14 problems

Consider the SPDE $du=(-u^3+Delta u)\,dt+\sqrt{2}\,dW_t$ and its semi-implicit Euler time discretization … Finite-time blowup conjecture. The above time discretization will exhibit…

The paper considers summation-by-parts (SBP) properties in discrete derivative operators and notes that the Active Flux method uses upwinding, with upwind SBP operators providing a…

Discrete Bernoulli instability conjecture. Such Riemann data, although they solve the discrete form of Bernoulli's equation, do not form a stable steady solution. The claim is a nu…

Consider the shallow water equations with discontinuous topography and a Riemann problem whose two states have constant Riemann invariants, so that the stationary contact wave is a…

In the semi-infinite problem on $[0,+\infty)$, let GKS stability mean that there exist $\alpha_0\geq 0$ and $C>0$ such that the estimate … for every $\alpha>\alpha_0$, for sufficie…

The absolute value function has derivative magnitude $1$ wherever it is differentiable. Gradient-explosion conjecture. Using the absolute value function in a training objective con…

Tadmor's strong-stability conjecture. The four-stage Runge–Kutta method fails strong stability: it is not uniformly similar to a contraction. Equivalently, there is no symmetrizer…

Let $(M,\bomega)$ be a compact connected Ke4hler manifold of complex dimension $n$, with Ke4hler form $\bomega$ and a class $[\bchi_0]\bin H^{1,1}(M;\bmathbb{R})$. The J-equation i…

The optimization uses an FMU to simulate design models while optimization algorithms explore values of the component parameters. Numerical-instability conjecture. The observed jump…

Let $H(u(\cdot,t_0),\tau)$ denote an exact solution operator mapping an initial condition $u(\cdot,t_0)$ to the solution at time $t_0+\tau$. Let $V$ be the finite-dimensional appro…

The defense uses a bounded condition number, and the optimization layer produces gradients during training. Bounded-condition-number stability conjecture. This occurs when the true…

CIR sparsity conjecture. The sparsity in the CIR measurements, arising from their impulse-like nature, compared with the smoother CSI measurements, causes numerical instabilities t…

The IDC-OS methods approximate the residual integral in their error equation using quadrature nodes; in the simulations described here, 13 uniformly distributed interior nodes are…

Standard-CFL conjecture. The CFL of scheme B always remains the standard one, whatever the damping profile is.