Dimitris

Mechanical engineer, co-founder of Delta Pi Systems

[Dimitris @DeltaPiSystems]

In mixed FEM for incompressible Stokes flow, velocity and pressure spaces must satisfy the discrete inf–sup (LBB) condition. Equal-order P1/P1 elements generally require stabilization; Taylor–Hood P2/P1 is a standard stable choice on suitable meshes.

[Howard Marks @hmarksofficial]

My latest memo assembles the critical questions one needs to consider to make intelligent, long-term investments in AI. While they can’t be answered definitively, investors mustn't let uncertainty result in inaction.

You can read it here: https://t.co/pbR1WZ0Qmx

[Dimitris @DeltaPiSystems]

Archimedes proved that the region between a parabola and a straight chord has 4/3 the area of an inscribed triangle with the same base and height. His proof used ever-smaller triangles and the method of exhaustion. #MathHistory

[Dimitris @DeltaPiSystems]

Will your feedback loop be stable once you close it?

Our new article explains the Nyquist criterion step by step, with an interactive tool: enter L(s), see the Nyquist plot, the margins and the verdict.

https://t.co/Myq5NlLLnz

#ControlEngineering

[Dimitris @DeltaPiSystems]

In trust-region optimization, ρₖ = [f(xₖ)−f(xₖ+pₖ)]/[mₖ(0)−mₖ(pₖ)] compares actual vs predicted reduction, assuming a positive denominator. Small ρₖ indicates a poor local model: reject the step and shrink the radius. Large ρₖ supports acceptance.

[Dimitris @DeltaPiSystems]

@TheUnjournaling Popper's falsifiability principle isn't itself falsifiable, but that's fine: it's a methodological proposal, not a scientific claim. Unfalsifiable ≠ meaningless, just non-scientific, so it doesn't refute itself. Its weak spot: science doesn't always work that way (Kuhn, Lakatos)

[Dimitris @DeltaPiSystems]

In explicit structural FEM, mass lumping replaces the consistent mass matrix with a diagonal one, avoiding a linear solve for accelerations. For undamped linear dynamics integrated by central differences, stability requires Δt ≤ 2/ω_max, the highest discrete natural frequency.

[Dimitris @DeltaPiSystems]

BFGS preserves a positive-definite inverse-Hessian approximation when yᵀs > 0, where s = xₖ₊₁ − xₖ and y = ∇f(xₖ₊₁) − ∇f(xₖ). For a descent direction, Wolfe line-search conditions guarantee this curvature inequality—even for nonconvex objectives.

[Dimitris @DeltaPiSystems]

For semidiscrete FEM diffusion, M du/dt + Ku = 0 (M symmetric positive definite, K positive semidefinite), backward Euler is stable for any Δt > 0. Yet large time steps can smear transients: unconditional stability does not imply unconditional accuracy.

[Dimitris @DeltaPiSystems]

In equality-constrained optimization, the KKT system couples primal and dual variables:
[H Aᵀ; A 0][p; λ]=−[∇f; c].
Its saddle-point structure is indefinite, so plain CG is unsuitable; MINRES, SQMR, or block preconditioners are common alternatives.

[Dimitris @DeltaPiSystems]

In Poisson FEM, Nitsche’s method enforces Dirichlet conditions weakly without Lagrange multipliers. Symmetric Nitsche adds boundary flux terms and a penalty scaling as γp²/h, where p is polynomial degree. Unlike pure penalty methods, it remains consistent for finite γ.

[Dimitris @DeltaPiSystems]

In ℓ₁-regularized optimization, proximal gradient descent uses xₖ₊₁ = soft(xₖ − α∇g(xₖ), αλ) to minimize g(x)+λ‖x‖₁. Soft-thresholding sets coordinates exactly to zero when |xₖ−α∇g(xₖ)|≤αλ, producing sparse solutions without combinatorial search.

[Dimitris @DeltaPiSystems]

In adaptive FEM, Dörfler marking selects elements M such that Σ(K∈M) η_K² ≥ θ Σ(K) η_K², where η_K is a local error indicator and 0<θ≤1. Larger θ marks more of the estimated error for refinement, trading computational cost against refinement aggressiveness.

[Dimitris @DeltaPiSystems]

For SPD quadratic objectives, conjugate gradient satisfies ‖eₖ‖ₐ/‖e₀‖ₐ ≤ 2[(√κ−1)/(√κ+1)]ᵏ, where κ=λmax(A)/λmin(A). Preconditioning improves convergence by reducing the effective condition number or clustering eigenvalues, often cutting iterations dramatically.

[Dimitris @DeltaPiSystems]

In high-order FEM, static condensation eliminates interior degrees of freedom before global assembly. Partitioning stiffness into boundary (b) and interior (i) blocks gives S = K_bb − K_bi K_ii⁻¹ K_ib. This reduces global system size while retaining exact algebraic equivalence.

[Dimitris @DeltaPiSystems]

In constrained optimization, ∇f(x*) need not vanish. For a closed convex feasible set C, first-order stationarity is equivalent to x* = Π_C(x* − α∇f(x*)) for any α > 0. The projected-gradient mapping measures stationarity even when constraints are active.

[Dimitris @DeltaPiSystems]

In FEM,nearly incompressible elasticity can cause volumetric locking in low-order displacement elements: the discrete space struggles to satisfy ∇·u≈0.Mixed displacement–pressure formulations can avoid locking, but require inf-sup-stable element pairs or suitable stabilization.

[Dimitris @DeltaPiSystems]

In trust-region optimization, step quality is measured by ρ = [f(x) − f(x+s)]/[m(0) − m(s)], the ratio of actual to predicted reduction. Small ρ signals a poor local model and motivates shrinking the trust region; ρ near 1 indicates good agreement.

[Dimitris @DeltaPiSystems]

In FEM, reduced integration can lower computational cost and alleviate locking, but may introduce zero-energy deformation modes (hourglassing). These modes produce little or no strain energy despite nonzero displacement, so underintegrated elements often require hourglass control

[Dimitris @DeltaPiSystems]

For nonlinear least squares f(x)=½‖r(x)‖², the exact Hessian is H=JᵀJ+Σᵢrᵢ∇²rᵢ. Gauss–Newton drops the second term, making its curvature approximation especially accurate near zero residuals—but potentially unreliable when residuals remain large.