Probability Density Alternation Method

• Probability Density Alternation (PDA) method is a novel technique that reformulates inverse integral equations by mapping kernels to probability densities, simplifying multi-dimensional problems.
• The method efficiently collapses high-order integrals into 1D histograms, enabling matrix inversion and reducing computational complexity in applications like AFM, OT, and SFA.
• PDA incorporates regularization strategies and iterative schemes, such as Tikhonov regularization and Landweber iteration, to improve stability in noisy and ill-conditioned scenarios.

The Probability Density Alternation (PDA) method is a unified approach for solving inverse problems governed by integral equations, particularly those arising in surface science contexts such as atomic force microscopy (AFM), optical tweezers (OT), and the surface force apparatus (SFA). PDA enables the reformulation of a general inverse integral equation as an inversion of a kernel constructed from a family of probability density functions derived from the integral’s geometry. While classical for single integrals, PDA is especially efficient for double or higher-order integrals, as it collapses high-dimensional tensor inversions into more tractable matrix problems. The method is grounded in a sequence of transformations: mapping the original kernel to a 1D probability density for each control parameter, assembling the resulting kernel matrix, and inverting this matrix to recover the unknown target function (Hashidate et al., 11 Jan 2026).

1. Reformulation of Inverse Integral Equations via Probability Densities

Consider an inverse problem defined by a Fredholm integral of the first kind: $$f(s)\;=\;\int_{x_a}^{x_b} g\bigl(h(x;s)\bigr)\;u(x)\;dx,\quad s\in[S_a,S_b],$$ where $g(h)$ is the unknown function to be recovered; $h(x;s)$ is a known injective transformation with range $h\in[h_a(s),h_b(s)]$; $u(x)$ is a weight function. The PDA method reformulates this as an expectation value of $g(h)$ under a probability density $P_s(h)$: $$P_s(h)\;=\;\frac{u\bigl(x(h;s)\bigr)}{\displaystyle\int_{h_a(s)}^{h_b(s)}u\bigl(x(h';s)\bigr)\,dh'}$$ for each fixed $s$, where $x(h;s)$ satisfies $h=h(x;s)$. This density is normalized: $\int_{h_a(s)}^{h_b(s)}P_s(h)\,dh=1.$ The kernel is then $M(s,h)\;=\;W(s)\,P_s(h)$, with

$W(s)\;=\;\int_{x_a}^{x_b}u(x)\,dx,$

yielding the compact integral equation: $$f(s)\;=\;\int_{h_{\min}}^{h_{\max}} M(s,h)\,g(h)\,dh,$$ where $h_{\min}=\min_{s\in[S_a,S_b]}h_a(s)$, $h_{\max}=\max_{s\in[S_a,S_b]}h_b(s)$. Upon discretization ($s\to\{s_i\}$, $h\to\{h_j\}$, $\Delta h_j$), the problem becomes a linear system: $\mathbf f\;=\;M\,\mathbf g,$ with $M_{ij}=M(s_i,h_j)\Delta h_j$, which is solved for $\mathbf g$ representing the discretized $g(h)$.

2. Alternation Solution, Regularization, and Iterative Schemes

The essential PDA algorithm consists of constructing $P_s(h)$ for each parameter $s$ and forming the kernel matrix $M$, followed by a single inversion step to solve for $\mathbf g$: $\mathbf g\;=\;M^{-1}\,\mathbf f.$ For problems where the kernel is ill-conditioned or the measurements $f(s)$ suffer from noise, iterative or regularized inversion schemes are applicable. An example is the Landweber iteration: $$\mathbf g^{(k+1)} = \mathbf g^{(k)} + \alpha M^T\bigl(\mathbf f - M \mathbf g^{(k)}\bigr),$$ with appropriate step size $\alpha$ for convergence. Tikhonov regularization or singular value decomposition (SVD) may also be employed, especially when the condition number of $M$ exceeds $10^8$. For noisy data, truncation or early stopping yields improved stability.

3. Applicability to Single Versus Higher-Order Integrals

For single integral equations, as often encountered in OT and AFM, conventional kernel inversion or direct algebraic manipulation may suffice, and PDA offers little computational advantage. For double or higher-order integral equations—leading to high-dimensional tensor equations—the direct approach is often computationally infeasible ($O(N^p)$ unknowns for a $p$-fold integral).

PDA collapses multi-dimensional integrals into 1D probability histograms by constructing $P_s(h)$ via: $$P_s(h) = \frac{1}{V}\int_{\Omega(s)}\delta\bigl(h-h(\mathbf x;s)\bigr)\,d\mathbf x,$$ where the delta function "pushes forward" the measure onto $h$, and $\Omega(s)$ is the domain of integration. The result is a 1D discretized $M$ with storage $O(N_s N_h)$ and operations dominated by histogram construction and matrix inversion ($O(N_s N_{\text{MC}})$ and $O(N_h^3)$, respectively). This reduction is especially powerful for surface force, solvation, and colloidal structure problems common in surface science (Hashidate et al., 11 Jan 2026).

4. Numerical Demonstration: Surface Force Apparatus (SFA) Example

An explicit example is given by reconstructing the pressure profile $\Phi(h)$ between crossed cylinders in the SFA system. The governing equation is a double integral: $$f(s) = \int_{x=0}^{r}\int_{y=0}^{r} \Phi\bigl(h(x,y;s)\bigr)\,dx\,dy,$$ with

$$h(x,y;s) = s + 2r - \sqrt{r^2 - x^2} - \sqrt{r^2 - y^2},$$

where $r = r_s + r_c$ and $s$ is the control parameter. PDA constructs

$$P_s(h) = \frac{1}{r^2}\int_{0}^{r}\int_{0}^{r}\delta\bigl(h-h(x,y;s)\bigr)\,dx\,dy,$$

and assembles

$M_{ij} = 4r^2 P_{s_i}(h_j)\Delta h,$

to yield the linear system solved for $\Phi(h_j)$. For $N_s = N_h = 1000$ and standard geometric parameters, the RMS error in the reconstructed profile is approximately $10^{-3}$, demonstrating both the accuracy and computational tractability.

\begin{figure}[h] \centering \includegraphics[width=0.8\linewidth]{SFA_reconstruction.pdf} \caption{Reconstructed vs. true pressure profile in the SFA system. The blue curve is the true benchmark $\Phi(h)$; the red dots are $\Phi_{\rm PDA}(h) = [M^{-1}f]_j$ at $h_j$. Parameters: $r_s=50\,$nm, $r_c=10^7\,$nm, $S_a=0$, $S_b=1000\,$nm, $N_s=N_h=1000$. RMS error $\approx10^{-3}$.} \label{fig:SFA_recon} \end{figure}

5. Implementation: Pseudocode, Complexity, and Best Practices

A summary of PDA implementation for general (single or higher-order) integrals is given below:

M = zeros((Ns, Nh)) # Ns x Nh kernel matrix for i in range(Ns): s = s_vals[i] X_samples = sample_domain(N_MC) # e.g., for (x,y) if double h_samples = h(X_samples; s) hist, _ = np.histogram(h_samples, bins=h_bins_edges, density=False) P_s = hist / (hist.sum() * Δh) # PDF over h W_s = total_weight(s) # e.g., ∫u(x)dx M[i,:] = W_s * P_s * Δh_j g_est = np.linalg.solve(M, f_vec) # Or Tikhonov or SVD if ill-conditioned

Key guidelines:

• Mesh sizes ($N_s, N_h$) typically range from 500–2000 for accurate results.
• For Monte Carlo histogramming, $N_{\text{MC}} = 10^4–10^5$ per $s$ is recommended to ensure smooth $P_s$.
• Storage is $O(N_s N_h)$. Building $M$ requires $O(N_s N_{\text{MC}})$ operations.
• Matrix inversion or regularization incurs an additional $O(N_h^3)$ cost, subdominant for moderate $N_h$.
• Condition numbers exceeding $10^8$ indicate regularization or SVD is required.
• For noisy measurements, early stopping in iterative inversion or singular value truncation is advised.

6. Context, Applications, and Significance

PDA was developed in the context of reconstructing pressure and density profiles in surface force and solvation measurements, notably for AFM, OT, and SFA data (Hashidate et al., 11 Jan 2026). In these applications, the physical observables are related via integral equations with geometric transformation $h(\cdot;s)$. PDA’s reduction to 1D kernel inversion for multiple integrals significantly decreases computational and memory requirements. This suggests PDA is particularly suited for surface- and colloidal-science inverse problems where measurement data are naturally parameterized and the physics are encoded in symmetric multi-dimensional domains. For single-order problems, PDA is mathematically equivalent to classical kernel inversion but offers a unified probabilistic interpretation.

A plausible implication is that PDA could extend to inverse problems in other domains with similar structural properties, provided the kernel can be mapped to a 1D histogram. Potential limitations remain for integral equations lacking injective or monotonic kernel mappings, or for sparse/noisy data, where regularization strategies are essential.

7. Summary

The Probability Density Alternation method is a systematic framework to convert multi-dimensional integral-equation inverse problems into tractable linear systems by exploiting a mapping onto probability density functions of the kernel variable. Particularly effective for double and higher-order integrals, PDA simplifies otherwise intractable tensor inversions and enables robust numerical reconstruction of hidden profiles in surface- and colloidal-science applications (Hashidate et al., 11 Jan 2026). For certain geometries and measurement configurations, the method streamlines inverse analysis, yielding high accuracy with moderate computational resources and direct regularization strategies.