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Finite Element Methods use fine meshes only where greater detail is needed, allowing the focusing behavior of a convex lens to be simulated accurately without the cost of refining the entire domain.

Mathelirium

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16,128 views • 2 months ago •via X (Twitter)

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Vikram Sarabhai Space Centre (VSSC), Indian Space Research Organisation, Thiruvananthapuram, is organising National Finite Element Developers' / FEAST Users’ meet (NAFED08) on February 01, 2025, at IIT Hyderabad. FEAST (Finite Element Analysis for STructures), a general purpose finite element software is completely an indigenous development complying with industry standard practices. To promote the usage of FEAST and enhance its capabilities through collaboration, NAFED is being conducted annually since 2016. The objective of the meet is to bring together domain experts and software developers within the country in the field of finite element and numerical methods. The central focus of the meet will be on the usage of FEAST software for solving complex engineering problems and consolidate usage feedbacks to enhance the software. Developers from academia and start-ups are given an opportunity to showcase their possible collaborative work and interact with FEAST software developers on how to interface their work with FEAST as external modules using python scripting. Events like a competition for design and analysis of structures using FEAST and a Quiz competition for students on finite element analysis are also planned as part of the meet. For further details visit the meet link:

Vikram Sarabhai Space Centre (VSSC), Indian Space Research Organisation, Thiruvananthapuram, is organising National Finite Element Developers' / FEAST Users’ meet (NAFED08) on February 01, 2025, at IIT Hyderabad. FEAST (Finite Element Analysis for STructures), a general purpose finite element software is completely an indigenous development complying with industry standard practices. To promote the usage of FEAST and enhance its capabilities through collaboration, NAFED is being conducted annually since 2016. The objective of the meet is to bring together domain experts and software developers within the country in the field of finite element and numerical methods. The central focus of the meet will be on the usage of FEAST software for solving complex engineering problems and consolidate usage feedbacks to enhance the software. Developers from academia and start-ups are given an opportunity to showcase their possible collaborative work and interact with FEAST software developers on how to interface their work with FEAST as external modules using python scripting. Events like a competition for design and analysis of structures using FEAST and a Quiz competition for students on finite element analysis are also planned as part of the meet. For further details visit the meet link:

ISRO

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A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work such as meshes, stencils, and stability tuning. Once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain.

A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work such as meshes, stencils, and stability tuning. Once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain.

Mathelirium

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A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.

A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.

Mathelirium

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The entire universe itself is finite (creation). Just like u having the ability to lift a cup of water (which is finite) whenever u wish via "kasb" and it wouldn't be shirk or kufr according to asharis, controlling the ENTIRE universe however one wishes with ALL its atoms ..

The entire universe itself is finite (creation). Just like u having the ability to lift a cup of water (which is finite) whenever u wish via "kasb" and it wouldn't be shirk or kufr according to asharis, controlling the ENTIRE universe however one wishes with ALL its atoms ..

Muwahhid

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Higuruma is the only character in the entire jjk verse to have grown faster than Yuji. He was able to use domain amplification, and RCT in the span of just three months of becoming a sorcerer. Bro rivals Gojo in terms of pure potential.

Higuruma is the only character in the entire jjk verse to have grown faster than Yuji. He was able to use domain amplification, and RCT in the span of just three months of becoming a sorcerer. Bro rivals Gojo in terms of pure potential.

𝙰𝚗𝚒𝚖𝚎 𝙵𝚕𝚒𝚡

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Do you actually know what convex optimization is in the geometric, guarantee-theoretic sense or have you only met it through solvers and loss curves? Convexity is rare comfort in optimization...there are no spurious local minima, no surprise traps, and inequalities you can use like tools instead of prayers. So, what is this convexity? Let x = (x₁, x₂) and let f(x) be convex. Plot the surface z = f(x). Pick a contact point x₀. The local slope is the gradient p = ∇f(x₀). That p is exactly the data that defines the supporting plane: z = f(x₀) + p · (x − x₀). Thus, f is said to be convex because for every x, f(x) ≥ f(x₀) + p · (x − x₀). So the plane at x₀ can slide under the surface, but it never slices through it. Not near the point...everywhere. Now for here is the interesting part: The slope becomes a coordinate system! Rewrite the same plane as z = p · x − b, where b is the offset. Because the plane passes through (x₀, f(x₀)), the offset is forced to be b = p · x₀ − f(x₀). And that number isn’t just geometry trivia. It’s the convex conjugate: f*(p) = sup over x ( p · x − f(x) ). At a differentiable contact point, the supporting plane touches f tightly enough that the supremum is achieved at x₀, giving the identity f*(p) = p · x₀ − f(x₀) when p = ∇f(x₀). So one moving contact point gives two linked readouts: primal position x₀ dual position (slope) p = ∇f(x₀) dual offset f*(p) One surface. Two worlds. #ConvexOptimization #Optimization #MachineLearning #SignalProcessing #AppliedMath #Engineering

Do you actually know what convex optimization is in the geometric, guarantee-theoretic sense or have you only met it through solvers and loss curves? Convexity is rare comfort in optimization...there are no spurious local minima, no surprise traps, and inequalities you can use like tools instead of prayers. So, what is this convexity? Let x = (x₁, x₂) and let f(x) be convex. Plot the surface z = f(x). Pick a contact point x₀. The local slope is the gradient p = ∇f(x₀). That p is exactly the data that defines the supporting plane: z = f(x₀) + p · (x − x₀). Thus, f is said to be convex because for every x, f(x) ≥ f(x₀) + p · (x − x₀). So the plane at x₀ can slide under the surface, but it never slices through it. Not near the point...everywhere. Now for here is the interesting part: The slope becomes a coordinate system! Rewrite the same plane as z = p · x − b, where b is the offset. Because the plane passes through (x₀, f(x₀)), the offset is forced to be b = p · x₀ − f(x₀). And that number isn’t just geometry trivia. It’s the convex conjugate: f(p) = sup over x ( p · x − f(x) ). At a differentiable contact point, the supporting plane touches f tightly enough that the supremum is achieved at x₀, giving the identity f(p) = p · x₀ − f(x₀) when p = ∇f(x₀). So one moving contact point gives two linked readouts: primal position x₀ dual position (slope) p = ∇f(x₀) dual offset f*(p) One surface. Two worlds. #ConvexOptimization #Optimization #MachineLearning #SignalProcessing #AppliedMath #Engineering

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MeshSplatting: Differentiable Rendering with Opaque Meshes Contributions: (i) An end-to-end optimization of mesh-based scene representations retains visual quality while training 2× faster than current state-of-the-art methods. (ii) Rather than a polygon soup, we generate a connected mesh by refining the vertex locations of a restricted Delaunay triangulation. (iii) Triangles are naturally connected to each other, and quantities stored within vertices are smoothly interpolated across each triangle. (iv) The optimization is aware that the triangles should be opaque, allowing direct high-quality rendering in standard game engines (see Fig. 1), opening the door for classical techniques like the use of depth buffers and occlusion culling [1, 22].

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Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Mathelirium

40,841 views • 6 months ago