发表机构
King Fahd University of Petroleum and Minerals; Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway; Algorithmiq S.r.l.（阿卜杜拉国王科技大学; 挪威阿尔塔大学希莱拉斯量子分子科学中心化学系; Algorithmiq有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出自适应多分辨率扩散算子变分框架，使扩散算子由自适应状态生成，并用于正则化逆问题，产生AMDI方法，数值实验验证其高效去噪性能。
AI 中文摘要
我们为自适应多分辨率表示上的状态相关扩散建立了一个变分框架，其中扩散算子由自适应状态本身生成。该状态由一个可容许的多分辨率树、其活动逼近空间与基，以及相应的系数表示组成。它确定了一个对称非负相互作用形式和一个相关的半正定内蕴扩散算子。与经典自适应小波方法（其中规定算子表示在演化的逼近空间上）相比，这里的细化和粗化同时修改表示、相互作用图和算子。由于自适应层级通过离散拓扑变化演化，耦合动力学通过时间离散变分原理而非固定空间上的微分演化来表述。我们证明了离散更新的存在性、离散能量不等式、系数空间零模态，以及冻结自适应状态的收缩性。对于正则化逆问题，该构造产生了自适应多分辨率扩散成像（AMDI），在状态相关能量中结合数据保真度、内蕴扩散、系数稀疏性和树复杂度。数值实验验证了组装的算子恒等式，检查了细化交换子衰减，并确认了离散能量耗散。自适应Haar和高阶多小波计算展示了在异构数据上的分辨率局部化。在去噪中，AMDI以不到完整活动表示的10%保持了高结构重建质量，并在保留的噪声实现中表现稳定。
英文摘要
We develop a variational framework for state-dependent diffusion on adaptive multiresolution representations in which the diffusion operator is generated by the adaptive state itself. The state consists of an admissible multiresolution tree, its active approximation space and basis, and the corresponding coefficient representation. It determines a symmetric nonnegative interaction form and an associated positive semidefinite intrinsic diffusion operator. In contrast to classical adaptive wavelet methods, where a prescribed operator is represented on an evolving approximation space, refinement and coarsening here modify simultaneously the representation, interaction graph, and operator. Because the adaptive hierarchy evolves through discrete topological changes, the coupled dynamics are formulated through a time-discrete variational principle rather than a differential evolution on a fixed space. We establish existence of the discrete updates, a discrete energy inequality, the coefficient-space null mode, and contractivity for frozen adaptive states. For regularized inverse problems, the construction yields Adaptive Multiresolution Diffusion Imaging (AMDI), combining data fidelity, intrinsic diffusion, coefficient sparsity, and tree complexity in a state-dependent energy. Numerical experiments verify the assembled operator identities, examine refinement-commutator decay, and confirm discrete energy dissipation. Adaptive Haar and higher-order multiwavelet calculations demonstrate localization of resolution on heterogeneous data. In denoising, AMDI retains high structural reconstruction quality with less than 10\% of the full active representation, with stable behavior across held-out noise realizations.
