MatLogica | Industrial Mathematics

Industrial Mathematics

Accelerate Scientific Computing with AAD

Speed up PDEs, optimization methods, and regressions by 6-100x using state-of-the-art automatic adjoint differentiation. With MatLogica, you save development time and reduce execution costs for large-scale scientific applications.

Focus on Science, Not Optimization

Our software allows scientists and engineers to focus on mathematical problems rather than performance optimization, achieving better performance and calculating adjoints automatically without manual derivative coding.

We can improve the performance of virtually any industrial mathematics application including:

  • Geophysics: Seismic inversion, full waveform inversion (FWI)
  • Weather & Climate: Data assimilation, forecasting models
  • Fluid Dynamics: Shape optimization, aerodynamic design
  • Medical Imaging: Electric impedance tomography, MRI reconstruction
  • Electromagnetics: Antenna design, scattering problems
  • Material Science: Topology optimization, inverse problems

Key Benefits

  • 6-100x faster than manual adjoint implementation
  • Automatic gradient computation - no hand-coding required
  • Large-scale PDEs: Millions of degrees of freedom
  • Reduced development time: Focus on physics, not derivatives
  • Lower compute costs: Faster optimization convergence

Gradient-Based Optimization

Automatically calculate the full Jacobian at a fraction of the usual computational cost. Essential for inverse problems, parameter estimation, and large-scale optimization in scientific computing.

Large-Scale PDE Sensitivity

MatLogica's AADC processes extremely large finite difference PDE schemes efficiently. It executes back-propagation quickly and flawlessly, computing sensitivities of PDE solutions relative to coefficients, boundary conditions, or initial data.

Automatic Adjoint Generation

No manual derivative coding required. AADC automatically generates efficient adjoint code for your PDE solvers, optimization routines, and regression models.

Applications Across Scientific Domains

Geophysics & Seismology

  • Full Waveform Inversion (FWI): Compute gradients for velocity model updates
  • Seismic Imaging: Sensitivity analysis for subsurface parameters
  • Reservoir Simulation: History matching and parameter estimation
  • Gravity/Magnetic Inversion: Fast gradient computation for geophysical inversions

Computational Fluid Dynamics

  • Shape Optimization: Aerodynamic design with gradient-based methods
  • Adjoint Flow Solvers: Navier-Stokes sensitivity analysis
  • Heat Transfer: Thermal optimization and inverse problems
  • Multiphase Flow: Parameter identification in complex flows

Weather & Climate Modeling

  • 4D-Var Data Assimilation: Efficient gradient computation for atmospheric models
  • Weather Forecasting: Adjoint sensitivity for initial conditions
  • Ocean Modeling: Parameter estimation in circulation models
  • Climate Sensitivity: Compute sensitivities to forcing parameters

Medical & Biomedical

  • Electric Impedance Tomography (EIT): Fast inverse problem solution
  • MRI Reconstruction: Gradient-based image optimization
  • Diffusion Imaging: Parameter estimation in tissue models
  • Biomechanics: Material property identification

Electromagnetics

  • Antenna Design: Shape optimization using adjoint methods
  • Scattering Problems: Inverse electromagnetic problems
  • Photonics: Device optimization with Maxwell solvers
  • Radar Imaging: Gradient-based reconstruction

Materials & Structures

  • Topology Optimization: Structural design with gradients
  • Material Identification: Inverse problems in mechanics
  • Composite Design: Multi-scale optimization
  • Fracture Mechanics: Parameter sensitivity analysis

Why AADC for Scientific Computing?

Computational Efficiency

Forward mode vs. Reverse mode (Adjoint): For problems with many inputs and few outputs (typical in inverse problems), the adjoint method computes all gradients in approximately the same time as one forward evaluation. AADC automates this process.

Accuracy

Machine precision derivatives: Unlike finite differences which suffer from truncation and round-off errors, AAD provides derivatives accurate to machine precision, crucial for ill-conditioned inverse problems.

Scalability

Handle massive grids: AADC efficiently processes PDEs with millions of degrees of freedom, making it suitable for 3D simulations and high-resolution models common in industrial applications.

Development Speed

Eliminate manual coding: Implementing adjoint methods manually can take months. AADC generates optimized adjoint code automatically, letting scientists focus on physics and mathematics.

Typical Performance Improvements

6-100x

Faster than manual implementations or finite differences

~1x

Adjoint cost vs. forward evaluation (theoretical ~2x)

106+

Degrees of freedom handled efficiently

Integration with Scientific Software

Compatible with:

  • Languages: C++, Fortran (via C++ wrappers), Python
  • PDE Frameworks: FEniCS, deal.II, custom finite difference/finite element codes
  • Optimization Libraries: IPOPT, NLopt, custom gradient-based optimizers
  • Linear Solvers: PETSc, Trilinos, Eigen, custom sparse solvers
  • HPC Environments: MPI parallel codes, GPU offloading (future)

Comparison: Gradient Computation Methods

Method Accuracy Cost (N inputs) Development Time
Finite Differences Low (truncation errors) N × forward cost Low
Manual Adjoint High ~2 × forward cost Very High (months)
AADC (Automatic) High (machine precision) ~2 × forward cost Low (days)

For inverse problems with N >> 1, automatic adjoint is the only practical approach.

Accelerate Your Scientific Computing

Let us benchmark AADC on your PDE solver or optimization problem

Schedule a Consultation

info@matlogica.com

Related topics: adjoint method PDEs, automatic differentiation scientific computing, full waveform inversion gradient, seismic inversion AAD, weather model data assimilation, computational fluid dynamics adjoint, shape optimization automatic differentiation, electric impedance tomography inverse problem, Jacobian calculation optimization, finite difference PDE sensitivity, gradient-based inverse problems, PDE parameter estimation, topology optimization AAD