Welcome to My Home Page

This is my personal webpage where I plan to share my work.

Research Work (and who did it)

• Quantim-Inspired Solver for Simulating Material Deformations: How to solve linear elasticity equations with fully TT-tensorized FEM. All work done by Siddhartha and me.
• SFFT-based Homogenization: Using Tensor Trains to Enhance FFT-Based Homogenization: How to speedup FFT homogenization of material microstructure with Tensor Networks. Most of the work was done by Sascha.
• Piecewise Polynomial Tensor Network Quantum Feature Encoding: How to encode piecewise linear functions as quantum features. All work done by me.
• A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer: How good are IBM quantum computers at solving PDEs (not good). Most of the work done by me.
• On the Ordering of Sites in the Density Matrix Renormalization Group using Quantum Mutual Information: A slightly more rigorous justification for the way sites are ordered in DMRG ab initio quantum chemistry ground state calculations. Solo work.
• Ranks of Tensor Networks for Eigenspace Projections and the Curse of Dimensionality: Can one show that eigenstates of high-dimensional unbounded operators have “small” tensor ranks (the short version)? Solo work.
• Approximation of Smoothness Classes by Deep Rectifier Networks: Any smoothness class can be optimally approximated with ReLU networks. An actual collaboration with Anthony.
• Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functions: Any multivariate smoothness class can be optimally approximated with tensor networks. An actual collaboration with Anthony.
• Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions — Part II: Any univariate smoothness class can be optimally approximated with tensor networks. An actual collaboration with Anthony.
• Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions – Part I: How to define approximation classes for 1D functions with tensor networks. An actual collaboration with Anthony.
• Tensor approximation and adaptivity within model order reduction: Tensor networks and reduced bases for differential equations (my PhD thesis).
• H1-Stability of the L2-Projection onto Finite Element Spaces on Adaptively Refined Quadrilateral Meshes: A result on stability of adaptive finite element methods on quadrilaterals for differential equations. All work done by Anja and me.
• Low-Rank Approximability and Entropy Area Laws for Ground States of Unbounded Hamiltonians: Can one show that eigenstates of high-dimensional unbounded operators have “small” tensor ranks (the extended version)? Solo work.
• Singular Value Decomposition in Sobolev Spaces: Part II: What is the closest thing to an SVD for infinite-dimensional weakly differentiable function (cont’d)? An actual collaboration with Anthony.
• Singular Value Decomposition in Sobolev Spaces: Part I: What is the closest thing to an SVD for infinite-dimensional weakly differentiable function? An actual collaboration with Anthony.
• HT-AWGM: A Hierarchical Tucker-Adaptive Wavelet Galerkin Method for High Dimensional Elliptic Problems: How to solve high-dimensional PDEs with adaptive Galerkin methods and tensor networks. Most of the work done by me.
• Reduced Basis Methods Based Upon Adaptive Snapshot Computations: How to make reduced basis work with adaptive mesh refinements. An actual collaboration with Kristina and Karsten.
• Reduced Basis Exact Error Estimates with Wavelets: How to do reduced basis with adaptively refined wavelet meshes. An actual collaboration with Karsten.

Feedback

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Contact

Checkout my projects on github. Email me at mazen.ali90@gmail.com.