Low-frequency waves have long wavelengths that wash over hidden objects, leaving reconstructed images blurry... Conversely, high-frequency waves have short wavelengths that bounce cleanly off boundaries... but they also make the underlying inverse problem exponentially more unstable.
<aside> 👏
<aside> <img src="/icons/download_yellow.svg" alt="/icons/download_yellow.svg" width="40px" />
Standard and Gradient Based Tikhonov Regularization in Helmholtz Inverse Scattering Sweeps
%%{
init: {
'theme': 'base',
'themeVariables': {
'darkMode': true,
'background': '#0B0F19',
'primaryColor': '#1E1B4B',
'primaryTextColor': '#F8FAFC',
'primaryBorderColor': '#6366F1',
'lineColor': '#A5B4FC',
'secondaryColor': '#0F4C5C',
'tertiaryColor': '#0F172A',
'clusterBkg': '#111827',
'clusterBorder': '#374151',
'defaultLinkColor': '#A5B4FC'
}
}
}%%
flowchart TD
%% Main Nodes
Start("Current Exploration:<br/><b>Acoustic Wave Inverse Scattering &</b><br/><b>Tikhonov Regularization</b>")
AdaptiveReg("1. Adaptive Regularization")
HigherOrder("2. Higher-Order & Anisotropic Smoothing")
NonHomogeneous("3. Extension to Non-Homogeneous Backgrounds")
Optimization("4. Algorithmic Optimization &<br/>Animation Enhancements")
Goal("Potential Orientation:<br/><b>Robust, High-Resolution Inversion &</b><br/><b>Advanced Visualization</b>")
%% Current Exploration Detail Subgraph
subgraph CurrentExpDetails [Current Exploration Focus]
direction TB
Focus1("Wavenumber Sweeps<br/>k = 1.0 to 8.0")
Focus2("Ill-Posedness & Exponential Decay")
Focus3("0th vs 1st Order Tikhonov")
Focus4("Localized Coordinate Mapping")
end
%% Connections from Current Exploration Details to Main Nodes
Start --> CurrentExpDetails
CurrentExpDetails --> AdaptiveReg
CurrentExpDetails --> HigherOrder
CurrentExpDetails --> NonHomogeneous
CurrentExpDetails --> Optimization
%% Adaptive Regularization Subgraph
subgraph AdaptiveRegDetails [1. Adaptive Regularization Details]
direction TB
Adapt1["Auto-parameter Tuning<br/>L-Curve and Discrepancy"]
Adapt2["Frequency-Scheduled Penalty"]
end
AdaptiveReg --> AdaptiveRegDetails
%% Higher-Order Smoothing Subgraph
subgraph HigherOrderDetails [2. Higher-Order Smoothing Details]
direction TB
High1["Total Variation and p-Laplacian"]
High2["Directional Anisotropy"]
end
HigherOrder --> HigherOrderDetails
%% Non-Homogeneous Background Subgraph
subgraph NonHomogeneousDetails [3. Non-Homogeneous Background Details]
direction TB
NonHom1["Variable Sound Speed Profiles"]
NonHom2["Multifrequency Joint Inversion"]
end
NonHomogeneous --> NonHomogeneousDetails
%% Algorithmic Optimization Subgraph
subgraph OptimizationDetails [4. Optimization Details]
direction TB
Opt1["Parallel Processing & GPU Acceleration"]
Opt2["Manim & Matplotlib 3D Vector Fields"]
end
Optimization --> OptimizationDetails
%% Connections to Final Goal
AdaptiveReg --> Goal
HigherOrder --> Goal
NonHomogeneous --> Goal
Optimization --> Goal
%% Styling Definitions
classDef startNode fill:#4C0519,stroke:#F43F5E,stroke-width:2px,color:#FFFFFF;
classDef mainBranch fill:#1E1B4B,stroke:#818CF8,stroke-width:2px,color:#FFFFFF;
classDef detailNode fill:#0F172A,stroke:#38BDF8,stroke-width:1.5px,color:#E2E8F0;
classDef goalNode fill:#064E3B,stroke:#34D399,stroke-width:2.5px,color:#FFFFFF;
%% Apply Classes
class Start startNode;
class AdaptiveReg,HigherOrder,NonHomogeneous,Optimization mainBranch;
class Goal goalNode;
class Focus1,Focus2,Focus3,Focus4,Adapt1,Adapt2,High1,High2,NonHom1,NonHom2,Opt1,Opt2 detailNode;
%% Subgraph Box Styling
style CurrentExpDetails fill:#111827,stroke:#4B5563,stroke-width:1.5px,color:#9CA3AF;
style AdaptiveRegDetails fill:#111827,stroke:#374151,stroke-width:1px,color:#9CA3AF;
style HigherOrderDetails fill:#111827,stroke:#374151,stroke-width:1px,color:#9CA3AF;
style NonHomogeneousDetails fill:#111827,stroke:#374151,stroke-width:1px,color:#9CA3AF;
style OptimizationDetails fill:#111827,stroke:#374151,stroke-width:1px,color:#9CA3AF;
</aside>