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Research article Special Issues

On an unsupervised method for parameter selection for the elastic net

  • Received: 24 May 2021 Accepted: 22 October 2021 Published: 24 November 2021
  • Despite recent advances in regularization theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov regularization parameter from noisy data. In this work, we improve their results and extend the analysis to the elastic net regularization. Furthermore, we design a data-driven, automated algorithm for the computation of an approximate regularization parameter. Our analysis combines statistical learning theory with insights from regularization theory. We compare our approach with state-of-the-art parameter selection criteria and show that it has superior accuracy.

    Citation: Zeljko Kereta, Valeriya Naumova. On an unsupervised method for parameter selection for the elastic net[J]. Mathematics in Engineering, 2022, 4(6): 1-36. doi: 10.3934/mine.2022053

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  • Despite recent advances in regularization theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov regularization parameter from noisy data. In this work, we improve their results and extend the analysis to the elastic net regularization. Furthermore, we design a data-driven, automated algorithm for the computation of an approximate regularization parameter. Our analysis combines statistical learning theory with insights from regularization theory. We compare our approach with state-of-the-art parameter selection criteria and show that it has superior accuracy.



    In this paper we consider solitary wave solutions of the time-dependent coupled nonlinear Schrödinger system with perturbation

    {iΦ1tΔΦ1=μ1|Φ1|2Φ1+β|Φ2|2Φ1+f1(x),xΩ, t>0,iΦ2tΔΦ2=μ2|Φ2|2Φ2+β|Φ1|2Φ2+f2(x),xΩ, t>0,Φ1(t,x)=Φ2(t,x)=0,xΩ, t>0, j=1,2, (1.1)

    where ΩRN is a smooth bounded domain, i is the imaginary unit, μ1,μ2>0 and β0 is a coupling constant. When N3, the system (1.1) appears in many physical problems, especially in nonlinear optics. Physically, the solution j denotes the j-th component of the beam in Kerr-like photorefractive media (see [1]). The positive constant μj is for self-focusing in the j th component of the beam. The coupling constant β is the interaction between the two components of the beam. The problem (1.1) also arises in the Hartree-Fock theory for a double condensate, that is, a binary mixture of Bose-Einstein condensates in two different hyperfine states, for more information, see [2].

    If we looking for the stationary solution of the system (1.1), i.e., the solution is independent of time t. Then the system (1.1) is reduced to the following elliptic system with perturbation

    {Δu+λ1u=μ1|u|2u+βuv2+f1(x),xΩ,Δv+λ2v=μ2|v|2v+βu2v+f2(x),xΩ,u=v=0,xΩ. (1.2)

    In the case where N3 and f1=f2=0, then the nonlinearity and the coupling terms in (1.2) are subcritical, and the existence of solutions has recently received great interest, for instance, see [3,4,5,6,7,8,9,10,11] for the existence of a (least energy) solution, and [12,13,14,15,16] for semiclassical states or singularly perturbed settings, and [17,18,19,20,21,22] for the existence of multiple solutions.

    In the present paper we consider the case when N=4 and p=2=4 is the Sobolev critical exponent. If f1=f2=0, the paper [23] proved the existence of positive least energy solution for negative β, positive small β and positive large β.Furthermore, for the case λ1=λ2, they obtained the uniqueness of positive least energy solutions and they studied the limit behavior of the least energy solutions in the repulsive case β, and phase separation is obtained. Later, the paper [24] studied the high dimensional case N5. The paper [25] proved the existence of sign-changing solutions of (1.2). Recently, the paper [26] considered the system (1.2) with perturbation in dimension N3. By using Nehari manifold methods, the authors proved the existence of a positive ground state solution and a positive bound state solution. To the best of our knowledge, the existence of multiple nontrivial solution to the system (1.2) with critical growth(N=4) is still unknown. In the present paper we shall fill this gap.

    Another motivation to study the existence of multiple nontrivial solution of (1.2) is coming from studying of the scalar critical equation. In fact, the second-order semilinear and quasilinear problems have been object of intensive research in the last years. In the pioneering work [27], Brezis and Nirenberg have studied the existence of positive solutions of the scalar equation

    {Δu=up+f,xΩ,u>0,xΩ,u=0,xΩ, (1.3)

    where Ω is a bounded smooth domain in RN, N3, p=N+2N2, f(x,u) is a lower order perturbation of up. Particularly, when f=λu, where λR is a constant, they have discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1.7) is highly sensitive to N, the dimension of the space. Precisely, the paper [27] has shown that, in dimension N4, there exists a positive solution of (1.3), if and only if λ(0,λ1); while, in dimension N=3 and when Ω=B1 is the unit ball, there exists a positive solution of (1.7), if and only if λ(λ1/4,λ1), where λ1>0 is the first eigenvalue of Δ in Ω. The paper [28] proved the existence of both radial and nonradial solutions to the problem

    {Δu=b(r)up+f(r,u),xΩ,r=|x|,u>0,xΩ,u=0,xΩ (1.4)

    under some assumptions on b(r) and f(r,u), p=N+2N2, where Ω=B(0,1) is the unit ball in RN. In the paper [29], G. Tarantello considered the critical case for (1.3). He proved that (1.3) has at least two solutions under some conditions of f: f0, fH1 and

    fH1<cNSN4,cN=4N2(N2N+2)N+24, (1.5)

    and

    S=infuH10(Ω){0}|u|22|u|24 (1.6)

    is the best Sobolev constant of the imbedding from H10(Ω) to Lp(Ω). For more results on this direction we refer the readers to [30,31,32,33,34,35] and the references therein.

    Motivated by the above works, in the present paper, we are interested in the critical coupled Schrödinger equations in (1.2) with λ1=λ2=0

    {Δu=μ1u3+βuv2+f1,xΩ,Δv=μ2v3+βu2v+f2,xΩ,u=v=0,xΩ. (1.7)

    where Ω is a smooth bounded domain in R4, Δ is the Laplace operator and p=2=4 is the Sobolev critical exponent, and μ1>0,μ2>0,0<βmin{μ1,μ2}.

    Obviously, the energy functional is denoted by

    I(u,v)=12Ω(|u|2+|v|2)dx14Ω(μ1|u|4+μ2|v|4+2βu2v2)dxΩ(f1u+f2v)dx (1.8)

    for (u,v)H=H10(Ω)×H10(Ω). So, the critical point of I(u,v) is the solution of the system (1.7). We shall fill the gap and generalize the results of [26] to the critical case. Our main tool here is the Nehari manifold method which is similar to the fibering method of Pohozaev's, which was first used by Tarantello [29].

    We define thee Nehari manifold

    N={(u,v)H|I(u,v)(u,v)=0}. (1.9)

    It is clear that all critical points of I lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed (u,v)H{(0,0)}, we set

    g(t)=I(tu,tv)=A2t2B4t4Dt,t>0.

    where

    A=Ω(|u|2+|v|2)dx,B=Ω(μ1|u|4+μ2|v|4+2βu2v2)dx,D=Ω(f1u+f2v)dx. (1.10)

    The mapping g(t) is called fibering map. Such maps are often used to investigate Nehari manifold for various semilinear problem. By using the relationship of I and g(t), we can divide N into three parts as follow:

    N+={(u,v)N|A3B>0},N0={(u,v)N|A3B=0},N={(u,v)N|A3B<0}.

    In order to get our results, we assume that fi satisfies

    fi0,fiL43(Ω),|fi|43<S3233K12,i=1,2, (1.11)

    where K=max{μ1,μ2}, S is defined in (1.6). Then we have the following main results.

    Theorem 1.1. Assume that 0<βmin{μ1,μ2}, and f1,f2 satisfies (1.11). Then

    infNI=infN+I=c0 (1.12)

    is achieved at a point (u0,v0)N. Furthermore, (u0,v0) is a critical point of I.

    Next we consider then following minimization problem

    infNI=c1. (1.13)

    Then we have the following result.

    Theorem 1.2. Assume that 0<βmin{μ1,μ2}, and f1,f2 satisfies (1.11). Then c1>c0 and the infimum in (1.13) is achieved at a point (u1,v1)N, which is the second critical point of I.

    Remark 1.3. We point out that to the best of our knowledge, the existence of multiple nontrivial solution to the system (1.2) with critical growth(N=4) is still unknown. In the present paper we shall fill this gap and generalized the results of [26] to the critical case.

    Throughout the paper, we shall use the following notation.

    ● Let (,) be the inner product of the usual Sobolev space H10(Ω) defined by (u,v)=Ωuvdx, and the corresponding norm is u=(u,u)12.

    ● Let S=infuH10(Ω){0}(|u|22/|u|24) be the best Sobolev constant of the imbedding from H10(Ω) to L4(Ω).

    |u|p is the norm of Lp(Ω) defined by |u|p=(Ω|u|pdx)1p, for 0<p<.

    ● Let (u,v)2=Ω(|u|2+|v|2)dx be the norm in the space of H=H10(Ω)×H10(Ω).

    ● Let C or Ci(i=1,2,...) denote the different positive constants.

    We shall use the variational methods to prove the main results. In this section we shall prove some basic results for the system (1.2). The next lemma states the purpose of the assuptions (1.11).

    Lemma 2.1. Assume that the conditions of Theorem 1.1 hold. Thenfor every (u,v)H{(0,0)}, there exists a uniquet1>0 such that (t1u,t1v)N. In particular, wehave

    t1>t0:=[A3B]12 (2.1)

    and g(t1)=maxtt0g(t), where A and B are given in (1.10). Moreover, if D>0, then there exists a unique t2>0, such that (t2u,t2v)N+, where D>0 is given in (1.10). In particular, one has

    t2<t0andI(t2u,t2v)I(tu,tv),t[0,t1]. (2.2)

    Proof. We first define the fibering map by

    g(t)=A2t2B4t4Dt,t>0.

    Then we have

    g(t)=AtBt3D=Φ(t)D.

    We deduce from Φ(t)=0 that

    t=t0=[A3B]12.

    If 0<t<t0, we have g(t)=Φ(t)>0, and if t>t0, one sees g(t)=Φ(t)<0. A direct computation shows that Φ(t) achieves its maximum at t0 and Φ(t0)=233A32B12.

    From the assumption (1.11), Sobolev's and Hölder's inequalities, we infer that

    D=Ω(f1u+f2v)dx|f1|43|u|4+|f2|43|v|4(|f1|243+|f2|243)(|u|24+|v|24)2max{|f1|43,|f2|43}(|u|24+|v|24)12<2S3233K12(|u|24+|v|24)12. (2.3)

    On the other hand, since 0<βmin{μ1,μ2}, it follows that

    Φ(t0)=233A32B12=233(Ω(|u|2+|v|2)dx)32(Ω(μ1|u|4+μ2|v|4+2βu2v2)dx)12233S32(|u|24+|v|24)32K12(Ω(|u|4+|v|4+2βu2v2)dx)122S3233K12(|u|24+|v|24)322(|u|44+|v|44)122S3233K12(|u|24+|v|24)32|u|24+|v|24=2S3233K12(|u|24+|v|24)12,

    where K=max{μ1,μ2}. Hence we get

    g(t0)=Φ(t0)D>0andg(t),ast+. (2.4)

    Thus, there exists an unique t1>t0 such that g(t1)=0. We infer from the monotonicity of Φ(t) that for t1>t0

    g(t1)=Φ(t1)<0,t21Φ(t1)=t21(A3Bt21)<0.

    This implies that (t1u,t1v)N. If D>0, then we have g(0)=Φ(0)D=D<0. Furthermore, there exists an unique t2[0,t0] such that g(t2)=0 and Φ(t2)=D. A direct computation shows that (t2u,t2v)N+ and I(t2u,t2v)I(tu,tv),t[0,t1].

    Next we study the structure of N0.

    Lemma 2.2. Let fi0(i=1,2) satisfy (1.11). Then for every(u,v)N{(0,0)}, we have

    Ω(|u|2+|v|2)dx3Ω(μ1|u|4+μ2|v|4+2βu2v2)dx0. (2.5)

    Hence we can get the conclusion that N0={(0,0)}.

    Proof. In order to prove that N0={(0,0)}, we only need to show that for (u,v)H{(0,0)}, g(t) has no critical point that is a turning point. We use contradiction argument. Assume that there exists (u,v)(0,0) such that (t0u,t0v)N0 and t0>0. Thus, we get

    g(t0)=At0Bt30D=0andg(t0)=A3Bt20=0.

    Then we have t0=[A3B]12. This contradicts (2.4). This finishes the proof.

    In the next lemma, we shall prove the properties of Nehari manifolds N.

    Lemma 2.3. Let fi0(i=1,2) satisfy (1.11). For (u,v)N{(0,0)}, then there exist ε>0 anda differentiable function t=t(w,z)>0,(w,z)H,(w,z)<ε, and satisfying the following conditions

    t(0,0)=1,t(w,z)((u,v)(w,z))N,(w,z)<ε,

    and

    <t(0,0),(w,z)>=2Ω(uw+vz)dx4Ω[μ1|u|2uw+μ2|v|2vz+β(uv2w+u2vz)]dxΩ(f1w+f2z)dxΩ(|u|2+|v|2)dx3Ω(μ1|u|4+μ2|v|4+2βu2v2)dx.

    Proof. We define F:R×HR by

    F(t,(w,z))=t(uw)22+t(vz)22t3Ω(μ1|uw|4+μ2|vz|4+2β(uw)2(vz)2)dxΩ(f1(uw)+f2(vz))dx.

    We deduce from Lemma 2.2 and (u,v)N that F(1,(0,0))=0. Moreover, one has

    Ft(1,(0,0))=Ω(|u|2+|v|2)dx3Ω(μ1|u|4+μ2|v|4+2βu2v2)dx0.

    By applying the implicit function theorem at point (1, (0, 0)), we can obtain the results.

    In this section we are devoted to proving Theorem 1.1. We begin the following lemma for the property of infI.

    Lemma 3.1. Let

    c0=infNI=infN+I.

    Hence I is bounded from below in N and c0<0.

    Proof. For (u,v)N, we have I(u,v),(u,v)=0. We infer from (1.10) that ABD=0. Thus, one deduces from (2.3) and Hölder inequality that

    D<C(|u|24+|v|24)12C1(|u|22+|v|22)12=C1A12.

    Hence, one deduces that

    I(u,v)=A2B4D=A43D4>A4C2A12.

    Thus, the infimum c0 in N+ is bounded from below. Next we prove the upper bound for c0. Let wiH10(Ω)(i=1,2) be the solution for Δw=fi,(i=1,2). So, for fi0 one sees that

    Ω(f1w1+f2w2)dx=|w1|22+|w2|22>0.

    We let t2=t2(u,v)>0 as defined by Lemma 2.1. Thus, we infer that (t2w1,t2w2)N+ and

    t22Ω(|w1|2+|w2|2)dxt42Ω(μ1|w1|4+μ2|w2|4+2βw21w22)dxt2Ω(f1w1+f2w2)dx=0.

    Furthermore, it follows from (2.2) that

    c0=inf(u,v)N+I(u,v)I(t2w1,t2w2))<I(0,0)=0.

    This completes the proof.

    The next lemma studies the properties of the infimum c0.

    Lemma 3.2.

    (1) The level c0 can be attained. That is, there exists(u0,v0)N+ such that I(u0,v0)=c0.

    (2) (u0,v0) is a local minimum for I in H.

    Proof. From Lemma 3.1, we can apply Ekeland's variational principle to the minimization problem, which gives a minimizing sequence {(un,vn)}N such that

    (i) I(un,vn)<c0+1n,

    (ii) I(w,z)I(un,vn)1n(|(wun)|2+|(zvn)|2),(w,z)N.

    For n large enough, by Lemma 3.1 and (i)-(ii) of the above, we can get

    C1>0,C2>0,0<C1|un|22+|vn|22C2.

    In the following we shall prove that I(un,vn)0 as n. In fact, we can apply Lemma 2.3 with (u,v)=(un,vn) and (w,z)=δ(Iu(un,vn),Iv(un,vn))I(un,vn)(δ>0). Then can find tn(δ) such that

    (wδ,zδ)=tn(δ)[(un,vn)δ(Iu(un,vn),Iv(un,vn))I(un,vn)]N.

    Thus, we infer from the condition (ii) that

    I(un,vn)I(wδ,zδ)1n(|(wδun)|2+|(zδvn)|2). (3.1)

    On the other hand, by using Taylor expansion we have that

    I(un,vn)I(wδ,zδ)=(1tn(δ))(I(wδ,zδ),(un,vn))+δtn(δ)(I(wδ,zδ),I(un,vn)I(un,vn))+o(δ).

    Dividing by δ>0 and letting δ0, we get

    1n(2+tn(0)(|un|2+|vn|2))tn(0)(I(un,vn),(un,vn))+I(un,vn)=I(un,vn). (3.2)

    Combining (3.1) and (3.2) we conclude that

    I(un,vn)Cn(2+tn(0)).

    We infer from Lemma 2.3 and (un,vn)N that tn(0) is bounded. That is,

    |tn(0)|C.

    Hence we obtain that

    I(un,vn)0asn. (3.3)

    Therefore, by choosing a subsequence if necessary, we have that

    (un,vn)(u0,v0)inHandI(u0,v0)=0,

    and

    c0I(u0,v0)=14(|u0|22+|v0|22)Ω(f1u0+f2v0)dxlimnI(un,vn)=c0.

    Consequently, we infer that

    (un,vn)(u0,v0)inH,I(u0,v0)=c0=infNI.

    From Lemma 2.1 and (3.3), we deduce that (u0,v0)N+.

    (2) In order to get the conclusion, it suffices to prove that (w,z)H,ε>0, if (w,z)<ε, then I(u0w,v0z)I(u0,v0). In fact, notice that for every (w,z)H with Ω(f1u+f2v)dx>0, we infer from Lemma 2.1 that

    I(su,sv)I(t1u,t1v),s[0,t0].

    In particular, for (u0,v0)N+, we have

    t2=1<t0=[Ω(|u0|2+|v0|2)dx3Ω(μ1|u0|4+μ2|v0|4+2βu20v20)dx]12.

    Let ε>0 sufficiently small. Then we infer that for (w,z)<ε

    1<[Ω(|(u0w)|2+|(v0z)|2)dx3Ω(μ1|u0w|4+μ2|v0z|4+2β(u0w)2(v0z)2)dx]12=˜t0. (3.4)

    From Lemma 2.3, let t(w,z)>0 satisfy t(w,z)(u0w,v0z)N for every (w,z)<ε. Since t(w,z)1 as (w,z)0, we can assume that

    t(w,z)<˜t0,(w,z)<ε.

    Hence we obtain that t(w,z)(u0w,v0z)N+ and

    I(s(u0w),s(v0z))I(t(w,z)(u0w),t(w,z)(v0z))I(u0,v0),0<s<˜t0.

    From (3.4) we can take s=1 and conclude

    I(u0w,v0z)I(u0,v0),(w,z)H,(w,z)<ε.

    This finishes the proof.

    Proof of Theorem 1.1. From Lemma 3.2, we know that (u0,v0) is the critical point of I.

    In this section we focus on the proof of Theorem 1.2. The main difficulty here is the lack of compactness(due to the embedding HL4(Ω)×L4(Ω) is noncompact). Motivated by previous works of [27,29,37], we shall seek the local compactness. Then by using the Mountain pass principle to find the second nontrivial solution of equation (1.7). The pioneering paper [29] has used this methods to find the second solution of the scalar Schrödinger equation. To this purpose, we first begin with the following lemma to find the threshold to recover the compactness.

    Lemma 4.1. For every sequence (un,vn)H satisfying

    (i) I(un,vn)cwithc<c0+14min{S2μ1,S2μ2}, where c0 is defined in (1.12), S is the best Sobolevconstant of the imbedding from H10(Ω) to L4(Ω),

    (ii)I(un,vn)0asn.

    Then {(un,vn)} has a convergent subsequence. This means thatthe (PS)c condition holds for all levelc<c0+14min{S2μ1,S2μ2}.

    Proof. From condition (i) and (ii), it is easy to verify that (un,vn) is bounded. So, for the subsequence {(un,vn)}(which we still call {(un,vn)}, we can find a (w0,z0)H such that (un,vn)(w0,z0) in H. Then from the condition (ii), we obtain that

    (I(w0,z0),(w,z))=0,(w,z)H.

    That is, (w0,z0) is a solution in H. Moreover, (w0,z0)N and I(w0,z0)c0. Let

    (un,vn)=(w0+wn,z0+zn).

    Then (wn,zn)(0,0) in H. Then it suffices to prove that

    (wn,zn)(0,0)inH. (4.1)

    We use the indirect argument. Assume that (4.1) does not hold. Then we divide the following three cases to find the contradiction.

    Case 1: wn0 and zn0 in H. Since (un,vn) is bounded, it follows that

    Ωw2nz2ndx=o(1).

    by (1.7), we can get

    c0+14min{S2μ1,S2μ2}>I(un,vn)=I(w0+wn,z0+zn)=I(w0,z0)+12Ω|zn|2dxμ24Ω|zn|4dx+o(1)c0+12|zn|22μ24|zn|44+o(1),

    and then

    12|zn|22μ24|zn|44<S24μ2. (4.2)

    We infer from the condition (ii) that

    o(1)=(I(un,vn),(un,vn))=(I(w0,z0),(w0,z0))+|zn|22μ2|zn|44+o(1).

    That is, we get

    |zn|22=μ2|zn|44+o(1).

    By using the embedding from H10(Ω) to L4(Ω), we get

    μ2|zn|44=|zn|22S|zn|24+o(1).

    Since zn0, we infer that |zn|24S/μ2++o(1). That is,

    |zn|44S2μ22+o(1).

    Hence we get

    12|zn|22μ24|zn|44=μ24|zn|44+o(1)14S2μ2. (4.3)

    This contradicts with the fact (4.2).

    Case 2: wn0 and zn0 in H. This can be accomplished by using same argument as in the proof of the Case 1.

    Case 3: wn0 and zn0 in H. Similar to the Case 1, we infer from condition (ii) that

    o(1)=(I(un,vn),(un,0))=|un|2μ1|un|44βΩu2nv2ndxΩf1undx=(I(w0,z0),(w0,0))+|wn|22μ1|wn|44βΩw2nz2ndx+o(1).

    Then we have

    |wn|22=μ1|wn|44+βΩw2nz2ndx+o(1). (4.4)

    One infers from Hölder and Sobolev inequality that

    S|wn|24|wn|22=μ1|wn|44+βΩw2nz2ndx+o(1)μ1|wn|44+β|wn|24|zn|24+o(1). (4.5)

    Since wn0, we have

    Sμ1|wn|24+β|zn|24+o(1)μ1(|wn|24+|zn|24)+o(1). (4.6)

    Similarly, we obtain that

    Sμ2|zn|24+β|wn|24+o(1)μ2(|wn|24+|zn|24)+o(1). (4.7)

    Thus, we conclude that

    |wn|24+|zn|24max{Sμ1,Sμ2}+o(1). (4.8)

    On the other hand, we infer from the condition (ii) that

    o(1)=(I(un,vn),(un,vn))=|un|2+|vn|2μ1|un|44μ2|vn|442βΩu2nv2ndxΩf1undxΩf2vndx=(I(w0,z0),(w0,z0))+|wn|22+|zn|22μ1|wn|44μ2|zn|442βΩw2nz2ndx+o(1). (4.9)

    From (4.6)-(4.9), we deduce that

    c0+14min{S2μ1,S2μ2}>I(un,vn)=I(w0+wn,z0+zn)=I(w0,z0)+12|wn|22+12|zn|2214(μ1|wn|44+μ2|zn|44+2βΩw2nz2ndx)+o(1)c0+14(μ1|wn|44+μ2|zn|44+2βΩw2nz2ndx)+o(1)c0+S4(|wn|24+|zn|24)+o(1)c0+14max{S2μ1,S2μ2}+o(1).

    This is a contradiction.

    In order to applying Lemma 4.1 to get the compactness, we need to prove the following inequality

    c1=infNI<c0+14min{S2μ1,S2μ2}.

    Let

    uε(x)=εε2+|x|2ε>0, xR4

    be an extremal function for the Sobolev inequality in R4. Let uε,a=u(xa) for xΩ and the cut-off function ξaC0(Ω) with ξa0 and ξa=1 near a. We set

    Uε,a(x)=ξa(x)uε,a(x),xR4.

    Following [37], we let Ω1Ω be a positive measure set such that u0>0,v0>0, where c0=I(u0,v0) is given in Theorem 1.1. Then we have the following conclusion.

    Lemma 4.2. For every R>0, and a.e. aΩ1, there existsε0=ε0(R,a)>0, such that

    min{I(u0+RUε,a,v0),I(u0,v0+RUε,a)}<c0+14min{S2μ1,S2μ2} (4.10)

    for every 0<ε<ε0.

    Proof. As in [37], a direct computation shows that

    I(u0+RUε,a,v0)=12|u0|22+RΩu0Uε,adx+R22|Uε,a|22+12|v0|22μ14(|u0|44+R4|Uε,a|44+4RΩu30Uε,adx+4R3ΩU3ε,au0dx)μ24|v0|44β2Ω(u20v20+2Ru0v20Uε,a+R2U2ε,av20)dxΩ(f1u0+f2v0)dxRΩfUε,adx+o(ε). (4.11)

    We infer from [27] that

    |Uε,a|22=F+O(ε2)and|Uε,a|44=G+O(ε4), (4.12)

    where

    F=R4|u1(x)|2dx,G=R4dx(1+|x|2)4,S=FG12

    If we let u0=0 outside Ω, then

    ΩU3ε,au0dx=R4u0ξa(x)ε3(ε2+|xa|2)3dx=εR4u0ξa(x)1ε4φ(xε)dx

    where

    φ(x)=1(1+|x|2)3L1(R4).

    Set

    E=R41(1+|x|2)3dx.

    Then we can derive

    R4u0ξa(x)1ε4φ(xε)dxu0(a)E.

    Since (u0,v0) is the critical point of I, it follows that

    Ω(|u0|2+|v0|2)dxΩ(μ1u40+μ2v40+2βu20v20)dxΩ(f1u0+f2v0)dx=0. (4.13)

    We infer from (4.11)-(4.13) that

    I(u0+RUε,a,v0)=I(u0,v0)+R22FR44μ1Gμ1R3ΩU3ε,au0dxβR22ΩU2ε,av20dx+o(ε)c0+R22FR44μ1Gμ1εR3Eu0(a)+o(ε). (4.14)

    In order to get the upper bound of (4.14), we define

    q1(s)=F2s2μ1G4s4kεs3,k=μ1Eu0(a)>0,

    and

    q2(s)=F2s2μ1G4s4.

    It is easy to get the maximum of q2(s) is achieved at s0=(Fμ1G)12. Let the maximum of q1(s) is achieved at sε, so we can let sε=(1δε)s0, and get δε0(ε0). Substituting sε=(1δε)s0 into q1(s)=0, we can get

    FF(1δε)2=3s0(1δε)kε.

    As in [29], we infer that

    δεε,ε0.

    Then we can get the upper bound estimation of I(u0+RUε,a,v0):

    I(u0+RUε,a,v0)c0+R22FR44μ1GkεR3+o(ε)c0+[(1δε)s0]22F[(1δε)s0]44μ1Gkε[(1δε)s0]3+o(ε)=c0+(s202Fs404μ1G)+(s40μ1Gs20F)δεkεs30+o(ε)<c0+S24μ1+o(ε).

    Thus, for ε0>0 small, we get

    I(u0+RUε,a,v0)<c0+S24μ1.

    Similarly, we obtain that

    I(u0,v0+RUε,a)<c0+S24μ2.

    So, we prove

    min{I(u0+RUε,a,v0),I(u0,v0+RUε,a)}<c0+14min{S2μ1,S2μ2},0<ε<ε0.

    This finishes the proof.

    Without loss of generality, from above Lemma 4.2 we can assume

    I(u0+RUε,a,v0)<c0+S24μ1,R>0,0<ε<ε0.

    Now we are ready to prove Theorem 1.2.

    Proof of Theorem 1.2. It is clear that there exists an uniqueness of t1>0 such that

    (t1u,t1v)NandI(t1u,t1v)=maxtt0I(t1u,t1v), (u,v)H, (u,v)=1.

    Moreover, t1(u,v) is a continuous function of (u,v), and N divides H into two components H1 and H2, which are disconnect with each other. Let

    H1={(u,v)=(0,0)or(u,v):(u,v)<t1((u,v)(u,v))}andH2={(u,v):(u,v)>t1((u,v)(u,v))}.

    Obviously, we have HN=H1H2. Furthermore, we obtain that N+H1 for (u0,v0)H1. We choose a constant C0 such that

    0<t1(u,v)C0,(u,v)=1.

    In the following we deduce that

    (w,z)=(u0+R0Uε,a,v0)H2, (4.15)

    where R0=(1F|C20(u0,v0)2)12+1. Since

    (w,z)2=(u0,v0)2+R20|Uε,a|2+2R0Ω|u0||Uε,a|dx=(u0,v0)2+R20F+o(1)>C20[t1((w,z)(w,z))]2

    for ε>0 small enough. We fix ε>0 small to make both (4.10) and (4.15) hold by the choice of R0 and aΩ1. Set

    Γ1={γC([0,1],H):γ(0)=(u0,v0),γ(1)=(u0+R0Uε,a,v0)}.

    We take h(t)=(u0+tR0Uε,a,v0). Then h(t)Γ1. From Lemma 4.1, we conclude that

    c=infhΓ1maxt[0,1]I(h(t))<c0+S24μ1.

    Since the range of every hΓ1 intersect N, we have

    c1=infNIc<c0+S24μ1. (4.16)

    Set

    Γ2={γC([0,1],H):γ(0)=(u0,v0),γ(1)=(u0,v0+R0Uε,a)}.

    By using the same argument, we can get similar results

    c=infhΓ2maxt[0,1]I(h(t))<c0+S24μ2.

    Moreover, since the range of every hΓ2 intersect N, we have

    c1=infNIc<c0+S24μ2. (4.17)

    Combining (4.16) and (4.17), we obtain that

    c1<c0+14min{S2μ1,S2μ2}.

    Next by using Mountain-Pass lemma(see [36]) to obtain that there exist {(un,vn)}N such that

    I(un)c1,I((un,vn))0.

    From Lemma 4.1, we can obtain a subsequence (still denote {(un,vn)}) of {(un,vn)}, and (u1,v1)H such that

    (un,vn)(u1,v1)inH.

    Hence, we get (u1,v1) is a critical point for I,(u1,v1)N and I(u1,v1)=c1.

    The authors thank the referees's nice suggestions to improve the paper. This work was supported by NNSF of China (Grants 11971202, 12071185), Outstanding Young foundation of Jiangsu Province No. BK20200042.

    The authors declare there is no conflicts of interest.



    [1] S. W. Anzengruber, R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Probl., 26 (2010), 025001. doi: 10.1088/0266-5611/26/2/025001
    [2] A. Astolfi, Optimization: An introduction, 2006.
    [3] F. Bauer, M. A. Lukas, Comparing parameter choice methods for regularization of ill-posed problems, Math. Comput. Simulat., 81 (2011), 1795–1841. doi: 10.1016/j.matcom.2011.01.016
    [4] M. Belkin, P. Niyogi, V. Sindhwani, Manifold regularization: A geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res., 7 (2006), 2399–2434.
    [5] R. Bhatia, Matrix analysis, New York: Springer-Verlag, 1997.
    [6] T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Probl., 25 (2009), 015015. doi: 10.1088/0266-5611/25/1/015015
    [7] A. Chambolle, R. DeVore, N. Lee, B. Lucier, Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319–335. doi: 10.1109/83.661182
    [8] A. Chambolle, P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167–188. doi: 10.1007/s002110050258
    [9] C. De Mol, E. De Vito, L. Rosasco, Elastic-net regularisation in learning theory, J. Complexity, 25 (2009), 201–230. doi: 10.1016/j.jco.2009.01.002
    [10] E. De Vito, M. Fornasier, V. Naumova, A machine learning approach to optimal Tikhonov regularization I: affine manifolds, Anal. Appl., 2021, in press.
    [11] C.-A. Deledalle, S. Vaiter, J. Fadili, G. Peyré, Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection, SIAM J. Imaging Sci., 7 (2014), 2448–2487. doi: 10.1137/140968045
    [12] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inform. Theory, 41 (1995), 613–627. doi: 10.1109/18.382009
    [13] D. L. Donoho, I. Johnstone, Ideal spatial adaptation via wavelet shrinkage, Biometrika, 81 (1994), 425–455. doi: 10.1093/biomet/81.3.425
    [14] B. Efron, T. Hastie, I. Johnstone, R. Tibshirani, Least angle regression, The Annals of Statistics, 32 (2004), 407–499.
    [15] Y. C. Eldar, Generalized SURE for exponential families: applications to regularization, IEEE Trans. Signal Process., 57 (2009), 471–481. doi: 10.1109/TSP.2008.2008212
    [16] H. W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Dordrecht: Kluwer Academic Publishers, 1996.
    [17] W. J. Fu, Nonlinear GCV and quasi-GCV for shrinkage models, J. Stat. Plan. Infer., 131 (2005), 333–347. doi: 10.1016/j.jspi.2004.03.001
    [18] R. Giryes, M. Elad, Y. C. Eldar, The projected GSURE for automatic parameter tuning in iterative shrinkage methods, Appl. Comput. Harmonic Anal., 30 (2011), 407–422. doi: 10.1016/j.acha.2010.11.005
    [19] G. H. Golub, M. Heath, G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215–223. doi: 10.1080/00401706.1979.10489751
    [20] P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561–580. doi: 10.1137/1034115
    [21] B. Hofmann, Regularization for applied inverse and ill-posed problems: a numerical approach, Springer-Verlag, 2013.
    [22] B. Jin, D. Lorenz, S. Schiffler, Elastic-net regularisation: error estimates and active set methods, Inverse Probl., 25 (2009), 115022. doi: 10.1088/0266-5611/25/11/115022
    [23] O. Lepskii, On a problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl., 35 (1991), 454–466. doi: 10.1137/1135065
    [24] A. Lorbert, P. Ramadge, Descent methods for tuning parameter refinement, In: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 2010,469–476.
    [25] V. A. Morozov, Methods for solving incorrectly posed problems, Springer Science & Business Media, 2012.
    [26] E. Novak, H. Woźniakowski, Optimal order of convergence and (in)tractability of multivariate approximation of smooth functions, Constr. Approx., 30 (2009), 457–473. doi: 10.1007/s00365-009-9069-8
    [27] C. M. Stein, Estimation of the mean of a multivariate normal distribution, Ann. Statist., 9 (1981), 1135–1151.
    [28] W. Su, M. Bogdan, E. Candés, False discoveries occur early on the Lasso path, Ann. Statist., 45 (2017), 2133–2150.
    [29] U. Tautenhahn, U. Hämarik, The use of monotonicity for choosing the regularization parameter in ill-posed problems, Inverse Probl., 15 (1999), 1487–1505. doi: 10.1088/0266-5611/15/6/307
    [30] A. Tikhonov, V. Glasko, Use of the best rate of adaptive estimation in some inverse problems, USSR Computational Mathematics and Mathematical Physics, 5 (1965), 93–107. doi: 10.1016/0041-5553(65)90150-3
    [31] A. N. Tikhonov, V. Y. Arsenin, Solutions of ill-posed problems, Washington, D. C.: V. H. Winston & Sons, New York: John Wiley & Sons, 1977.
    [32] J. A. Tropp, An introduction to matrix concentration inequalities, Now Foundations and Trends, 2015.
    [33] R. Vershynin, High-dimensional probability: An introduction with applications in data science, Cambridge University Press, 2018.
    [34] E. De Vito, S. Pereverzyev, L. Rosasco, Adaptive kernel methods using the balancing principle, Found. Comput. Math., 10 (2010), 455–479. doi: 10.1007/s10208-010-9064-2
    [35] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600–612. doi: 10.1109/TIP.2003.819861
    [36] S. N. Wood, Modelling and smoothing parameter estimation with multiple quadratic penalties, J. Roy. Stat. Soc. B, 62 (2000), 413–428. doi: 10.1111/1467-9868.00240
    [37] H. Zou, T. Hastie, Regularisation and variable selection via the elastic net, J. Roy. Stat. Soc. B, 67 (2005), 301–320. doi: 10.1111/j.1467-9868.2005.00503.x
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