arXiv subjects

Luis Espath

Publications and source records attributed to Luis Espath.

At least 19 recordsLinked to original sources

arXiv2026

A Unified Gradient Theory for Frame-Indifferent Rates of Tensorial Internal Variables

We develop a thermodynamically consistent framework for weakly nonlocal continua with tensor-valued internal variables. Let $\boldsymbol{L}=\operatorname{grad}\boldsymbol{v}$, with stretching tensor $\boldsymbol{D}=\operatorname{sym}\boldsymbol{L}$ and spin tensor $\boldsymbol{W}=\operatorname{skw}\boldsymbol{L}$. We introduce the generators $\boldsymbol{\Gamma}_{\alpha}=\boldsymbol{W}+\alpha\boldsymbol{D}$, $\alpha\in\{0,1\}$, which unify corotational and upper-convected transport. The resulting kinematic structure induces canonical frame-indifferent evolutions of both the internal variable and its spatial gradient, thereby providing a closure for gradient-dependent theories. Starting from the balances of linear momentum and microforces, together with an internal power expenditure depending on the internal variable and its gradient, we derive a local free-energy imbalance for incompressible isothermal processes. Under isotropy and inherited symmetry assumptions, this imbalance admits a canonical decomposition into contributions associated with $\boldsymbol{D}$, $\operatorname{grad}\boldsymbol{L}$, the generator-induced rate $\mathfrak{D}_{\alpha}\boldsymbol{J}$, and its gradient $\mathfrak{D}^{\nabla}_{\alpha}(\operatorname{grad}\boldsymbol{J})$. This decomposition yields explicit constitutive restrictions ensuring thermodynamic consistency and identifies the induced higher-order stress contributions arising from gradient dependence. Finally, we construct a coupled gradient theory combining viscoelasticity and constrained orientational order, in which distinct internal variables evolve under different transport mechanisms. The framework extends classical theories with tensorial internal variables, including Oldroyd-B and Landau-de Gennes-type models.

physics.flu-dyn↗

arXiv2026

Compressible fluids with distinct mass and linear-momentum transport

We formulate a thermodynamically consistent continuum theory for compressible, viscous, heat-conducting fluids in which the velocity entering the balance of mass is distinguished from the specific linear momentum entering the balances of linear momentum and energy. Starting from balances of mass, linear momentum, angular momentum, and internal energy, together with a power identity and the Clausius--Duhem inequality, we derive the mechanical and thermodynamic consequences of allowing these fields to differ. From local angular-momentum balance, we show that the Cauchy stress need not be symmetric and we determine its skew part. From the dissipation inequality, we obtain an admissible internal-energy flux and a closure in which the relative transport between mass and linear momentum is proportional to the pressure gradient rather than to the mass-density gradient. We also derive a free-enthalpy imbalance across shocks and a reduced wall dissipation inequality for rigid, impermeable walls undergoing prescribed rigid motion, together with simple admissible wall laws for temperature-controlled and heat-flow-controlled settings. For ideal gases, we write the governing equations in conservative dimensionless form, recover the classical compressible Navier--Stokes--Fourier theory when relative transport vanishes, and identify a distinguished low-Mach regime in which mass transport and linear-momentum transport remain distinct at leading order.

physics.flu-dyn↗

arXiv2025

Inexact Uzawa-Double Deep Ritz Method for Weak Adversarial Neural Networks

Residual minimization in dual norms is central to Weak Adversarial Neural Network (WAN) approaches for solving partial differential equations (PDEs). This framework naturally leads to saddle-point problems whose numerical solutions can be highly unstable depending on the underlying iterative scheme. Motivated by this structure, we propose and analyze the Uzawa Double Deep Ritz Method, a deep PDE solver that integrates neural network approximations with the classical Uzawa iteration. The proposed method is built around two coupled update rules performed at each iteration: a residual update, obtained by minimizing a Ritz functional associated with the dual problem, and a solution update, obtained by minimizing a Ritz functional driven by the current residual. Both variables are represented by neural networks, mirroring the classical Uzawa architecture for saddle-point problems. By replacing the adversarial min-max optimization of WAN with a sequence of Deep Ritz minimization problems, our study theoretically proves that the proposed method acts as an iterative scheme for solving the WAN formulation. Furthermore, we establish a comprehensive convergence theory for an inexact Uzawa scheme where both subproblems are solved approximately. This analysis extends to practical gradient-based implementations, providing rigorous stability and convergence guarantees for both single and multiple-gradient step update strategies. Numerical experiments validate our theoretical findings and demonstrate the robustness of the proposed approach.

math.NA↗

arXiv2025

A Phase-field Model for Apoptotic Cell Death

The process of programmed cell death, namely apoptosis, is a natural mechanism that regulates healthy tissue, multicellular structures, and homeostasis. An improved understanding of apoptosis can significantly enhance our knowledge of biological processes and systems. For instance, pathogens can manipulate the apoptotic process to either evade immune detection or to facilitate their spread. Furthermore, of particular clinical interest is the ability of cancer cells to evade apoptosis, hence allowing them to survive and proliferate uncontrollably. Thus, in this work, we propose a phase-field framework for simulating intrinsic or extrinsic apoptosis induced by an activation field, including deriving the configurational mechanics underlying such phenomena. Along with exploring varying conditions needed to initiate or reduce apoptosis, this can serve as a starting point for computational therapeutic testing. To showcase model capabilities, we present simulations exhibiting different types of cellular dynamics produced when varying the mechanisms underlying apoptosis. The model is subsequently applied to probe different morphological transitions, such as cell shrinkage, membrane blebbing, cavity formation and fragmentation. Lastly, we compare the characteristics observed in our simulations to electron microscopy images, providing additional support for the model.

q-bio.CB↗

arXiv2025

Efficient Stochastic BFGS methods Inspired by Bayesian Principles

Quasi-Newton methods are ubiquitous in deterministic local search due to their efficiency and low computational cost. This class of methods uses the history of gradient evaluations to approximate second-order derivatives. However, only noisy gradient observations are accessible in stochastic optimization; thus, deriving quasi-Newton methods in this setting is challenging. Although most existing quasi-Newton methods for stochastic optimization rely on deterministic equations that are modified to circumvent noise, we propose a new approach inspired by Bayesian inference to assimilate noisy gradient information and derive the stochastic counterparts to standard quasi-Newton methods. We focus on the derivations of stochastic BFGS and L-BFGS, but our methodology can also be employed to derive stochastic analogs of other quasi-Newton methods. The resulting stochastic BFGS (S-BFGS) and stochastic L-BFGS (L-S-BFGS) can effectively learn an inverse Hessian approximation even with small batch sizes. For a problem of dimension $d$, the iteration cost of S-BFGS is $\mathcal{O}(d^2)$, and the cost of L-S-BFGS is $\mathcal{O}(d)$. Numerical experiments with a dimensionality of up to $30,720$ demonstrate the efficiency and robustness of the proposed method.

math.OC↗

arXiv2024

Multilevel randomized quasi-Monte Carlo estimator for nested integration

Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}\boldsymbol{x}\right)\mathrm{d}\boldsymbol{y}$, for nonlinear $f$, making them computationally challenging, particularly in high-dimensional settings. Although widely used for single integrals, traditional Monte Carlo (MC) methods can be inefficient when encountering complexities of nested integration. This work introduces a novel multilevel estimator, combining deterministic and randomized quasi-MC (rQMC) methods to handle nested integration problems efficiently. In this context, the inner number of samples and the discretization accuracy of the inner integrand evaluation constitute the level. We provide a comprehensive theoretical analysis of the estimator, deriving error bounds demonstrating significant reductions in bias and variance compared with standard methods. The proposed estimator is particularly effective in scenarios where the integrand is evaluated approximately, as it adapts to different levels of resolution without compromising precision. We verify the performance of our method via numerical experiments, focusing on estimating the expected information gain of experiments. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds. The results reveal that the proposed multilevel rQMC estimator outperforms existing MC and rQMC approaches, offering a substantial reduction in computational costs and offering a powerful tool for practitioners dealing with complex, nested integration problems across various domains.

math.NA↗

arXiv2023

Laplace-based strategies for Bayesian optimal experimental design with nuisance uncertainty

Finding the optimal design of experiments in the Bayesian setting typically requires estimation and optimization of the expected information gain functional. This functional consists of one outer and one inner integral, separated by the logarithm function applied to the inner integral. When the mathematical model of the experiment contains uncertainty about the parameters of interest and nuisance uncertainty, (i.e., uncertainty about parameters that affect the model but are not themselves of interest to the experimenter), two inner integrals must be estimated. Thus, the already considerable computational effort required to determine good approximations of the expected information gain is increased further. The Laplace approximation has been applied successfully in the context of experimental design in various ways, and we propose two novel estimators featuring the Laplace approximation to alleviate the computational burden of both inner integrals considerably. The first estimator applies Laplace's method followed by a Laplace approximation, introducing a bias. The second estimator uses two Laplace approximations as importance sampling measures for Monte Carlo approximations of the inner integrals. Both estimators use Monte Carlo approximation for the remaining outer integral estimation. We provide four numerical examples demonstrating the applicability and effectiveness of our proposed estimators.

math.NA↗

arXiv2023

A bulk-surface continuum theory for fluid flows and phase segregation with finite surface thickness

In this continuum theory, we propose a mathematical framework to study the mechanical interplay of bulk-surfaces materials undergoing deformation and phase segregation. To this end, we devise a principle of virtual powers with a bulk-surface dynamics, which is postulated on an arbitrary part $\mathcal{P}$ where the boundary $\partial\mathcal{P}$ may lose smoothness, that is, the normal field may be discontinuous at an edge $\partial^2\mathcal{P}$. The final set of equations somewhat resemble the Navier--Stokes--Cahn--Hilliard equation for the bulk and the surface. Aside from the systematical treatment based on a specialized version of the virtual power principle and free-energy imbalances for bulk-surface theories, we consider two additional ingredients: an explicit dependency of the apparent surface density on the surface thickness and mixed boundary conditions for the velocity, chemical potential, and microstructure.

physics.flu-dyn↗

arXiv2023

Scalable method for Bayesian experimental design without integrating over posterior distribution

We address the computational efficiency in solving the A-optimal Bayesian design of experiments problems for which the observational map is based on partial differential equations and, consequently, is computationally expensive to evaluate. A-optimality is a widely used and easy-to-interpret criterion for Bayesian experimental design. This criterion seeks the optimal experimental design by minimizing the expected conditional variance, which is also known as the expected posterior variance. This study presents a novel likelihood-free approach to the A-optimal experimental design that does not require sampling or integrating the Bayesian posterior distribution. The expected conditional variance is obtained via the variance of the conditional expectation using the law of total variance, and we take advantage of the orthogonal projection property to approximate the conditional expectation. We derive an asymptotic error estimation for the proposed estimator of the expected conditional variance and show that the intractability of the posterior distribution does not affect the performance of our approach. We use an artificial neural network (ANN) to approximate the nonlinear conditional expectation in the implementation of our method. We then extend our approach for dealing with the case that the domain of experimental design parameters is continuous by integrating the training process of the ANN into minimizing the expected conditional variance. Through numerical experiments, we demonstrate that our method greatly reduces the number of observation model evaluations compared with widely used importance sampling-based approaches. This reduction is crucial, considering the high computational cost of the observational models. Code is available at https://github.com/vinh-tr-hoang/DOEviaPACE.

math.NA↗

arXiv2023

Double-loop randomized quasi-Monte Carlo estimator for nested integration

Nested integration of the form $\int f\left(\int g(\bs{y},\bs{x})\di{}\bs{x}\right)\di{}\bs{y}$, characterized by an outer integral connected to an inner integral through a nonlinear function $f$, is a challenging problem in various fields, such as engineering and mathematical finance. The available numerical methods for nested integration based on Monte Carlo (MC) methods can be prohibitively expensive owing to the error propagating from the inner to the outer integral. Attempts to enhance the efficiency of these approximations using the quasi-MC (QMC) or randomized QMC (rQMC) method have focused on either the inner or outer integral approximation. This work introduces a novel nested rQMC method that simultaneously addresses the approximation of the inner and outer integrals. The method leverages the unique nested integral structure to offer a more efficient approximation mechanism. As the primary contribution, we derive asymptotic error bounds for the bias and variance of our estimator, along with the regularity conditions under which these bounds can be attained. Incorporating Owen's scrambling techniques, we address integrands exhibiting infinite variation in the Hardy--Krause sense, enabling theoretically sound error estimates. Moreover, we derive a truncation scheme for applications in the context of expected information gain estimation. We verify the estimator quality through numerical experiments by comparing the computational efficiency of the nested rQMC method against standard nested MC estimation to highlight the computational savings and enhanced applicability of the proposed approach.

math.NA↗

arXiv2023

Physics-informed Spectral Learning: the Discrete Helmholtz--Hodge Decomposition

In this work, we further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. \cite{Esp21} based on a discrete $L^2$ projection to solve the discrete Hodge--Helmholtz decomposition from sparse data. Within this physics-informed statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. Moreover, our PiSL computational framework enjoys spectral (exponential) convergence. We regularize the minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the divergence- and curl-free constraints become a finite set of linear algebraic equations. The proposed computational framework combines supervised and unsupervised learning techniques in that we use data concomitantly with the projection onto divergence- and curl-free spaces. We assess the capabilities of our method in various numerical examples including the `Storm of the Century' with satellite data from 1993.

cs.LG↗

arXiv2022

Deep NURBS -- Admissible Physics-informed Neural Networks

In this study, we propose a new numerical scheme for physics-informed neural networks (PINNs) that enables precise and inexpensive solution for partial differential equations (PDEs) in case of arbitrary geometries while strictly enforcing Dirichlet boundary conditions. The proposed approach combines admissible NURBS parametrizations required to define the physical domain and the Dirichlet boundary conditions with a PINN solver. The fundamental boundary conditions are automatically satisfied in this novel Deep NURBS framework. We verified our new approach using two-dimensional elliptic PDEs when considering arbitrary geometries, including non-Lipschitz domains. Compared to the classical PINN solver, the Deep NURBS estimator has a remarkably high convergence rate for all the studied problems. Moreover, a desirable accuracy was realized for most of the studied PDEs using only one hidden layer of neural networks. This novel approach is considered to pave the way for more effective solutions for high-dimensional problems by allowing for more realistic physics-informed statistical learning to solve PDE-based variational problems.

math.NA↗

arXiv2022

Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in stochastic optimization

Using quasi-Newton methods in stochastic optimization is not a trivial task given the difficulty of extracting curvature information from the noisy gradients. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise. We propose a Bayesian approach to obtain a Hessian matrix approximation for stochastic optimization that minimizes the secant equations residue while retaining the extreme eigenvalues between a specified range. Thus, the proposed approach assists stochastic gradient descent to converge to local minima without augmenting gradient noise. We propose maximizing the log posterior using the Newton-CG method. Numerical results on a stochastic quadratic function and an $\ell_2$-regularized logistic regression problem are presented. In all the cases tested, our approach improves the convergence of stochastic gradient descent, compensating for the overhead of solving the log posterior maximization. In particular, pre-conditioning the stochastic gradient with the inverse of our Hessian approximation becomes more advantageous the larger the condition number of the problem is.

math.OC↗

arXiv2022

A continuum framework for phase field with bulk-surface dynamics

This continuum mechanical theory aims at detailing the underlying rational mechanics of dynamic boundary conditions proposed by Fischer, Maass, & Dieterich [1], Goldstein, Miranville, & Schimperna [2], and Knopf, Lam, Liu & Metzger, [3]. As a byproduct, we generalize these theories. These types of dynamic boundary conditions are described by the coupling between the bulk and surface partial differential equations for phase fields. Our point of departure within this continuum framework is the principle of virtual powers postulated on an arbitrary part $\mathcal{P}$ where the boundary $\partial\mathcal{P}$ may lose smoothness. That is, the normal field may be discontinuous along an edge $\partial^2\mathcal{P}$. However, the edges characterizing the discontinuity of the normal field are considered smooth. Our results may be summarized as follows. We provide a generalized version of the principle of virtual powers for the bulk-surface coupling along with a generalized version of the partwise free-energy imbalance. Next, we derive the explicit form of the surface and edge microtractions along with the field equations for the bulk and surface phase fields. The final set of field equations somewhat resembles the Cahn--Hilliard equation for both the bulk and surface. Lastly, we provide a suitable set of constitutive relations and thermodynamically consistent boundary conditions.

math-ph↗

arXiv2021

Small-noise approximation for Bayesian optimal experimental design with nuisance uncertainty

Calculating the expected information gain in optimal Bayesian experimental design typically relies on nested Monte Carlo sampling. When the model also contains nuisance parameters, which are parameters that contribute to the overall uncertainty of the system but are of no interest in the Bayesian design framework, this introduces a second inner loop. We propose and derive a small-noise approximation for this additional inner loop. The computational cost of our method can be further reduced by applying a Laplace approximation to the remaining inner loop. Thus, we present two methods, the small-noise Double-loop Monte Carlo and small-noise Monte Carlo Laplace methods. Moreover, we demonstrate that the total complexity of these two approaches remains comparable to the case without nuisance uncertainty. To assess the efficiency of these methods, we present three examples, and the last example includes the partial differential equation for the electrical impedance tomography experiment for composite laminate materials.

math.NA↗

arXiv2021

A spatio-temporal adaptive phase-field fracture method

We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in an Eulerian-Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical formulation adapts in space and time to bound the errors, solving the mesh-bias issues these models typically suffer. The time-step adaptivity estimates the temporal truncation error of the partial differential equation that governs the solid equilibrium. The second-order generalized-$\alpha$ time-marching scheme evolves the dynamic system. We estimate the temporal error by extrapolating a first-order approximation of the present time-step solution using previous ones with backward difference formulas; the estimate compares the extrapolation with the time-marching solution. We use an adaptive scheme built on a residual minimization formulation in space. We estimate the spatial error by enriching the discretization with elemental bubbles; then, we localize an error indicator norm to guide the mesh refinement as the fracture propagates. The combined space and time adaptivity allows us to use low-order linear elements in problems involving complex stress paths. We efficiently and robustly use low-order spatial discretizations while avoiding mesh bias in structured and unstructured meshes. We demonstrate the method's efficiency with numerical experiments that feature dynamic crack branching, where the capacity of the adaptive space-time scheme is apparent. The adaptive method delivers accurate and reproducible crack paths on meshes with fewer elements.

math-ph↗

arXiv2021

On the equivalence of different adaptive batch size selection strategies for stochastic gradient descent methods

In this study, we demonstrate that the norm test and inner product/orthogonality test presented in \cite{Bol18} are equivalent in terms of the convergence rates associated with Stochastic Gradient Descent (SGD) methods if $\epsilon^2=\theta^2+\nu^2$ with specific choices of $\theta$ and $\nu$. Here, $\epsilon$ controls the relative statistical error of the norm of the gradient while $\theta$ and $\nu$ control the relative statistical error of the gradient in the direction of the gradient and in the direction orthogonal to the gradient, respectively. Furthermore, we demonstrate that the inner product/orthogonality test can be as inexpensive as the norm test in the best case scenario if $\theta$ and $\nu$ are optimally selected, but the inner product/orthogonality test will never be more computationally affordable than the norm test if $\epsilon^2=\theta^2+\nu^2$. Finally, we present two stochastic optimization problems to illustrate our results.

math.OC↗

arXiv2021

Statistical Learning for Fluid Flows: Sparse Fourier divergence-free approximations

We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free (SFdf) approximation based on a discrete $L^2$ projection. Within this physics-informed type of statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. We regularize our minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the incompressibility (divergence-free) constraint becomes a finite set of linear algebraic equations. We couple our spatial approximation with the truncated Singular Value Decomposition (SVD) of the flow measurements for temporal compression. Our computational framework thus combines supervised and unsupervised learning techniques. We assess the capabilities of our method in various numerical examples arising in fluid mechanics.

physics.flu-dyn↗