A Minisymposium in Computational Mathematics at the
SIAM Annual Meeting AN25
July 28-August 1, 2025
Montreal, Canada


Advances in Mathematical and Numerical Analysis of Partial Differential Equations for Application-Oriented Computations


ABSTRACT. The latest advances in computational analysis for partial differential equations with realistic applications are the focus in this mini-session. We bring together investigators from various backgrounds to present and discuss theory and methods of application-oriented computations for PDE. The latest machine learning challenges in these fields will also be explored. This mini-session serves as a platform to exchange new ideas, extend academic networks, and incubate future cooperation. Graduate students are encouraged to participate in the special event.


Organizers:


Important Deadlines


Instructions for Submission


Part I Confirmed Speakers

  1. Time: Not specified
    Title: The Efficient Computation of Accurate Continuous Numerical Solutions of Nonlocal Two-Point Boundary Value Problems and Parabolic Problems in One Space Variable.

    Presenter: Graeme Fairweather
    Affiliation: Mathematical Review, American Mathematical Society.
    Co-Author: Paul Muir
    Affiliation: Mathematics and Computer Science, Saint Mary’s University.

    Abstract: Ordinary and partial differential equations containing terms or coefficients that depend on an integral of a function involving one of the solution components are termed nonlocal. We first discuss the reformulation of nonlocal two-point boundary value problems to enable their solution using widely available software packages. Similarly, we consider nonlocal parabolic problems in one-space variable in which an integral of a function of the solution appears in one or more of the boundary conditions. The reformulation then involves replacing such a boundary condition with an ordinary differential equation. For each case, we use a software package that provides an error-controlled continuous numerical solution. Numerical examples are provided to demonstrate the efficacy of our approach and its advantages over numerous existing techniques.

  2. Time: Not specified
    Title: A fourth-order exponential time differencing scheme with dimensional splitting for multidimensional Black-Scholes Equation.
    Presenter: Emmanuel Asante-Asamani
    Affiliation: Clarkson University

    Abstract: An L-stable, fourth-order exponential time differencing (ETD) Runge-Kutta scheme with dimensional splitting is developed to solve nonsmooth, multidimensional reaction-diffusion equations. Our scheme uses an L-acceptable Padé (0,4) rational function to approximate the matrix exponentials in the dimensionally split ETDRK4 scheme. The resulting scheme, ETDRK4P04-IF, is verified empirically to be fourth-order accurate for several RDEs and demonstrated to be more efficient than competing fourth order schemes for solving the multidimensional linear Black-Scholes equation.

  3. Time: Not specified
    Title: Efficient Option Pricing Using Deep Bsde Solvers: A High-Dimensional Approach to the Black-Scholes Model
    Presenter: Guangming Yao (Ibraheem Yahayah, Olaoluwa Ogunleye, Kalani Rubasinghe)
    Affiliation: Clarkson University

    Abstract: In this paper, we apply backward stochastic differential equations (BSDEs) and deep neural networks to solve the Black-Scholes equations for option pricing. Using the Deep BSDE solver introduced by Han et al. (2017), we demonstrate its effectiveness in both low- and high-dimensional settings. This deep learning-based approach reformulates the PDE problem into a BSDE framework, enabling efficient numerical solutions for complex financial derivatives. We also extend our analysis to generalized Black-Scholes models incorporating transaction costs and stochastic volatility. To the best of our knowledge, this study is among the first to systematically apply Deep BSDE solvers to the Black-Scholes framework across multiple dimensions. Numerical experiments highlight key patterns in option pricing and the trade-offs between accuracy and computational efficiency.

  4. Time: Not specified
    Title: Dimensional splitting of a fourth-order exponential time differencing scheme with a real and distinct poles rational approximation for solving nonlinear systems of reaction diffusion equations
    Presenter: Wisdom Attipoe
    Affiliation: Clarkson University

    Abstract: Reaction-diffusion systems are mathematical models that describe the spatiotemporal dynamics of chemical substances as they diffuse and react. Variety of time discretization schemes have been developed to solve the stiff ODE system resulting from the spatial-discretization of the PDE. The quest to develop more efficient and accurate schemes to handle multidimensional, stiff problems with non-smooth data is ever increasing. In this work, we develop a dimensionally split, fourth-order, L-stable, parallelizable exponential time differencing scheme (ETD) by approximating the matrix exponentials in the class of ETD Runge-Kutta schemes with a fourth-order non-Padé rational function having real distinct poles (RDP). A variety of multidimensional, non-linear reaction-diffusion systems having Dirichlet, Neumann boundary conditions are used to empirically validate the order of convergence of the scheme and compare its performance with existing fourth schemes.

Part II Confirmed Speakers

  1. Time: Not specified
    Title: A Numerical Technique for Analyzing Critical Domains in Coupled Quenching Problems
    Presenter: W.Y. Chan
    Affiliation: Texas A&M University, Texarkana

    Abstract: In this talk, we introduce a coupled-parabolic quenching problem within square domains. The size of the critical domains depends on the existence of a steady-state solution. The integral solution is represented in terms of Green’s function and a conformal mapping is employed to transform the square domain into a circle. Our objective is to numerically determine the critical domains associated with this problem.

  2. Time: Not specified
    Title: On a New Symmetry in Splitting Methods
    Presenter: Fernando Casas
    Affiliation: Universitat Jaume I, Castellon

    Abstract: Splitting methods constitute a powerful tool for the numerical integration of differential equations, either arising directly from dynamical systems or from partial differential equations of evolution previously discretized in space. Efficient high-order schemes have been designed that provide accurate solutions whilst preserving some of the most salient qualitative features of the system. The presence of negative coefficients in methods of order greater than two, however, restricts their application to, e.g., equations defined in semigroups, thus motivating the exploration of splitting methods with complex coefficients with positive real part. Among them, we will present and analyze a new class of schemes which exhibit a good long time behavior when applied to linear unitary and integrable linear Hamiltonian systems. Some of the new methods show better efficiency than state-of-the-art splitting methods with real coefficients for several classes of problems. This work is done in collaboration with Joackim Bernier (Nantes), Sergio Blanes (Valencia) and Alejandro Escorihuela-Tom\`as (Castell\'on).

  3. Time: Not specified
    Title: Artificial Neural Networks for the Epidemiological Analysis of Mumps Model Via Mittag-Leffler Kernel
    Presenter: Jerry Y. Shih
    Affiliation: National Chung Hsing University Taiwan

    Abstract: This study develops a fractional-order mathematical model utilizing the fractal fractional Mittag-Leffler operator to analyze Mumps virus transmission. The model undergoes qualitative and quantitative assessments, emphasizing Ulam-Hyers stability, as well as the existence, uniqueness, and well-posedness of solutions, along with an analysis of disease-free equilibrium points and treatment sensitivity. Fixed-point theory ensures the constraints on solutions, while an advanced numerical approach highlights the significance of fractional dynamics. Simulations demonstrate convergent behavior across population classes, accurately capturing Mumps transmission. A comparative study affirms the superiority of fractional-order modeling over integer-order approaches. Synthetic data generated using the Newton two-step solver with the Mittag-Leffler kernel are analyzed through a supervised learning framework utilizing artificial neural networks. The results from the neural network closely correlate with numerical findings, presenting minimal errors validated through convergence metrics and regression outputs. This framework offers a robust methodology for the epidemiological analysis of Mumps virus dynamics.

  4. Time: Not specified
    Title: Novel Machine Learning Investigation to the Mathematical Fractional Model Analysis Breast Cancer and Chemotherapy Heart
    Presenter: Sanaullah Saqib
    Affiliation: National Chung Hsing University Taiwan

    Abstract: The world's second-most common cause of death for women is breast cancer. Cancer can be treated by removing cancer cells surgically, killing them, or stopping them from receiving the signal necessary for cell division. Patients who receive cancer treatment may not always have beneficial effects. Treatment for breast cancer can affect the cardiovascular system. We created a fractional mathematical model. The model is built using a set of fractional differential equations. We used a modified ABC-fractional order to explain the fractional breast cancer model. A population is subdivided into five subsets. Stages include 1 and 2 (A), 3 and 4 (B), disease-free (D), and cardiotoxic (E). We have demonstrated the existence, uniqueness, and positivity of model simulation. We run simulations and use neural networks to provide a graphical comparison study and validate the numerical simulation. The findings also show how various model parameters affect the system, offering greater insight and a more effective solution for real-world issues. The efficacy, stability, precision, dependability, and relevance of the proposed approach are confirmed by the error distribution using histograms, obtaining low MSE,RMSE, MAE,MAPE,NSE, absolute error and linear regression outputs for the breast cancer model. We conclude with computational simulations that allow us to visualize our theoretical results.

Part III Confirmed Speakers

  1. Time: Not specified
    Title: An Rbf-Etd Approach for High-Dimensional Black-Scholes Equations with Non-Smooth Data.
    Presenter: Ibraheem Yahayah
    Affiliation: Clarkson University

    Abstract: The Black-Scholes equation is a bedrock in financial mathematics, widely used for pricing options and derivatives. However, its application faces major challenges, like non-smooth initial conditions, high dimensionality in multi-asset scenarios, and nonlinearities arising from factors like transaction costs or volatility dependent on option price gradients. To address these complexities, we propose an RBF-ETD scheme that combines Radial Basis Functions (RBF) for spatial approximation with Exponential Time Differencing (ETD) for efficient time integration. This approach leverages the flexibility of RBFs in solving high-dimensional PDEs and ETD's ability to handle nonlinearities and non-smooth data without iterative solvers. We validate the method by comparing it with established schemes such as RBF with Backward Euler and RBF with Crank Nicolson to examine its comparative accuracy, stability, and computational efficiency. The proposed RBF-ETD scheme promises a robust tool for solving complex Black-Scholes models in real-world financial applications.

  2. Time: Not specified
    Title: Dubious No More: Action of the Matrix Exponential and $\varphi$-functions via Differential Equations
    Presenter: Siqi Wei
    Affiliation: University of Saskatchewan

    Abstract: The computation of these $\varphi$ functions contributes to the main cost of an exponential integrator. Accordingly, matrix-free strategies are popular to compute the $\varphi$ functions of matrices that are large and sparse. Methods such Krylov subspaces methods have been developed for computing the linear combination of $\varphi$ functions. However, their scalability in parallel computing is not ideal. In this talk, we explore the use of embedded Runge--Kutta methods to compute the linear combination of the $\varphi$ functions. This method allows us to exploit the special structure of the ODE to achieve an efficient evaluation of the $\varphi$ functions and provides better strong scalability compared with Krylov subspace methods.

  3. Time: Not specified
    Title: Nonlinear Spike-and-Slab Sparse Coding for Intepretable Image Encoding
    Presenter: Jacquelyn Shelton
    Affiliation: Hong Kong Polytechnic University

    Abstract: Sparse coding is a popular approach to model natural images but has faced two main modelling challenges: low-level image components (e.g. edge-like structures and their occlusions) and varying pixel intensities. Traditionally, images are modelled as a sparse linear superposition of dictionary elements, where the probabilistic view of this problem is the coefficients follow a Laplace or Cauchy distribution. We present a novel probabilistic generative model, instead with spike-and-slab prior and nonlinear combination of components. The prior enables the represention of exact zeros for eg the absence of an image component, eg an edge, and a distribution over non-zero pixel intensities. The max in the data likelihood targets occlusions, ie dictionary elements correspond to image components that can occlude each other. Because these model modifications result in highly multimodal posterior distributions, parameter optimization is analytically and computationally intractable. Thus we design a Markov Chain Monte Carlo sampling approach based on the exact form of the posterior by observing that the data likelihood with the nonlinear max can be written as right and left piecewise functions, which we can apply to higher dimensional data using latent variable preselection. Results on natural and artificial occlusion-rich data with known sparse structure show our approach can successfully learn ground truth model parameters and approximate/characterize the data generation process.


Contact

Email address: bruce.wade@louisiana.edu

UL Lafayette:
Dr. Bruce Wade, Professor & Department Head
215 Maxim Doucet Hall
337.482.5173
bruce.wade@louisiana.edu

Mailing address:
Department of Mathematics
P.O. Box 43568
University of Louisiana at Lafayette
Lafayette, LA 70504-3568

University of Louisiana at Lafayette Logo