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Akwum Onwunta

Publications and source records attributed to Akwum Onwunta.

18 recordsLinked to original sources
arXiv2026

Supervised Deep Multimodal Matrix Factorization for Interpretable Brain Network Analysis

We present Supervised Deep Multimodal Matrix Factorization (SD3MF), an interpretable framework for integrative brain network analysis that generalizes Symmetric Nonnegative Matrix Tri-Factorization (SNMTF) from unsupervised single-graph clustering to supervised prediction over populations of multimodal graphs. SD3MF learns deep hierarchical factorizations for each modality together with a shared latent representation that aligns subjects across views. An encoder-decoder formulation jointly optimizes graph reconstruction and supervised prediction, while adaptive weights enable data-driven multimodal fusion. By representing each subject through community-level interaction matrices, the model yields interpretable and discriminative features. Experiments on multimodal connectome datasets show that SD3MF consistently outperforms strong deep learning baselines such as CNNs and GNNs, while enabling biologically interpretable insights. Code for reproducibility is available at: https://github.com/amjadseyedi/SD3MF.

cs.LG↗
arXiv2025

Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization

We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element discretization, stochastic Galerkin projection, and advanced time-discretization schemes, the approach addresses challenges of scaling large and ill-conditioned linear systems arising in uncertainty quantification. By exploiting sparsity of linear systems in stochastic Galerkin method, we formulate hierarchical preconditioners based on truncated stochastic expansion that strike an effective balance between computational cost and preconditioning quality. Numerical experiments demonstrate that the proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods, providing robust and efficient solvers for both steady-state and time-dependent optimal control problems under uncertainty.

math.OC↗
arXiv2025

Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they provide a flexible and convenient way to encode priors and build efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., for plug-and-play methods for learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferential of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton-Jacobi partial differential equations (HJ PDEs) to develop novel deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present several numerical results that demonstrate the efficiency of the proposed method in high dimensions.

math.NA↗
arXiv2025

Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations

Linear dissipative differential equation is a fundamental model for a large number of physical systems, such as quantum dynamics with non-Hermitian Hamiltonian, open quantum system dynamics, diffusion process and damped system. In this work, we propose efficient quantum algorithms for simulating linear dissipative differential equations. The key idea of our algorithms is to perform the simulation only over an effective time period when the dynamics has not significantly dissipated yet, rather than over the entire physical evolution period. We conduct detailed analysis on the complexity of our algorithms and show that, while maintaining low state preparation cost, our algorithms can completely eliminate the time dependence. This is a more than exponential improvement compared to the previous state-of-the-art quantum algorithms.

quant-ph↗
arXiv2025

Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces

The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the standard potential as well as two extreme settings: Potentials with quadratic convex part (uniform positive curvature), and potentials which are concave between the potential wells and either linear or infinite outside (highly concentrated curvature). In all three scenarios, the 'effective time step size' of the scheme scales with the square of the small parameter $\varepsilon$ governing the width of transition layers. A weaker 'slow motion' result is proved under much more general assumptions. Thus, stability is achieved by effectively 'freezing' the interfaces in place. The time step limitation is not geometric in origin, but depends on the phase-field parameter $\varepsilon$. Along the way, we establish a new link between an Allen-Cahn type equation and a thresholding approximation of mean curvature flow.

math.NA↗
arXiv2024

Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations. We obtain the singular limit of the evolution equations as the length parameter of the phase fields tends to zero by formal expansions and numerically confirm its validity for circles in two dimensions. Our analysis is complemented by numerical experiments for planar curves, surfaces in three-dimensional space, and semi-supervised learning tasks on graphs.

math.AP↗
arXiv2024

Tensor train solution to uncertain optimization problems with shared sparsity penalty

We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of dimensionality we use tensor product approximations. To handle the non-smoothness of the objective function we employ a smoothed version of the shared sparsity objective. We consider both a benchmark elliptic PDE constraint, and a more realistic topology optimization problem in engineering. We demonstrate that the error converges linearly in iterations and the smoothing parameter, and faster than algebraically in the number of degrees of freedom, consisting of the number of quadrature points in one variable and tensor ranks. Moreover, in the topology optimization problem, the smoothed shared sparsity penalty actually reduces the tensor ranks compared to the unpenalised solution. This enables us to find a sparse high-resolution design under a high-dimensional uncertainty.

math.OC↗
arXiv2024

Fast-forwarding quantum algorithms for linear dissipative differential equations

We establish improved complexity estimates of quantum algorithms for linear dissipative ordinary differential equations (ODEs) and show that the time dependence can be fast-forwarded to be sub-linear. Specifically, we show that a quantum algorithm based on truncated Dyson series can prepare history states of dissipative ODEs up to time $T$ with cost $\widetilde{\mathcal{O}}(\log(T) (\log(1/\epsilon))^2 )$, which is an exponential speedup over the best previous result. For final state preparation at time $T$, we show that its complexity is $\widetilde{\mathcal{O}}(\sqrt{T} (\log(1/\epsilon))^2 )$, achieving a polynomial speedup in $T$. We also analyze the complexity of simpler lower-order quantum algorithms, such as the forward Euler method and the trapezoidal rule, and find that even lower-order methods can still achieve $\widetilde{\mathcal{O}}(\sqrt{T})$ cost with respect to time $T$ for preparing final states of dissipative ODEs. As applications, we show that quantum algorithms can simulate dissipative non-Hermitian quantum dynamics and heat processes with fast-forwarded complexity sub-linear in time.

quant-ph↗
arXiv2023

Deep Nonnegative Matrix Factorization with Beta Divergences

Deep Nonnegative Matrix Factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales. However, all existing deep NMF models and algorithms have primarily centered their evaluation on the least squares error, which may not be the most appropriate metric for assessing the quality of approximations on diverse datasets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that $\beta$-divergences offer a more suitable alternative. In this paper, we develop new models and algorithms for deep NMF using some $\beta$-divergences, with a focus on the Kullback-Leibler divergence. Subsequently, we apply these techniques to the extraction of facial features, the identification of topics within document collections, and the identification of materials within hyperspectral images.

cs.LG↗
arXiv2023

Complexity analysis of regularization methods for implicitly constrained least squares

Optimization problems constrained by partial differential equations (PDEs) naturally arise in scientific computing, as those constraints often model physical systems or the simulation thereof. In an implicitly constrained approach, the constraints are incorporated into the objective through a reduced formulation. To this end, a numerical procedure is typically applied to solve the constraint system, and efficient numerical routines with quantifiable cost have long been developed for that purpose. Meanwhile, the field of complexity in optimization, that estimates the cost of an optimization algorithm, has received significant attention in the literature, with most of the focus being on unconstrained or explicitly constrained problems. In this paper, we analyze an algorithmic framework based on quadratic regularization for implicitly constrained nonlinear least squares. By leveraging adjoint formulations, we can quantify the worst-case cost of our method to reach an approximate stationary point of the optimization problem. Our definition of such points exploits the least-squares structure of the objective, and provides new complexity insights even in the unconstrained setting. Numerical experiments conducted on PDE-constrained optimization problems demonstrate the efficiency of the proposed framework.

math.OC↗
arXiv2023

Smoothed Moreau-Yosida Tensor Train Approximation of State-constrained Optimization Problems under Uncertainty

We propose an algorithm to solve optimization problems constrained by partial (ordinary) differential equations under uncertainty, with almost sure constraints on the state variable. To alleviate the computational burden of high-dimensional random variables, we approximate all random fields by the tensor-train decomposition. To enable efficient tensor-train approximation of the state constraints, the latter are handled using the Moreau-Yosida penalty, with an additional smoothing of the positive part (plus/ReLU) function by a softplus function. In a special case of a quadratic cost minimization constrained by linear elliptic partial differential equations, and some additional constraint qualification, we prove strong convergence of the regularized solution to the optimal control. This result also proposes a practical recipe for selecting the smoothing parameter as a function of the penalty parameter. We develop a second order Newton type method with a fast matrix-free action of the approximate Hessian to solve the smoothed Moreau-Yosida problem. This algorithm is tested on benchmark elliptic problems with random coefficients, optimization problems constrained by random elliptic variational inequalities, and a real-world epidemiological model with 20 random variables. These examples demonstrate mild (at most polynomial) scaling with respect to the dimension and regularization parameters.

math.OC↗
arXiv2022

TTRISK: Tensor Train Decomposition Algorithm for Risk Averse Optimization

This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the KKT system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large-scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.

math.NA↗
arXiv2021

Optimal Control, Numerics, and Applications of Fractional PDEs

This article provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.

math.OC↗
arXiv2021

Novel Deep neural networks for solving Bayesian statistical inverse

We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require several thousands of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates and illustrate the efficiency of our approach via several numerical examples.

math.NA↗
arXiv2019

Reduced-order modeling for nonlinear Bayesian statistical inverse problems

Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally challenging because one would then need to solve the forward problem at every sample point. In this paper, the use of the discrete empirical interpolation method (DEIM) is considered for the forward solves within an MCMC routine. A preconditioning strategy for the DEIM model is also proposed to accelerate the forward solves. The preconditioned DEIM model is applied to a finite difference discretization of a nonlinear PDE in the MCMC model. Numerical experiments show that this approach yields accurate forward results. Moreover, the computational cost of solving the associated statistical inverse problem is reduced by more than 70%.

math.NA↗
arXiv2017

An inexact Newton-Krylov method for stochastic eigenvalue problems

This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high dimensional systems with tensor product structure when discretized with the stochastic Galerkin method. Here, we exploit this inherent tensor product structure to develop a globalized low-rank inexact Newton method with which we tackle the stochastic eigenproblem. We illustrate the effectiveness of our solver with numerical experiments.

math.NA↗
arXiv2017

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

math.NA↗