Part 3 — Randomized Algo and Spectral Decomposition for High-Dimensional Fractional Laplacians

In the ambit of mathematical and computational sciences, solving ultra high-dimensional partial differential equations (PDEs) has always been a formidable challenge. Traditional spectral decomposition methods falter under the sheer weight of computational requirements as dimensions soar. This is where the ingenuity of randomized algorithms comes into play. These sophisticated tools are reshaping our approach to the spectral decomposition of PDEs in high-dimensional spaces.

In this blog, I explore the theoretical possibilities of dimensionality reduction of high dimensional PDEs using randomized algorithms and spectral decomposition. This a continuation of the following:

Part 1 — Fractional Elliptic Problems and Artificial General Intelligence

Part 2 — Fractional Laplacian and Cognitive Modeling

Clarifying the Differences Between Randomized Algorithms and Algorithms Using Random Inputs

1. Randomized Algorithms: These are algorithms that make decisions at some points during their execution based on random choices. The randomness is intrinsic to the algorithm’s strategy and is used to achieve performance gains in terms of speed, simplicity, or both. The solutions provided by randomized algorithms are typically approximate rather than exact, but they offer advantages in terms of computational cost, particularly when dealing with complex or intractable problems. They are widely used in scenarios like hashing (randomized hash functions) and machine learning models.
2. Algorithms Using Random Inputs: These are typically used in probabilistic analysis or simulations where the algorithm itself is deterministic, but the inputs are sampled from a random distribution. The goal here is to understand the behavior or performance of the algorithm under a variety of conditions, which are modeled by the random inputs. For instance, Monte Carlo simulations use random inputs to model complex systems and predict their behavior over time. The algorithm processes these inputs deterministically, and the randomness does not guide the algorithmic logic itself but rather influences the variety of input scenarios.
3. Dimensionality Reduction: While randomized algorithms can be used for dimensionality reduction (e.g., random projection), it’s not exclusive or primary to their application. Randomized techniques in dimensionality reduction specifically help in dealing with high-dimensional data by reducing its volume while trying to preserve important traits such as distances or data variances.

Thus, the primary distinction lies in how randomness is utilized: randomized algorithms use randomness as an integral part of their operational logic to simplify computation and enhance performance, whereas algorithms that take random numbers as inputs are generally deterministic algorithms that are analyzed under random conditions to assess their behavior or robustness.

The Complexity of Ultra High-Dimensional PDEs

Traditional spectral methods, which involve eigenvalue and eigenvector calculations, become increasingly intractable as the problem’s dimensionality climbs into the thousands. The matrices from discretizations of PDEs in such dimensions are typically large, dense, and computationally intensive to handle.

The Spectral Decomposition of the Fractional Laplacian

For a function u defined in an appropriate Sobolev space, the fractional Laplacian can be described spectrally in 𝑅𝑑 as follows:

where ⟨⋅,⋅⟩ denotes the inner product in L2(Rd). Each eigenfunction ϕk​ corresponds to an eigenvalue λk​, and this series expansion forms the basis for numerically solving PDEs involving fractional derivatives.

Advancing into Fractional Sobolev Spaces

The domain of fractional Sobolev spaces Ws,p(Rd) is where the solutions to fractional PDEs exhibit their fractional smoothness. These spaces are defined by the following norm:

where [u]Ws,p(Rd)​ is the Gagliardo seminorm:

This seminorm is crucial for measuring the fractional rate of change of the function u over the domain.

The Role of Randomized Algorithms in Spectral Decomposition: A Mathematical Exposition

Randomized algorithms serve as a cornerstone for the spectral analysis of high-dimensional operators, particularly in the context of PDEs where direct computation is infeasible. These algorithms harness randomness to efficiently approximate eigenvalues and eigenvectors, which are central to understanding the behavior of PDEs in complex domains.

Randomized Estimation of Eigenvalues and Eigenvectors

The eigenvalues (λk​) and eigenvectors (vk​) of a large d×d matrix A corresponding to the discretized PDE operator can be approximated using a lower-dimensional projection. This projection is achieved by employing a random matrix Ω∈R(d×k), with k≪d, whose columns are random vectors drawn from a probability distribution tailored to the problem’s geometry, such as the Gaussian distribution. The steps are as follows:

Random Sampling:

Generate the random sampling matrix Ω whose columns ωi​ are random vectors:

Range Approximation:

Construct a matrix Y which captures the action of A on the range of Ω:

Orthogonalization:

Perform QR decomposition on Y to obtain an orthonormal basis Q:

where Q∈R(d×k) has orthonormal columns.

Eigenvalue Estimation:

Form the matrix T from the orthonormal basis Q:

and solve the smaller eigenproblem

where Σ is a diagonal matrix with the approximate eigenvalues λ~k​ and V^ contains the corresponding eigenvectors. The eigenvalues of A are then approximated by λ~k​, and eigenvectors by QV^.

The Rayleigh quotient for an approximation v~ of an eigenvector is given by:

which provides an approximation λ~ for the associated eigenvalue.

Fractional Sobolev Norm Estimation

The fractional Sobolev norm is defined on a function u by the Ws,p norm and involves a Gagliardo seminorm term. Randomized algorithms facilitate its computation by approximating the integrals that constitute the Gagliardo seminorm.

For a given function u, the Ws,p(Rd) norm is:

with the second term constituting the Gagliardo seminorm. Randomized approaches estimate this term through Monte Carlo integration, where samples (xi​,yi​) are drawn from a distribution over Rd×Rd, giving an estimated seminorm [u]~​Ws,p​:

where N is the number of sampled points, and xi​,yi​ are sampled point pairs.

Computational Efficiency in High Dimensions

The implementation of these methods dramatically reduces computational complexity from the intractable O(d^3) to a feasible O(dk^2), assuming k is chosen judiciously relative to d and the desired approximation accuracy.

Applications and Implications

The use of randomized algorithms in the spectral decomposition of PDEs has profound implications across multiple fields, from quantum mechanics to financial modeling. In each case, the algorithms open the door to solving problems that were previously intractable due to high dimensionality.

Future Directions and Open Discussions

The field of randomized algorithms in spectral decomposition is rich with research opportunities. Future work might focus on adaptive sampling techniques, the development of error estimation metrics, and hybrid algorithms that combine deterministic and stochastic methods.

Randomized algorithms have fundamentally altered the landscape of spectral decomposition in high-dimensional PDEs. They have turned the impossible into the possible, allowing us to tackle problems of a scale and complexity that were once beyond our computational reach. By continuing to refine these techniques and exploring new applications, we pave the way for exciting advancements in mathematics, science, and engineering.

This blog is but a foray into the intricate world of randomized algorithms and spectral methods, and there is much more to be explored and understood. I invite further discussion and collaboration to deepen our collective knowledge and push the boundaries of what we can achieve.

Disclaimer

Freedom Preetham is an AI Researcher with background in math and quantum physics and working on genomics in particular. You are free to use and expand on this research idea as applicable to other domains. Attribution to Freedom Preetham is welcome if you find it useful.