At the heart of all quasi-Newton methods is an update rule that enables us to gradually improve the Hessian approximation using the already available gradient evaluations. Theoretical results show that the global performance of optimization algorithms can be improved with higher-order derivatives. This motivates an investigation of generalizations of quasi-Newton update rules to obtain for example third derivatives (which are tensors) from Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments.

Coupled differential equations generally present an important algebraic structure.
For example in the incompressible Navier-Stokes equations, the velocity is affected only by the selenoidal part of the applied force.
This structure can be translated naturally by the notion of complex.
One idea is then to exploit this complex structure at the discrete level in the creation of numerical methods.

The goal of the presentation is to expose the notion of complex by motivating its uses. 
We will present in more detail the creation of a scheme for the Navier-Stokes equations and study its properties.
 

The recent advancements in additive manufacturing have enabled the creation of lattice structures with intricate small-scale details. This has led to the need for new techniques in the field of topology optimization that can handle a vast number of design variables. Despite the efforts to develop multi-scale topology optimization techniques, their high computational cost has limited their application. To overcome this challenge, a new technique called dehomogenization has shown promising results in terms of performance and computational efficiency for optimizing compliance problems.

In this talk, we extend the application of the dehomogenization method to stress minimization problems, which are crucial in structural design. The method involves homogenizing the macroscopic response of a proposed family of microstructures. Next, the macroscopic structure is optimized using gradient-based methods while orienting the cells according to the principal stress components. The final step involves dehomogenization of the structure. The proposed methodology also considers singularities in the orientation field by incorporating singular functions in the dehomogenization process. The validity of the methodology is demonstrated through several numerical examples.

Sketch-and-precondition techniques are popular for solving large overdetermined least squares (LS) problems. This is when a right preconditioner is constructed from a randomized 'sketch' of the matrix. In this talk, we will see that the sketch-and-precondition technique is not numerically stable for ill-conditioned LS problems. We propose a modifciation: using an unpreconditioned iterative LS solver on the preconditioned matrix. Provided the condition number of A is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a comparable backward error to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to provide a convincing argument that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems.