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Distributed-order PDEs are tractable mathematical models for complex multiscaling anomalous transport, where derivative orders are distributed over a range of values. We develop a fast and stable Petrov-Galerkin spectral method for such…
A finite-element method dependant adjoint-based procedure to determine the temperature field of structures based on measured displacements or strains and a set of standard loads is developed and tested. Given a series of force and…
A fast inverse heat conduction model (IHCM) is developed for estimating unknown properties of multi-layer composites considering internal heat generation. This work builds on the validated analytical forward models presented in Part I.…
Thermodynamics (in concert with its sister discipline, statistical physics) can be regarded as a data reduction scheme based on partitioning a total system into a subsystem and a bath that weakly interact with each other. The ubiquity and…
Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We…
We consider the standard thermodynamic processes with constraints, but with additional uncertainty about the control parameters. Motivated by inductive reasoning, we assign prior distribution that provides a rational guess about likely…
An equation based reduced order model applicable to generalized heat equation and thermal simulations of power electronics systems developed in commercial CFD tools, is presented in this work. The model considers the physics of heat…
Differential equations are commonly used to model dynamical deterministic systems in applications. When statistical parameter estimation is required to calibrate theoretical models to data, classical statistical estimators are often…
We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions,…
A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based…
Mode-based model-reduction is used to reduce the degrees of freedom of high dimensional systems, often by describing the system state by a linear combination of spatial modes. Transport dominated phenomena, ubiquitous in technical and…
In this paper, we resolve several long standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by…
Parametric partial differential equations (PDEs) serve as fundamental mathematical tools for modeling complex physical phenomena, yet repeated high-fidelity numerical simulations across parameter spaces remain computationally prohibitive.…
The Projected Gradient Descent (PGD) algorithm is a widely used and efficient first-order method for solving constrained optimization problems due to its simplicity and scalability in large design spaces. Building on recent advancements in…
Modal analysis has long been consolidated as a basic tool to interpret dynamics and build low-order models of mechanical, thermal, and fluid systems. Eigenmodes arising from the spectral decomposition of the underlying linearized dynamics…
Accurate reconstruction of ambient temperature at death scenes is crucial for estimating the postmortem interval (PMI) in forensic science. Typically, this is done by correcting weather station temperatures using measurements from the…
The aim of this paper is to establish the framework of the enclosure method for some class of inverse problems whose governing equations are given by parabolic equations with discontinuous coefficients. The framework is given by considering…
The assessment of the thermal properties of walls is essential for accurate building energy simulations that are needed to make effective energy-saving policies. These properties are usually investigated through in-situ measurements of…
We develop an optimization-based approach to the problem of reconstructing temperature-dependent material properties in complex thermo-fluid systems described by the equations for the conservation of mass, momentum and energy. Our goal is…
We propose a parameter-minimal neural architecture for solving differential equations by restricting the hypothesis class to Horner-factorized polynomials, yielding an implicit, differentiable trial solution with only a small set of…