【摘要】：分裂可行性問(wèn)題是出現(xiàn)在信號(hào)處理,放射治療和醫(yī)學(xué)圖像重建等現(xiàn)實(shí)問(wèn)題中的一類(lèi)重要的逆問(wèn)題.設(shè)H1,H2是兩個(gè)實(shí)Hilbert空間,C(?)H1,Q(?)H2是兩個(gè)非空閉凸集,A:→H2是一個(gè)有界線性算子.分裂可行性問(wèn)題可表述為:找一點(diǎn)x∈C使得Ax ∈ Q.為了解該問(wèn)題,許多作者已經(jīng)給出了各種各樣的算法.2012年,Moudafi對(duì)分裂可行性問(wèn)題進(jìn)行了推廣,提出了分裂等式問(wèn)題.設(shè)H1,H2,H3是三個(gè)實(shí)Hilbert空間,C(?)H1,Q(?)H2是兩個(gè)非空閉凸集,A:H1→H3,B:H2 →H3是兩個(gè)有界線性算子.分裂等式問(wèn)題可表述為:找點(diǎn)x∈C,y∈Q使得Ax=By.顯然,當(dāng)H2=H3,B=I(單位算子)時(shí),分裂等式問(wèn)題就簡(jiǎn)化為分裂可行性問(wèn)題.針對(duì)分裂等式問(wèn)題,許多作者也已給出了相應(yīng)的算法.鑒于分裂等式問(wèn)題及其相關(guān)問(wèn)題在現(xiàn)實(shí)世界中的重要應(yīng)用,值得我們對(duì)其進(jìn)一步研究.在這篇文章中,我們主要研究了分裂等式問(wèn)題的幾類(lèi)相關(guān)問(wèn)題:一,多集分裂等式問(wèn)題:設(shè)H1,H2,H3是三個(gè)實(shí)Hilbert空間,{Ci}(i=1)t(?)H1,{Qj}(j=1)r(?)H2是兩組非空閉凸集,A:H1→H3,B:H2→H3是兩個(gè)有界線性算子,則多集分裂等式問(wèn)題可表述為:找點(diǎn)x ∈∩(i=1)t Ci,y∈∩(j=1)r Qj使得Ax=By.并給出了相應(yīng)的迭代解法-有自適應(yīng)步長(zhǎng)的算法和構(gòu)造方向法,算法的主要思想在于減少計(jì)算量和提高收斂速度;二,分裂等式不動(dòng)點(diǎn)問(wèn)題:設(shè)H1,H2,H3是三個(gè)實(shí)Hilbert空間,T1:H1→H1,T2:H2 →H2是兩個(gè)非線性算子,Fix(T1),Fix(T2)分別是算子T1,T2的不動(dòng)點(diǎn)集且Fix(T1)≠(?),Fix(T2)≠(?),A:H1→H3,B:H2→H3是兩個(gè)有界線性算子,則分裂等式不動(dòng)點(diǎn)問(wèn)題可表述為:找點(diǎn)x∈Fix(T1),y∈Fix(T2)使得Ax=By.相應(yīng)的迭代解法不需要相關(guān)算子的半緊性而具有強(qiáng)收斂性;三,將分裂等式問(wèn)題與變分包含問(wèn)題結(jié)合的分裂等式變分包含問(wèn)題:設(shè)H1,H2,H3是三個(gè)實(shí)Hilbert空間,U:H1→2H1:H2→2H2 是兩個(gè)集值極大單調(diào)算子,A:H1 →H3,B:H2→H3是兩個(gè)有界線性算子.則分裂等式變分包含問(wèn)題可表述為:找點(diǎn)x∈H1,y∈H2使得0 ∈U(x),0∈K(y),Ax=By.我們給出的迭代解法強(qiáng)收斂到分裂等式變分包含問(wèn)題的最小范數(shù)解.
[Abstract]:The splitting feasibility problem is an important inverse problem in signal processing, radiotherapy and medical image reconstruction. Let H _ 1O _ 2 be two real Hilbert spaces C _ (?) H _ (1) Q (?) H _ 2 are two nonempty closed convex sets A: h _ 2 is a bounded linear operator. The problem of splitting feasibility can be expressed as: finding a point x 鈭，

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