11] that the L2 errors of the approximate solutions are O(ω−1/2) for the case with multiple waves as ω . It is clear that the L2 errors O(ω−1/2) of the approximate → ∞ solutions are not small unless ω is very large, which limits the applications of the method. The unsatisfactory L2 errors come from the low accuracy O(ω−1/2) of the approximate wave directions computed by the NMLA. In this paper, inspired by the ideas in [2], we design a new algorithm for computing local wave directions to improve their accuracy. We show that the accuracy of the computed local wave direction d as well as the best L2 approximate errors of the resulting plane wave space can achieve O(ω−1) (= O(h) if hω = const.), which is much smaller than O(ω−1/2) for a large ω. Then, by combining the ideas proposed in [24], we apply the constructed plane wave spaces to the discretization of the nonhomogeneous Helmholtz equation (1.1). We test two examples to confirm that the L2 errors of the resulting approximate solutions linearly decay when the mesh size h decreases or ω increases (choosing h such that hω = const.). The paper is organized as follows. In Section 2, we derive basic expressions of the new plane wave type basis functions by the geometrical optics ansatz. In section 3, we describe algorithms for approximately computing local wave directions and investigate the accuracies of the approximate wave directions and prove the best L2 error of the resulting finite element spaces . Finally, we report some numerical results on the proposed methods in Section 4.

2. Geometric optics ansatz-based plane wave basis functions

For a given h > 0, we divide the domain Ω into a union of quasi-uniformly polygonal elements with the size h. Let denote the resulting partition. For Th convenience, we separate ω from κ: κ(r)= ω ξ(r) with ξ(r)=1/c2(r).

2.1. Geometric optics ansatz. If the solutionp of the equation (1.2) corresponds to a simple wave, according to the geometric optics ansatz, the solution of (1.2) can be expressed as the L¨uneberg-Kline expansion [4]: r u(ω, r)= eiωφ( )A(r), (2.1) where φ is called the phase function satisfying the eikonal equation

φ(r) 2 = ξ(r), (2.2) |∇ | and A(r) is called the amplitude function that can be written as ∞ A (r) A(r)= s (2.3) (iω)s s=0 X 4 with A ∞ satisfying a recursive system of PDEs: { s}s=0 2 φ A + A ∆φ = ∆A − (2.4) ∇ · ∇ s s − s 1 for s =0, 1, , with A− 0. ··· 1 ≡ The key features of the geometric optics ansatz are: A ∞ and φ are independent of the frequency ω; •{ s}s=0 A ∞ and φ depend on c(r) (and f(r) if (1.1) is considered). •{ s}s=0 When more waves are involved in the solution of the equation (1.2), the generic solution of (1.2) should be locally deﬁned as a ﬁnite sum of terms like (2.1). Hence, in general crossing waves, we use N(r) to denote the number of crossing waves at the position r and expresse the solution of the Helmholtz equation (1.2) as

N(r)

u(ω, r)= un(ω, r), (2.5) n=1 X where each un(ω, r) has its ansatz form as (2.1) ∞ r r A (r) u (ω, r)= A (r)eiωφn( ) = eiωφn( ) n,s . (2.6) n n (iω)s s=0 X And for the n-wave ansatz (n = 1, ,N(r)), the ω-independent phase function ··· φ (r) and A (r) ∞ satisfy the eikonal equation (2.2) and the corresponding n { n,s }s=0 system (2.4) respectively.

2.2. Construction of plane wave type basis functions. Recall that the solution of (1.2) can be written as (see (2.5) and (2.6))

N(r) iωφn(r) u(ω, r)= un(ω, r), un(ω, r)= An(ω, r)e n=1 X r and the phase functions φ (r) N( ) are independent of ω. If the phase functions { n }n=1 φn(r) are known, then we need only to determine the amplitude functions An(ω, r), which can be approximated by ﬁnite element functions with smaller degrees of freedom. We need only to consider the case with a single wave. For a suﬃciently large ω, by (2.1) and (2.3) we have

iωφ(r) −1 u(ω, r)= A0(r)e + O(ω ). Motivated by the above expression, we deﬁne a plane wave type basis function as ϕ(r)= p(r)eiωτ(r), where τ is a real polynomial approximately satisfying (2.2) and a is a complex polynomial deﬁned by τ(r).

We consider a generic element K0 with the diameter h and the barycenter r0 =

(x0,y0). By the Taylor formula, φ can be written as φ(r)= φ(r ) + ( φ)(r ) (r r )+ O( r r 2). 0 ∇ 0 · − 0 | − 0| 5

Then 2 iωφ(r0) iω(∇φ)(r0)·(r−r0)+O(ωh ) −1 u(ω, r)= A0(r)e e + O(ω ). It follows by (2.2) that ( φ)(r ) 2 = ξ(r ), | ∇ 0 | 0 which means that ( φ)(r ) can be written as ( φ)(r )= ξ(r )(cos θ, sin θ) with ∇ 0 ∇ 0 0 some unknown direction angle θ [0, π). Let θ be a good approximation of θ. A ∈ h p natural idea is to choose

τ (r)= ξ(r )(cos θ sin θ ) (r r ), h 0 h h · − 0 which can be regarded as anp approximation of φ(r) φ(r ). It is easy to see that − 0 such τh(r) approximately satisfies the eikonal equation in the sense that ( τ )(r) 2 = ξ(r)+ O( r r ), r K . | ∇ h | | − 0| ∈ 0 For this linear polynomial τh, we look for a linear polynomial ph(r) such that ph(r) satisfies a similar equation with (2.4) that A0(r) meets (we can omit the constant iωφ(r0) e ). Replacing φ and A0 in (2.4) (for l = 0) by τh and ph respectively and using the fact ∆τh = 0, we can define a linear polynomial ph(r) by τ p =0, on K . ∇ h · ∇ h 0 We have two independent choices of ph(r) satisfying the above equation, namely, pτh (r) = 1+(sin θ cos θ ) (r r ) and pτh (r)=1 (sin θ cos θ ) (r r ). 1 h − h · − 0 2 − h − h · − 0 They correspond to two plane wave basis functions

τh iωτh(r) τh iωτh(r) ϕ1(r)= p1 (r)e and ϕ2(r)= p2 (r)e . For the case with more waves, we can similarly deﬁne plane wave-type basis functions. For an element K0 with the barycenter r0, deﬁne φ (r ) d := ∇ n 0 = c(r ) φ (r ) (n =1, ,N(r )). n φ (r ) 0 ∇ n 0 ··· 0 |∇ n 0 | By the eikonal equation, we have d = 1 and so it can be written as d = | n| n (cos θ cos θ )t with θ [0, π). We call d and θ n-th ray direction (or wave n n n ∈ n n direction) of the wave fronts at r0 and the direction angle of the n-th ray, respectively.

Let θh,n be an approximation of θn. Deﬁne d = (cos θ cos θ )t and τ = ξ(r )d (r r ), h,n h,n h,n h,n 0 h,n · − 0 which can be regarded as an approximation of φ (rp) φ (r ). Deﬁne n − n 0 pτh,n (r)=1+ d⊥ (r r ) and pτh,n (r)=1 d⊥ (r r ) 1 h,n · − 0 h − h,n · − 0 which satisfy the equation

τ pτh,n =0, on K . ∇ h,n · ∇ j 0 6

Then we deﬁne 2N(r0) plane wave basis functions on K0 as follows

τ r ϕ (ω, r)= p h,n (r)eiωτh,n( ), n =1, ,N(r ); j =1, 2 (r K ). n,j j ··· 0 ∈ 0

The core task of this article is to compute approximate direction angles θh,n (n = 1, ,N(r )). ··· 0

Remark 2.1. In most applications, there are only several rays , i.e., N(r0) is small. Then the number of the local basis functions ϕ is less than that of the { n,j } p version of the plane wave methods, so the plane wave method with the proposed − basis functions is cheaper than the p version of the plane wave methods for (1.2). − In the existing works [3] and [10], the factor ph(r) was directly chosen as complete linear polynomials, which corresponds to three independent basis functions. Here we have used the geometric optics ansatz (2.4) to reduce the number of basis functions on each element.

3. An adaptive plane wave method based on approximate direction angles

N(r0) Since the ray angles θn n=1 described in Subsection 2.2 are not known in appli- { } r cations, we hope to ﬁnd a cheap way to compute good approximations θ N( 0) { h,n}n=1 of them for a large ω . An important observation made in [10, 11] is that the ray N(r0) angles θn n=1 are independent of ω, so we can use an approximate solution of { } r low-frequency problem to compute θ N( 0). { h,n}n=1 Choosing positive numbersω ˜ ω, and consider auxiliary Helmholtz equations ≪ (∆+ω ˜2ξ(r))˜u(˜ω, r)=0, r = (x, y) Ω, ∈ (3.1) ((∂n + iω˜ ξ(r))u(˜ω, r)= g(˜ω, r), r ∂Ω. ∈ The solution of (3.1) can be writtenp as

N(r) iωφ˜ n(r) u˜(˜ω, r)= uñ(˜ω, r), uñ(˜ω, r)= Añ(˜ω, r)e . (3.2) n=1 X Sinceω ˜ ω (for example,ω ˜ = √ω), the Helmholtz equation (3.1) can be nu- ≪ ω˜ merically solved more cheaper than the original equation (1.2). Let uh denote an approximate solution of (3.1). ω˜ By using the approximation uh , one can probe a good approximation dh,n = (cos θh,n, sin θh,n) of the n-th ray direction dn by the method proposed in [2]. How- ever, the accuracy of the resulting approximation is unsatisfactory. In this section, inspired by the ideas in [2], we will use a different method from [2] to probe ap- r proximate ray angles θ N( 0). { h,n}n=1 Let us first introduce some common notations repeatedly used in this section. We use r to denote the space gradient operator with respect to the (x, y) ∇ 2 coordinate and r as the Hessen operator with respect to the space variable r. If ∇ 7

2 the applied variable only depends on space, we also simplify r and r as and ∇ ∇ ∇ 2 respectively. Furthermore, denote ∇ ru(r ) := ( ru(r)) r r . ∇ 0 ∇ | = 0 Consider a reference point r . For 0 < ρ 1, the circle neighborhood of r with 0 ≪ 0 radio ρ is denoted by

O(r ,ρ) := r : r r <ρ = r : r = r + r(cos θ, sin θ), r<ρ;0 θ 2π . 0 { | − 0| } { 0 ≤ ≤ } We make the following assumption on Ω.

Assumption 3.1. We assume that Ω can be decomposed into several non-overlapping simply connected polygon region. On each region, there is only one kind of medium and ξ is assumed to be suﬃciently smooth. Hence N(r) is a constant on each region.

In this section we give an adaptive ray learning method for general wave solution. 3.1. Approximate ray angles θ determined from an analytic solution { n} of the low-frequency problem. For a point r O(r ,ρ), we write r = r + rd with d = (cos θ, sin θ). A circle ∈ 0 0 θ θ centered at r0 with radius ρ is denoted by

S (r ) := r : r r = ρ = r : r = r + ρd , 0 θ< 2π . ρ 0 { | − 0| } { 0 θ ≤ } We ﬁrst recall the NMLA method developed in [2], which can be used to compute ray directions with low accuracy. NMLA. Setω ˜ = √ω and letu ˜(˜ω, r) be the solution of the low-frequency problem (3.1) (˜u(˜ω, r) was represented by (3.2)). Choosing r > 0 such thatωr ˜ 2 = (1). 0 0 O Deﬁne an auxiliary function (which was called impedance quantity) c(r ) U (θ) := (1 + 0 ∂ )˜u(˜ω, r + r d ), (3.3) ω˜ iω˜ r 0 0 θ which removes any possible ambiguity due to resonance [10] and improves the ro- bustness to noise for solutions of the Helmholtz equation.

The NMLA method sample the impedance quantity Uω˜ (θ) on the circle Sr0 (r0). To this end, let ( U ) denote the ℓ-th Fourier coeﬃcient of U , namely, F ω˜ ℓ ω˜ 1 2π ( U ) = U (θ)e−iℓθdθ. F ω˜ ℓ 2π ω˜ Z0 1 Setω ˜0 = r ωr˜ 0. Then apply the ﬁltering operator to the impedance quantity c( 0) B Uω˜ M ω˜0 iℓθ 1 ( Uω˜ )ℓe Uω˜ (θ)= ℓ F ′ (3.4) B 2Mω˜ +1 i (Jℓ(˜ω0) iJ (˜ω0)) 0 ℓ=−M ℓ Xω˜0 − where M = max(1, [˜ω ], [˜ω + (˜ω1/3 2.5)]). Notice that the number of waves ω˜0 0 0 0 − N(r ) is just the number of sharp peaks in the graph of function U (θ). 0 B ω˜ 8

iωφ˜ n(r0) Define Bñ(˜ω, r)= Añ(˜ω, r)e . It was shown in [1] that

N(r0)

Uω˜ (θ)= B˜n(˜ω, r0)SM (θ θn), (3.5) B ω˜0 − n=1 X where S (θ)= sin([2k+1]θ/2) . As a consequence, whenω ˜ we have k [2k+1] sin(θ/2) 0 → ∞

B˜n(˜ω, r0), if θ = θn, Uω˜ (θ)= (3.6) B (0, otherwise. ∗ ˜∗ For a suﬃciently largeω ˜, we compute θn and Bn(˜ω, r0) by ˜∗ ∗ Bn(˜ω, r0)= Uω˜ (θn)= max Uω˜ (θ). B θ∈(0,2π] B Then (see [1])

θ∗ θ (˜ω−1/2), B˜∗(˜ω, r ) B˜ (˜ω, r ) (˜ω−1/2). (3.7) | n − n| ∼ O | n 0 − n 0 | ∼ O As we will see, the accuracy of the approximate ray angles are lower than the expected accuracy ( (˜ω−2)), which determines the approximation error of the ray- O based FEM method. To improve the accuracy of the approximate ray angle, we propose a post- processing method where the data obtained from the NMLA is used as the initial data for a routine that tries to fit some information of the wave u(˜ω, r). Post-processing. We need to investigate how to useu ˜ (if it is known) to learn more exact ray angles. Take a sampling circle S (r ) with r satisfyingωr ˜ = (1). r1 0 1 1 O Define two dual impedance quantities on the circle Sr1 (r0) as c(r ) U +(θ)=(1+ 0 ∂ )˜u(˜ω, r + r d ), ω˜ iω˜ r 0 1 θ  (3.8)  − c(r0) U (θ)=(1 ∂r)˜u(˜ω, r0 + r1dθ). ω˜ − iω˜ 1  Setω ˜1 = r ωr˜ 1. Choosing a positive integer Mω˜1 satisfying Mω˜1 ω˜1. For c( 0) ≥ each integer l satisfying l M , define a sampling quantities by | |≤ ω˜1 + − 1 [ Uω˜ (θ)]ℓ [ Uω˜ (θ)]ℓ Uℓ = F ′ F ′ . iℓ iω˜1 (J (˜ω ) iJ (˜ω )) + J (˜ω ) − iω˜1 (J (˜ω )+ iJ (˜ω )) J (˜ω ) 2 ℓ 1 − ℓ 1 ℓ 1 2 ℓ 1 ℓ 1 − ℓ 1 e Before describing the post-processing, we give an expansion of Uℓ. Set

1 i T 1 i T a−1 = ( , ) , a1 = ( , ) . e 2 −2 2 2 For n =1, ,N(r ); j = 1, 1, deﬁne the parameters ··· 0 −

b = B˜ (˜ω, r ), b = r a− rB˜ (˜ω, r ) n,0 n 0 n,j 1 j · ∇ n 0 and T 2 b = r c(r )B˜ (˜ω, r )a φ (r )a− , n,2j 1 0 n 0 −j ∇ n 0 j 9

r where B˜ (˜ω, r) = A˜ (˜ω, r)eiωφn( 0) (see the last part). Besides, for ℓ M we n n | | ≤ ω˜1 deﬁne 2 2 2 2Jℓ (˜ω1) ω˜1(Jℓ+j (˜ω1)+ Jℓ (˜ω1)) 2(ℓ + j)Jℓ+j (˜ω1)Jℓ(˜ω1) µℓ,0 = + − , µℓ,j = j +− − (j = 1) aℓ aℓ aℓ aℓ ± and 2 Jℓ+j(˜ω1) Jℓ+2j (˜ω1)Jℓ(˜ω1) µℓ,2j = ij(l + j)˜ω1 − + − (j = 1). aℓ aℓ ± Here iω˜ ′ iω˜ ′ a+ = 1 (J (˜ω ) iJ (˜ω )) + J (˜ω ), a− = 1 (J (˜ω )+ iJ (˜ω )) J (˜ω ). ℓ 2 ℓ 1 − ℓ 1 ℓ 1 ℓ 2 ℓ 1 ℓ 1 − ℓ 1 Lemma 3.1. For each integer l satisfying l M , the number U has the ex- | | ≤ ω˜1 ℓ pansion for large ω˜

N(r0) 2 e −i(l+k)θn Uℓ = bn,kµℓ,ke . (3.9) n=1 − X kX= 2 e Proof. By the Taylor expansion on r (0, ω˜−1], we have ∈ 2 Añ(˜ω, r)= Añ(˜ω, r0 + rdθ)= Añ(˜ω, r0)+ r rAñ(˜ω, r0) dθ + (r ), ∇ 2 · O dn dθ r T 2 3 φn(r)= φn(r0 + rdθ)= φn(r0)+ r · + dθ φn(r0)dθ + (r ). c(r0) 2 ∇ O By the definition ofu ˜ and using the above expansions, we deduce that

N(r0) u˜(˜ω, r + rd ) = B˜ (˜ω, r )+ r rB˜ (˜ω, r ) d 0 θ n 0 ∇ n 0 · θ n=1 X2 ωr˜ d d r T 2 i r n· θ −2 + iω˜ B˜ (˜ω, r )d φ (r )d e c( 0) + (˜ω ). 2 n 0 θ ∇ n 0 θ O Then we have

N(r0) + 1 U (θ)= B˜ (˜ω, r )(1 + d d )+ r rB˜ (˜ω, r ) d (1 + d d + ) ω˜ n 0 n · θ 1∇ n 0 · θ n · θ iω˜ n=1 1 X iωr˜ 1 d d +r B˜ (˜ω, r )dT 2φ (r )d [ (1 + d d )+ c(r )] eiω˜1 n· θ + (˜ω−2) 1 n 0 θ ∇ n 0 θ 2 n · θ 0 O and

N(r0) − 1 U (θ)= B˜ (˜ω, r )(1 d d )+ r rB˜ (˜ω, r ) d (1 d d ) ω˜ n 0 − n · θ 1∇ n 0 · θ − n · θ − iω˜ n=1 1 X iωr˜ 1 d d +r B˜ (˜ω, r )dT 2φ (r )d [ (1 d d ) c(r )] eiω˜1 n· θ + (˜ω−2). 1 n 0 θ ∇ n 0 θ 2 − n · θ − 0 O iθ −iθ Substituting dθ = e a−1 + e a1 into the above two equalities and using the 2-D Jacobi-Anger expansion ∞ iω˜1dn·dθ ℓ −iℓθn+iℓθ e = i Jℓ(˜ω1)e , −∞ ℓ=X 10 we can directly verify that the coeﬃcients of two Fourier transformation are

N(r0) 1 ′ [ U +(θ)] = (˜ω−2)+ b (J (˜ω ) iJ (˜ω ))e−iℓθn iℓ F ω˜ ℓ O n,0 ℓ 1 − ℓ 1 n=1 1 X ′ + b [(1 + )J (˜ω ) iJ (˜ω )]e−i(ℓ+j)θn n,j iω˜ ℓ+j 1 − ℓ+j 1 j=−1,1 1 X iω˜ ω˜ ′ +2r c(r )B˜ (˜ω, r )aT 2φ (r )a [( 1 + 1)J (˜ω )+ 1 J (˜ω )]e−iℓθn 1 0 n 0 −1∇ n 0 1 2 ℓ 1 2 ℓ 1 iω˜ ω˜ ′ + b [( 1 + 1)J (˜ω )+ 1 J (˜ω )]e−i(ℓ+2j)θn n,2j 2 ℓ+2j 1 2 ℓ+2j 1 j=−1,1 X and

N(r0) 1 ′ [ U −(θ)] = (˜ω−2)+ b (J (˜ω )+ iJ (˜ω ))e−iℓθn iℓ F ω˜ ℓ O n,0 ℓ 1 ℓ 1 n=1 1 X ′ + b [(1 )J (˜ω )+ iJ (˜ω )]e−i(ℓ+j)θn n,j − iω˜ ℓ+j 1 ℓ+j 1 j=−1,1 1 X iω˜ ω˜ ′ +2r c(r )B˜ (˜ω, r )aT 2φ (r )a [( 1 1)J (˜ω ) 1 J (˜ω )]e−iℓθn 1 0 n 0 −1∇ n 0 1 2 − ℓ 1 − 2 ℓ 1 iω˜ ω˜ ′ + b [( 1 1)J (˜ω ) 1 J (˜ω )]e−i(ℓ+2j)θn . n,2j 2 − ℓ+2j 1 − 2 ℓ+2j 1 j=−1,1 X

Then, by the deﬁnition of Uℓ, we obtain

N(r0) e −iℓθn −i(ℓ+j)θn Uℓ = bn,0µ0,ℓe + bn,−jµj,ℓe n=1 j=−1,1 X X e −i(l+2j)θn −2 + b − µ e + (˜ω ), (3.10) n, 2j 2j,ℓ O j=−1,1 X where ′ ′ 2 Jℓ(˜ω1) iJℓ(˜ω1) Jℓ(˜ω1)+ iJℓ(˜ω1) 2Jℓ (˜ω1) µℓ,0 = −′ ′ = , iω˜1 (J (˜ω ) iJ (˜ω )) + J (˜ω ) − iω˜1 (J (˜ω )+ iJ (˜ω )) J (˜ω ) a+a− 2 ℓ 1 − ℓ 1 ℓ 1 2 ℓ 1 ℓ 1 − ℓ 1 ℓ ℓ

1 ′ 1 ′ (1 + )Jℓ+j (˜ω1) iJ (˜ω1) (1 )Jℓ+j (˜ω1)+ iJ (˜ω1) iω˜1 ℓ+j iω˜1 ℓ+j µℓ,j = ′ − − ′ iω˜1 iω˜1 2 (Jℓ(˜ω1) iJℓ(˜ω1)) + Jℓ(˜ω1) − 2 (Jℓ(˜ω1)+ iJℓ(˜ω1)) Jℓ(˜ω1) − ′ ′ − Jℓ+j(˜ω1)Jℓ(˜ω1) ω˜1Jℓ+j (˜ω1)Jℓ(˜ω1)+˜ω1Jℓ+j(˜ω1)Jℓ(˜ω1) = − − + − aℓ aℓ 2 2 ω˜1(Jℓ+j (˜ω1)+ Jℓ (˜ω1)) 2(ℓ + j)Jℓ+j (˜ω1)Jℓ(˜ω1) = j +− − , aℓ aℓ

′ ′ iω˜1 iω˜1 ( 2 + 1)Jℓ+2j (˜ω1)+˜ω1Jℓ+2j (˜ω1) ( 2 1)Jℓ+2j (˜ω1) ω˜1Jℓ+2j (˜ω1) µℓ,2j = ′ − ′ − iω˜1 iω˜1 2 (Jℓ(˜ω1) iJℓ(˜ω1)) + Jℓ(˜ω1) − 2 (Jℓ(˜ω1)+ iJℓ(˜ω1)) Jℓ(˜ω1) 2 ′ − ′ − iω˜1 2 (Jℓ+2j (˜ω1)Jℓ(˜ω1) Jℓ+2j (˜ω1)Jℓ(˜ω1)) = + −− aℓ aℓ 11

2 Jℓ+j (˜ω1) Jℓ+2j (˜ω1)Jℓ(˜ω1) = ij(l + j)˜ω1 − + − . aℓ aℓ The equality (3.10) is just the desired expansion (3.9).

Remark 3.1. When the Fourier transformer is applied to only one of the two + − impedance quantities Uω˜ (θ) and Uω˜ (θ), we obtain complicated nonlinear models containing the blue terms as in the NMLA method. In order to remove the blue terms, we deﬁne new sampling quantities U˜l, which can simplify the ﬁtted model.

We need to determine the 6N(r ) unknowns θ and b in (3.9) ( j 2; n = 0 n n,j | | ≤ 1, ,N(r )). Thus at least 6N(r ) equations are needed therein, which implies ··· 0 0 that M 3N(r ). Therefore the best choice would beω ˜ = M = 3N(r ), ω˜1 ≥ 0 1 ω˜1 0 which means that the sampling radius r1 can be chosen as r1 =3N(r0)c(r0)/ω˜. Let ϑ denote the column vector composed of θ and b ( j 2; n =1, ,N(r )). n n,j | |≤ ··· 0 Deﬁne the function

N(r0) 2 −i(ℓ+k)θn Gℓ(ϑ)= µℓ,kbn,ke . n=1 − X kX= 2 Then, from (3.9) we have U = G (ϑ)+ (˜ω−2). (3.11) ℓ ℓ O Although the number U can be computed, the vector ϑ is unknown. ℓ e By introducing 6N(r ) parameters θ¯ and χ , which corresponds to the 0 { n} { n,j } parameters θ and b ,e let λ denote the vector composed of θ¯ and χ ( j n n,j { n} { n,j } | |≤ 2; n =1, ,N(r )), and deﬁne ··· 0 N(r0) 2 −i(ℓ+k)θ¯n Gℓ(λ)= µℓ,kχn,ke . n=1 − X kX= 2

Deﬁne the functional (Mω˜1 =3N(r0))

Mω˜1 J(λ)= U G (λ) 2. (3.12) | ℓ − ℓ | ℓ=−M Xω˜1 e We need to minimize the functional J(λ) to determine the unknown λ:

min J(λ). (3.13) λ We solve the minimization problem (3.13) by the damped least-squares (DLS) ∗ ˜∗ algorithm [29]. Let θn and Bn(˜ω, r0) be the low-accuracy ray angle and the ray amplitudes computed by NMLA. Then we use the preliminary values as starting values θ¯0 = θ∗ , χ0 = B˜∗(˜ω, r ), χ0 =0, j = 2, 1, 1, 2. (3.14) n n n,0 n 0 n,j − − (n =1, ,N(r )) ··· 0 12

Usually a few iterations gives a dramatic improvement to the accuracy of θ∗ . { n} Let θpost denote the ray angles generated by this post-processing method. We { n } describe the ﬁnal result as follows.

Theorem 3.1. Given the analytical solution u(˜ω, r) (˜ω = √ω) of the low frequency problem (3.1), the post-processing process (3.13) can improve the approximation error of the ray directions to be (ω−1), namely, O θpost θ Cω˜−2 = Cω−1 (n =1, ,N(r )) (3.15) | n − n|≤ ··· 0 for a suﬃciently large ω, where the constant C depends only on θ (and N(r )). { n} 0 Proof. Let θ and θpost denote the N-dimensional column vector composed of θ , ,θ 1 ··· N and θpost, ,θpost, respectively. Namely, 1 ··· N post θ1 θ1 . post . θ =  .  and θ =  .  . post  θN   θ     N      Let ϑ− denote the (5N)-dimensional column vector composed of b ( j 2; n = n,j | |≤ 1, ,N(r )). Deﬁne the (6N)-dimensional column vector ··· 0 post post θ λ− = − . ϑ ! Norice that J is a real-valued function, and θ and θpost are real vectors. Using the Taylor formula yields 1 J(ϑ) J(λpost)= J(λpost) (θ θpost)+ H (ξ)(θ θpost) (θ θpost), − − ∇θ − · − 2 θ − · − post where ξ is a mediate value between ϑ and λ− , and Hθ(ξ) denotes the Hesse matrix of J(λ) with the variable θ at the point ξ. Since J(λpost) = 0, the above equality ∇θ − becomes 1 J(ϑ) J(λpost)= (θ θpost)tH (ξ)(θ θpost). − − 2 − θ − It follows by (3.11)-(3.12) that J(ϑ) = (˜ω−4). Then, noticing J(λpost) 0, the O − ≥ above equality leads to

(θ θpost)tH (ξ)(θ θpost) Cω˜−4. (3.16) − θ − ≤ It suﬃces to prove that

(θ θpost)tH (ξ)(θ θpost) c θ θpost 2. (3.17) − θ − ≥ k − k Let λpost, which is a (6N)-dimensional column vector, denote a solution of the minimization problem (3.13). We ﬁrst prove that λ satisﬁes

ϑ λpost (˜ω−1/2). (3.18) k − k2 ≤ O 13

Let λk be the approximation generated by the k-th iteration of the DLS method for (3.13). By (3.7) and (3.14), the initial value λ0 satisﬁes

ϑ λ0 (˜ω−1/2). k − k2 ≤ O The DLS method ensures that (see [29])

ϑ λk Cρk ϑ λ0 Cρkω˜−1/2 k − k2 ≤ k − k2 ≤ with 0 ρ 1. Hence we get ≤ ≤ k −1/2 ϑ lim λ 2 Cω˜ , k − k→∞ k ≤ which implies (3.18). It follows by (3.18) that

ϑ ξ = (˜ω−1). k − k2 O Using this, together with J(ϑ)= (˜ω−4) and r = (˜ω−1), we can verify that (for O 1 O a suﬃciently largeω ˜)

3N ∂2J = ℓ2µ2 (b b e−iℓ(θn−θk) + b b eiℓ(θn−θk))+ (˜ω−1). ∂θ ∂θ |ξ ℓ,0 n,0 k,0 n,0 k,0 O n k − ℓ=X3N

(ℓ) −iℓθn Deﬁne an = bn,0e . The above formula can be written as

2 3N ∂ J 2 2 (ℓ) (ℓ) (ℓ) (ℓ) −1 = ℓ µ (a a + an a )+ (˜ω ). ∂θ ∂θ |ξ ℓ,0 n k k O n k − ℓ=X3N For convenience, we set ε = θ θpost and ε = θ θpost. Then n n − n − 3N N N t 2 2 (ℓ) 2 (ℓ) 2 −1 2 ε H (ξ)ε = ℓ µ a ε + an ε + (˜ω ) ε . θ ℓ,0 | n n| | n| O k k ℓ=−3N n=1 n=1 X X X 2 Notice that µℓ,0 has a positive lower bound independent ofω ˜, the above equality gives N N εtH (ξ)ε c a(ℓ)ε 2 + (˜ω−1) ε 2. (3.19) θ ≥ | n n| O k k n=1 Xℓ=1 X N −iℓθ1 −iℓθN t Deﬁne the vectors αℓ = (e e ) and the matrix A = αℓαℓ. In ··· ℓ=1 addition, let Λ denote the diagonal matrix with the diagonal entries Pbn,0 . It is { } easy to check that

N N N a(ℓ)ε 2 = α (Λε) 2 = (Λε)tA(Λε). | n n| | ℓ | n=1 Xℓ=1 X Xℓ=1 14

Notice that the direction angles θ are diﬀerent each other. Then the vectors { n} α , , α are linearly independent, and so the matrix A is Hermitian positive 1 ··· N deﬁnite. Moreover, the minimal eigenvalue of A is independent ofω ˜. Thus

N N a(n)ε 2 c Λε 2 c ε 2. | ℓ n| ≥ k k ≥ k k n=1 Xℓ=1 X Here we have used the fact that b has a lower bound independent ofω ˜. Then {| n,0|} the constant c depends on θ only. Substituting the above inequality into (3.19) { n} yields (3.17), which, combing (3.16), gives the desired result.

If we replace u(˜ω, r) by an approximate solution, we can design the corresponding numerical method.

3.2. Numerical method. For a ﬁxed large wave number ω, set ω = √ω. Using the given function ξ and g in problem (1.1), we consider the following Helmholtz equation with much lower frequency e

∆u + ω2ξ(r)u =0, in Ω, (3.20) ((∂n + iω ξ(r))u = g(ω, r), on ∂Ω, e p and use u(ω, r) to denote its analyticale solution.e Based on a good approximate solution of u(ω, r), we can compute high accuracy ray angles by the following five steps: e Step 1. Lete be a quasi-uniformly triangular partition of the domain Ω Th0 with the mesh size h satisfying ωh2 1. We apply the GOPWDG method to the 0 0 ≈ discretization of (3.20) on , and use u (ω, r) to denote the resulting approximate Th0 h0 solution. e e Step 2. On each element of h0 with the barycenter r0, we apply the NMLA T r Nh0 ( 0) method to uh0 (ω, r). We compute the number Nh0 (r0) of rays, ray angles θh0,n n=1 N (r ) { } and the low-accuracy amplitudes B h0 0 as in the first part of Subsection { h0,n}n=1 3.1. e Step 3. Let e be a uniformly refining triangular mesh of with the mesh Th Th0 size h˜ satisfying ωh 3N (r )c(r ). We apply the GOPWDG method to the ≈ h0 0 0 discretization of (3.20) on e, and use ue(ω, r) to denote the resulting approximate Th h solution. ee r r ∗ Nhe ( 0) ∗ Nhe ( 0) Step 4. Consider every element of e. Lete Ne(r ), θe and Be Th h 0 { h,n}n=1 { h,n}n=1 N (r ) N (r ) be the natural interpolation of N (r ), θ h0 0 and B h0 0 , re- h0 0 { h0,n}n=1 { h0,n}n=1 e spectively. We replace the function u(ω, r) in (3.8) by uh(ω, r) to get two new impedance quantities, and further sample the quantities to compute the numbers

Ue ℓ≤3Ne (r ). We use these numbers toe deﬁne the minimizatione problem (3.13), { h,ℓ} h 0 e 15

r post Nhe ( 0) and solve it by the DLS method to obtain high-accuracy ray angles θe , h,n n=1 ∗ ∗ { } where the initial guesses are chosen as θe and Be . h,n h,n e Step 5. Let h be a uniformly reﬁning triangulation of h with the mesh size h T Nh(r0) T satisfying ωh 1. Let Nh(r0) and θh,n n=1 denote the natural interpolations ≈ r { } post Nhe ( 0) of Ne(r0) and θe on h, respectively. h { h,n }n=1 T e Suppose we have obtained exact ray numbers (Nh(r0)= N(r0)) on each element. We give the approximation property of the resulting ray angles in the following theorem.

Theorem 3.2. The resulting ray angles of the above process have the following approximation error

θpost θ = (ω−1), n =1, ,N(r ). (3.21) | h,n − n| O ··· 0 Proof. Notice that the GOPWDG approximate solutions of the low-frequency prob- post lem possess suﬃcient high accuracy. Applying (3.15) to every θe and using the h,n deﬁnitions given in the above Step 5, we obtain the approximate error of the resulting ray angles.

3.3. Approximation property of the adaptive plane wave space. Deﬁne d = (cos θpost, sin θpost),n =1, ,N(r ). On the element K with the barycen- h,n h,n h,n ··· 0 0 tric points denoted by r0, our adaptive GOPW basis functions are chosen as

τ r ϕ (ω, r)= p h,n (r)eiωτh,n( ), n =1, ,N(r ), j =1, 2. (3.22) n,j j ··· 0

τh,n with τh,n(r) and pj (r) (j =1, 2) be the polynomials determined as in Subsection 2.2 by replacing the discrete plane wave direction dn with dh,n. Our adaptive GOPW space adapted to solve the high-frequency problem is deﬁned as

V (K )= span ϕ (ω, r): n =1, ,N(r ); j =1, 2 . ray 0 n,j ··· 0 In order to investigate approximate properties of this space, we deﬁne

2(K )= v H2(K ): v satisﬁes (∆ + κI)=0on K . H 0 ∈ 0 0 Theorem 3.3. Let u 2(K ) be the analytic solution of (1.2). Assume that ∈ H 0 φ and A in the optics ansatz (2.5)-(2.6) of u satisfy φ C2(K ) and A n n,s n ∈ 0 n,s ∈ C1(K ). Then there exists an interpolation operator π : 2(K ) V (K ) such 0 h H 0 → ray 0 that −1 2 u π u 2 C(ω + h + ωh ), (3.23) k − h kL (K0) ≤ where the constant C depends only on the upper bound of φ 2 and A 1 . k nkH (K0) k n,skH (K0) In particular, when choosing h as h ω−1, we have ∼ −1 u π u 2 C(ω )= Ch. k − h kL (K0) ≤ 16

Proof. Since the basis functions in Vray(K0) are independent for different wave directions, we can only consider the case that u has only one wave direction on K0, i.e., n = 1. For ease of notation, we simply write (see Subsection 2.1) ∞ r u(r)= A(r)eiωφ( ), A(r)= (iω)−sA (r) r K , (3.24) s ∈ 0 s=0 X t Let θ be the exact direction angle defined by φ and set dθ = (cosθ sin θ) , which satisfies φ(r )= c−1(r )d since φ 2 = ξ = c−2. Then, by the Taylor formula, ∇ 0 0 θ |∇ | the phase function φ can be written as φ(r)= φ(r )+ P θ(r)+ O( r r 2), r K , 0 1 | − 0| ∈ 0 where P θ(r) is a linear polynomial of x x and y y and it can be written as 1 − 0 − 0 P θ(r)= c−1(r )d (r r )= c−1(r )(cosθ(x x ) + sin θ(y y )). 1 0 θ · − 0 0 − 0 − 0 Let θh be the approximation direction angle defined in the last subsection, which satisfies θ θ = O(ω−1). (3.25) | h − | By the definitions of τ (see Subsection 2.2), we have τ (r)= P θh (r) (r K ) with h h 1 ∈ 0 P θh (r)= c−1(r )d (r r )= c−1(r )(cosθ (x x ) + sin θ (y y )). 1 0 θh · − 0 0 h − 0 h − 0 Moreover, from the definition of the polynomials pτh 2 (see Subsection 2.2), we { j }j=1 have pτh (r)=1+ d⊥ (r r ) and pτh (r)=1 d⊥ (r r ). 1 θh · − 0 2 − θh · − 0

Let γK0 (A0) denote the integration average of A0 on K0, and deﬁne x1 = x2 = 1 2 γK0 (A0). Then τh τh x1p1 (r)+ x2p2 (r)= γK0 (A0). Thus A (r) x pτh,n (r)+ x pτh,n (r) = A (r) γ (A ) , r K , | 0 − 1 1 2 2 | | 0 − K0 0 | ∈ 0 which implies that 2 r r τ A eiωφ( 0) eiωφ( 0) x p h,n = A (r) γ (A ) , on K . (3.26) | 0 − j j | | 0 − K0 0 | 0 j=1 X Deﬁne 2 iωφ(r0) τh iωτh,n(r) πhu(r)= e xj pj (r)e . j=1 X Notice that θ 2 iωφ(r0) iωP +O(ωh ) −1 u(r)= A0e e 1 + O(ω ). Then 2 θ 2 iωφ(r ) iωφ(r ) τh iωP +O(ωh ) u(r) π u(r) = A e 0 e 0 x p e 1 − h 0 − j j j=1 X 17

2 θ 2 iωφ(r ) τh iωP +O(ωh ) iωτ (r) −1 + e 0 x p e 1 e h + O(ω ). j j − j=1 X Therefore, by (3.26) we have

u(r) π u(r) C A (r) γ (A ) + γ (A ) τ (r) P θ(r) +O(ω−1). (3.27) | − h |≤ | 0 − K0 0 | | K0 0 || h − 1 | It follows by (3.25) that

θ −1 2 τ (r) P (r) C φ 2 (ω h)+ O(h ), r K . | h − 1 |≤ k kH (K0) ∈ 0 Substituting this into (3.27), we obtain

2 −1 u π u 2 C A 1 + φ 2 (h + ωh )+ O(ω ), k − h kL (K0) ≤ k 0kH (K0) k kH (K0) which gives the desired result.

−1 Remark 3.2. Under the assumption that h ω and the upper bound of φ 2 ∼ k nkH (K0) and A 1 is independent of ω, the adaptive plane wave space V (K ) pos- k n,skH (K0) ray 0 sesses 1 order convergence with respect to h or ω−1 and has much better conver- − gence than existing discrete spaces for Helmholtz equations with large wave numbers.

3.4. A discretization method of (1.1). The adaptive plane wave space Vray(K0) was designed for homogeneous Helmholtz equations on the element K0. In order to use this space to the discretization of the nonhomogeneous Helmholtz equation (1.1), we need to adopt the plane wave method combined with local spectral elements (PW-LSFE) first presented in [24], which was extended to the case with variable wave numbers in [21]. To shorten the length of this paper, here we only describe the basic idea of the method (more details can be found in [21], where only a different plane wave space was used). Assume that f is defined in a slightly large domain containing Ω as its subdomain and the domain Ω is strictly star-shaped. As usual, let Ω be decomposed into the union of some elements Ω N , which constitute quasi-uniform and and shape- { k}k=1 regular triangulations h with mesh sizes h. For each element Ωk, we choose a disc ∗ T domain Ωk that has almost the same size of Ωk and contains Ωk as its subdomain. (1) 1 ∗ Let uk H (Ωk) be the solution of the restriction of the nonhomogeneous ∈ ∗ Helmholtz equation (1.1) on Ωk, with homogeneous Robin boundary condition on ∗ (1) 2 (1) (1) (2) (1) ∂Ωk. Define u L (Ω) as u Ωk = uk Ωk for each Ωk. Set u = u u , (2) ∈ | | − then u satisfies a homogeneous Helmholtz equation on every element Ωk. Notice that u(2) satisfies two transmission conditions depending on u(1) on the common edge of two neighboring elements. ∗ Let m be a positive integer, and let Sm(Ωk) denote the set of polynomials defined on Ω∗, whose orders are less or equal to m. We use u(1) S (Ω∗) to denote k k,h ∈ m k the standard spectral element solution of the nonhomogeneous Helmholtz equation satisfied by u(1), and define u(1) N S (Ω ) by u(1) = u(1) . k h ∈ k=1 m k h |Ωk k,h|Ωk Q 18

We use V ( ) to denote the adaptive plane wave space spanned by the local ray Th basis functions ϕ (ω, r). Namely, the space V ( ) is defined as n,j ray Th V ( )= v L2(Ω) : v V (K), K . ray Th { h ∈ h|K ∈ ray ∀ ∈ Th} Let u(2) V ( ) be the approximation of u(2), where u(2) are defined by the h ∈ ray Th h discontinuous Galerkin method with V ( ) for the local homogeneous Helmholtz ray Th equation satisfied by u(2). The final approximate solution u L2(Ω) is defined by u = u(1) + h ∈ h|Ωk h |Ωk u(2) . For convenience, we call this method as ray-GOPWDG-LSFE method. In h |Ωk most situations, the number N(r0) of the wave directions is small, so the space V ( ) has much smaller degrees of freedom than the standard plane wave space ray Th V ( ), where p must increase when ω increases. p Th

4. Numerical experience

In this section we apply the ray-GOPWDG-LSFE method to solve the nonhomogeneous Helmholtz equations with variable wave numbers ∆u ω2ξ2u = f in Ω, − − ∂ (4.1) ( + iωξ)u = g on ∂Ω,  ∂n and we report some numerical results to confirm the effectiveness of the proposed methods. Let Ω be divided into small rectangles. Each rectangle has the same mesh size h, where h is the length of the longest edge of the elements. The resulting uniform triangulation is denoted by . We fix the number of elements per wavelength for Th the experiments made in this subsection, i.e., we choose h = ω−1. Moreover, we only consider two examples where the ray number is a constant among the whole computed area, then the number of basis functions on every element is fixed. We introduce the relative L2 error

uex uh 2 Err. = k − kL (Ω) , u 2 k exkL (Ω) where uex is the analytic solution and uh is the numerical solution. Deﬁne δ by log(Err /Err ) δ = 2 1 , log(ω2/ω1) which can measure the “pollution eﬀect” of a numerical method. ˆ ˆF ˆG Let θ(dex), θ(dh ) and θ(dωe ) denote the exact ray angles, the numerical ray angles computed in ray-FEM method and the numerical ray angles computed by the post-processing method in ray-GOPWDG-LSFE method, respectively. Denote F G by u the numerical solution solved by the ray-FEM method. Let u and udˆ h dˆωe ex denote the numerical solution solved by the ray-GOPWDG-LSFE method used 19

ˆG ˆ by the direction dωe and dex, respectively. Let “DOFs” denotes the computing complexity of the considered numerical methods. We compare the performances of the ray-GOPWDG-LSFE method and the ray- FEM method for the high-frequency numbers ω = 400, 625, 900.

4.1. Example 1 (Single wave in a heterogeneous medium). We consider an example in a heterogeneous medium in the domain Ω = [0, 1] [0, 1] (see [14]): 1 × Define ξ(r) = c(r)2 with the velocity field c(r) as a smooth converging lens with a Gaussian profile at the center (x0,y0)=(1/2, 1/2)

4 1 c(x, y)= 1 exp 32 (x r )2 + (y r )2 . 3 − 8 − − 1 − 2 The analytic solution of the problem is given by

iωxy uex(x, y)= c(x, y)e

2 Then the source term is fex = ∆u ω ξ(r)u and the boundary function is chosen ∂ − − as gex = ( ∂n + iω)uex. Following the process Step 1-Step 2 in Subsection 3.2, we ﬁrst solve the corresponding low frequency-problem ω = 20, 25, 30 on a coarsen mesh h, respectively. The NMLA sampling results shows that there is only one ray locally at each ele- e e

1.8 sampling the numerical solution sampling the analytical solution 1.6

1.4

1.2 | 0 α 1 |BU

0.8

0.6

0.4

0.2

0 -1 0 1 2 3 4 5 6 7 θ

Figure 1. NMLA sampling at the location (19/20, 3/20) ment (see Figure 1). We then use the post-processing method locally to calculate the ray directions in each element. The maximum iteration steps cost by the DLS method are reported in Table 1. 20

Table 1. Maximum iteration steps of the DLS method in the post-processing method

ω 400 625 900 iter(s) 33 34 31

∞ In Table 2, we list L errors θ(dˆ e) θ(dˆ ) ∞ of the ray angles. k ω − ex k

Table 2. Complexity and ray angle approximation errors of the ray-GOPWDG-LSFE method and ray-FEM method

G F ω Comp. θ(dˆ e ) θ(dˆ ) ∞ Order Comp. θ(dˆ ) θ(dˆ ) ∞ Order k ω − ex k k h − ex k 400 9.7e+5 4.321e-3 1.3e+8 5.242e-3 − − 625 2.4e+6 2.635e-3 1.11 4.4e+8 3.435e-3 0.95 900 4.9e+6 1.839e-3 0.98 1.2e+9 2.384e-3 1.00

The relative L2 errors of the approximated solutions and the total DOFs needed in the ray-GOPWDG-LSFE method are reported in the Table 3.

Table 3. DOFs. and Approximation errors of the ray- GOPWDG-LSFE method and ray-FEM method

G F ω DOFs udˆ u 0,Ω Order DOFs uh u 0,Ω Order k ωe − k k − k 400 6.6e+4 5.817e-4 1.3e+7 7.113e-4 − − 625 1.6e+5 3.592e-4 1.08 4.1e+7 4.615e-4 0.96 900 3.4e+5 1.671e-4 1.04 6.7e+7 3.214e-4 0.99

It shows that for the single wave solution, the approximation error of the numerical solutions of the both methods can have an optimal convergence of (ω−1). O However, the total DOFs of the ray-GOPWDG-LSFE method is (ω2), which is O also optimal in solving the two-dimensional Helmholtz equation and much less than the total DOFs needed by the ray-FEM method.

4.2. Example 2. (Constant gradient of slowness squared ξ). We provide an example in a heterogeneous medium with wave speed of constant gradient (see [14]): ξ(r) = c2 +2G (r r ) with parameters c = 1, G = (0.1, 0.2) and 0 0 · − 0 0 0 − r = ( 0.1, 0.1). Referring to [14], there are two rays crossing in the domain 0 − − Ω = [0, 1] [0, 1]. The two phase functions are known analytically and they are × given by G 2 φ =¯cσ | 0| σ3, j =1, 2, (4.2) j j − 6 j 21 where j 2 2 2 2(¯c + ( 1) c¯ G0 r r0 ) σ = − − | | | − | , j =1, 2, (4.3) j q G 2 p| 0| with c¯ = c + G (r r ). (4.4) 0 0 · − 0 Then the analytic solution is given by

2 2 uex = exp(iωφ1)/(xy +1i)+ exp(iωφ2)/((x + y +1i)). (4.5)

2 The source term is fex = ∆u ω ξ(r)u and the boundary function is chosen as ∂ − − gex = ( ∂n + iω)uex. Following the process Step 1-Step 2 in Subsection 3.2, we ﬁrst solve the corresponding low frequency-problem ω = 20, 25, 30 on a coarsen mesh h, respectively. It shows that we can learn nearly-accurate ray directions based on a numerical e e

1.5

sampling the numerical solution sampling the analytical solution

1 | 0 α |BU

0.5

0 -1 0 1 2 3 4 5 6 7 θ

Figure 2. NMLA sampling at the location (19/20, 3/20) solution generated by the GOPWDG-LSFE method on the mesh h = (ω−1) (see O Figure 2). We then use the post-processing method locally to calculate thee ray directionse in each element. The maximum iteration steps needed in the DLS method are listed in Table 4.

Table 4. Maximum iteration steps of the DLS method in the post-processing method

ω 400 625 900 iter(s) 36 35 33

The L∞ errors of the ray directions are reported in Table 5. 22

Table 5. Complexity and ray angle approximation errors of the ray-GOPWDG-LSFE method and ray-FEM method

G F ω Comp. θ(dˆ e ) θ(dˆ ) ∞ Order Comp. θ(dˆ ) θ(dˆ ) ∞ Order k ω − ex k k h − ex k 400 1.3e+6 8.433e-3 1.9e+8 1.205e-1 − − 625 3.1e+6 5.133e-3 1.11 6.3e+8 9.238e-2 0.60 900 6.5e+6 3.482e-3 1.06 1.7e+9 7.662e-2 0.51

The relative L2 errors of the approximated solutions and the total DOFs needed of the ray-GOPWDG-LSFE method are shown in the Table 6.

Table 6. Dofs. and approximation errors of the ray-GOPWDG- LSFE method and ray-FEM method

G F ω DOFs udˆ u 0,Ω Order DOFs uh u 0,Ω Order k ωe − k k − k 400 1.3e+6 3.934e-4 1.9e+7 3.391e-3 − − 625 3.1e+6 2.327e-4 1.176 6.3e+7 2.516e-3 0.67 900 6.5e+6 1.562e-4 1.094 1.7e+7 2.049e-3 0.56

It shows that our ray-GOPWDG-LSFE method are more eﬃcient than the ray- FEM method for high-frequency Helmholtz problem and the relative L2 errors have the optimal convergence of (ω−1). The total DOFs of the ray-GOPWDG-LSFE O method is (ω2), which is also optimal in solving the two-dimensional Helmholtz O equation.

5. Conclusion

In this paper we have introduced an adaptive ray-based GOPW method for the high-frequency Helmholtz equation in smooth media. We have developed diﬀerent ray-learning method for the single wave as well as the multiple wave. We have derived an interpolation error of the ray-GOPW spaces. The numerical results shows that the ray GOPW method can achieve asymptotic convergence rate of (ω−1) for multiple waves. The computing complexity can be also optimal as O (ω2) in two dimensions. O References

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