From 600ee485eb92de0b1d0d75383317fe1d7e511b88 Mon Sep 17 00:00:00 2001 From: DotNaos Date: Thu, 17 Apr 2025 13:18:15 +0200 Subject: [PATCH] Organisation --- .../formulas/Zusammenfassung.md | 288 ++++++++++++++++++ .../tasks/Arbeitsauftrag - Finanzen.pdf | Bin 0 -> 76811 bytes .../tasks/Aufgaben.md | 146 +++++++++ .../tasks/Präsentation Finanzen.md | 52 ++++ .../Notes/01.11.24.canvas | 9 + .../Notes/11.10.24.md | 40 +++ .../Notes/20.09.24.md | 6 + .../Notes/25.10.24.md | 6 + .../Notes/27.09.2024.md | 88 ++++++ .../Notes/4.10.24 | 23 ++ .../Notes/Canvas.canvas | 4 + .../code/fld.py | 226 ++++++++++++++ .../formulas/Zusammenfassung.md | 279 +++++++++++++++++ .../formulas/Zusammenfassung.pdf | Bin 0 -> 1131414 bytes .../tasks/Aufgabe 5.md | 8 + 15 files changed, 1175 insertions(+) create mode 100644 CDS1011-Introduction-to-Data-Science/formulas/Zusammenfassung.md create mode 100644 CDS1011-Introduction-to-Data-Science/tasks/Arbeitsauftrag - Finanzen.pdf create mode 100644 CDS1011-Introduction-to-Data-Science/tasks/Aufgaben.md create mode 100644 CDS1011-Introduction-to-Data-Science/tasks/Präsentation Finanzen.md create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/01.11.24.canvas create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/11.10.24.md create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/20.09.24.md create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/25.10.24.md create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/27.09.2024.md create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/4.10.24 create mode 100644 CDS1012-Introduction-to-Computational-Science/Notes/Canvas.canvas create mode 100644 CDS1012-Introduction-to-Computational-Science/code/fld.py create mode 100644 CDS1012-Introduction-to-Computational-Science/formulas/Zusammenfassung.md create mode 100644 CDS1012-Introduction-to-Computational-Science/formulas/Zusammenfassung.pdf create mode 100644 CDS1012-Introduction-to-Computational-Science/tasks/Aufgabe 5.md diff --git a/CDS1011-Introduction-to-Data-Science/formulas/Zusammenfassung.md b/CDS1011-Introduction-to-Data-Science/formulas/Zusammenfassung.md new file mode 100644 index 0000000..4f85b3e --- /dev/null +++ b/CDS1011-Introduction-to-Data-Science/formulas/Zusammenfassung.md @@ -0,0 +1,288 @@ +# 1. Teilgebiete von AI + +![AI Begriffe](image.png) + +1. **Machine Learning (ML)** – Ein System lernt aus Daten, Muster zu erkennen und Vorhersagen zu treffen, ohne explizit programmiert zu sein. + + - **Supervised Learning**: Trainiert mit gelabelten Daten (z. B. Klassifikation & Regression). + - **Unsupervised Learning**: Findet Muster in ungelabelten Daten (z. B. Clustering, Anomalie-Erkennung). + - **Reinforcement Learning (RL)**: Agent lernt durch Belohnungen und Strafen (z. B. AlphaGo, Robotik). +2. **Deep Learning (DL)** – Eine Unterkategorie von ML, die neuronale Netzwerke mit vielen Schichten nutzt. Besonders leistungsfähig bei Bild-, Sprach- und Textverarbeitung (z. B. CNNs für Bilder, RNNs für Sprache). + + +Jede dieser Disziplinen entwickelt sich rasant weiter und wird oft kombiniert eingesetzt. +# 2. Machine Learning +![[Pasted image 20250129121715.png]] +## Supervised Learning +- Klassifikation + - Training: Objekt: Label gegeben + - Test: Objekt: Label zuordnen + - Methoden +- Regression + - >Training + - >X: Datenpunkt Y gegeben + - Test: X: Y vorhersagen +### Examples +**Klassifikation** +- K-Nearest-Neighbors +- Naïve Bayes +- Support Vector Machines +- Decision Trees + +**Regression** +- Polynomregression +- Lineare Regression +- Random Forrest Regression +- Support Vector Regression + +## Unsupervised Learning +- Clustering +- Anomalieerkennung +- Visualisierung +- Dimensionsereduktion +- Lernen von Assoziationsregeln + +## Semi-supervised Learning +![[Pasted image 20250129121927.png]] + +## Others +**Batch-Learning** +- Einmal hin alles drin + +**Online-Learning** +- Stück für Stück, mit Web scraper +- Immer mal wieder kommt was neues + +# 3. Daten ( -> Herausforderungen ) +## Datenmenge +Deep Learning braucht viel Daten, skaliert aber extrem gut bis ins unendliche (Eher fehlt die Rechenleistung) +- Mindestdatenmenge muss erreicht sein + +## Datenqualität +**Wird schlechter durch:** +- Rauschen (Daten die ungewollt im Datensatz sind) +- Fehler ( Daten die Falsch sind z.B. sensoren, User geben falsche angaben) +- Ausreißer: Daten die extrem abweichen +- Empty Values: z.B. NaN / Null im Datensatz + +## Datenrepräsentation +- Korrelation =/= Kausalität +- zu wenige Daten + +## Overfitting +- Alles ist ein Muster +- Analog: CEO der jeden Müll hyped + +**Lösung**: +- Einfacheres Modell +- Weniger Features +- Bessere Daten (siehe Datenqualität) +- (Basically alles weniger komplex machen) + +## Underfitting +- Nichts ist ein Muster +- Analog: Person die die Augen verschließt ("nothing ever happens") + +**Lösung**: +- Siehe Overfitting (aber invertiert) +- Die richtigen Features finden + +## Data Snooping Bias +In der Datenvoranalyse (Data Snooping) wird versucht sich ein bild vom Datensatz zu machen und voreilige Schlüsse gezogen. Diese können sich durch die ganze Forschung ziehen und sie negativ beeinflussen. + +## Stratified Sampling +- Klustern von Datenpunkten +- Zufällige Auswahl von Datenpunkten aus den Clustern + +![[Pasted image 20250129121441.png]] +## Train-Test Split +- Trainingsdaten: Modell trainieren +- Testdaten: Modell testen + +Normalerweise 70% / 30% + +## Parameter +### Hyperparameter +- Entkoppelt vom Modell +- Beispiel: Temperature bei LLM's + +### Modelparameter +- Wird Trainiert + + +# 4. Qualitätsmetriken +## Root Mean Squared Error +> Es berechnet die Differenz für alle Datenpunkte. +> Diese Werden Quadriert um größere Differzenzen zu hervorzuheben. ( und auch für positives Vorzeichen ) +> Daraus wird der Durchschnitt berechnet und die Wurzel gezogen (um das Quadrieren rückgängig zu machen). + +$$ \text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2} $$ + +**Variablen:** +- $y_i$: Tatsächliche Werte +- $\hat{y}_i$: Vorhergesagte Werte ($f(x_i)$) +- $n$: Anzahl der Datenpunkte + +**Schritte:** +1. Berechne die Differenzen: $y_i - \hat{y}_i$ +2. Quadriere die Differenzen: $(y_i - \hat{y}_i)^2$ +3. Berechne den Durchschnitt: $\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$ +4. Ziehe die Quadratwurzel: $\sqrt{\dots}$ + +## Mittlerer absoluter Fehler MAE +Siehe RMSE aber ohne Quadrieren. +> Das Quadrieren wegzulassen bedeutet das die Skala für die Differenzen linear ist und nicht exponentiell. D.h. Ausreißer haben weniger Einfluss auf das Ergebnis. +**Formel:** +$$ +\text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i| +$$ + +- $y_i$: Tatsächlicher Wert +- $\hat{y}_i$: Vorhergesagter Wert +- $n$: Anzahl der Datenpunkte + +## Arithmetisches Mittel +> Summe aller Werte durch die Anzahl der Werte +$$ \text{AM} = \frac{1}{n} \sum_{i=1}^{n} x_i $$ + +## Median +> Der Wert der in einer Sortierten Liste in der Mitte steht bei ```ungeraden n```. Bei ```geraden n``` der Durchschnitt der beiden mittleren Werte. + +$$ \text{Median} = \begin{cases} x[\frac{n+1}{2}] & \text{für ungerade } n \\ \frac{1}{2} (x[\frac{n}{2}] + x[\frac{n}{2}+1]) & \text{für gerade } n \end{cases} $$ + +## Streuungsmasse +### Mittlere absolute abweichung (MAD) +> gleiches Konzept wie bei MAE, aber der "gewünschte" Wert ist der Erwartungswert. + +>INFO: Da hier nicht mit Wahrscheinlichkeiten gerechnet wird, ist der Erwartungswert = der Durchschnittswert. (Weil alle Werte gleich wahrscheinlich sind, und die Verteilung symmetrisch) + +$$ \text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \hat{x}_i| $$ + +### Varianz und Standardabweichung +> **Varianz**: Durchschnittliche quadratische Abweichung (ohne Wurzel) \ +> **Standardabweichung**: Wurzel der Varianz (Root Mean Squared Error für X mit n-1) + +$$ \sigma^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \hat{x})^2 $$ + +$$ \sigma = \sqrt{\sigma^2} $$ +## Pearson Korrelationskoeffizient +$$ +r = \frac{\sum_{i=1}^n + \bigl(x_i - \overbrace{\bar{x}}^{\text{Mittelwert}}\bigr) + \bigl(y_i - \overbrace{\bar{y}}^{\text{Mittelwert}}\bigr)} + {(n - 1)\,\underbrace{s_x \cdot s_y}_{\text{Standardabweichung}}}. +$$ +- **Zusammenhang zwischen zwei numerischen Variablen** + +- **$r \in [-1; +1]$** + - **1**: stark positive Korrelation + - **0**: keine lineare Korrelation + - **–1**: stark negative Korrelation + +- **Nichtlinearer Zusammenhang** trotzdem möglich + - exponentiell, quadratisch + +- **Korrelation ≠ Kausalität** + + +# 6. Regression +![[Pasted image 20250129121337.png]] +## Lineare Regression +**Lineare Regression** +$$ +\begin{align*} +\textrm{Lineare Gleichung: } +\quad &y = c + m \cdot x \\[6pt] +\textrm{Steigung: } +\quad &m = \frac{\sum (x - \bar{x}) (y - \bar{y})}{\sum (x - \bar{x})^2} \\[6pt] +\textrm{Intercept: } +\quad &c = \bar{y} - m \,\bar{x} +\end{align*} +$$ +**R2** +$$ +\begin{align*} +\textrm{Bestimmtheitsmaß:} \quad +&R^2 = \frac{SSR}{SST} \\[6pt] +\textrm{Sum of Squares Regression (SSR):} \quad +&SSR = \sum_{i} \bigl(\hat{y}_i - \bar{y}\bigr)^2 \\[6pt] +\textrm{Total Sum of Squares (SST):} \quad +&SST = \sum_{i} \bigl(y_i - \bar{y}\bigr)^2 +\end{align*} +$$ + + + +# 9. Supervised Learning +## Kunfusionsmatrix +![[Pasted image 20250129121620.png]] + +### Precision (Relevanz) +$precision=\frac{tp}{tp+fp}$ + +### Recal (Sensibilität) +$recall=\frac{tp}{tp+fn}$ + +### F1 +$f1=2\cdot\frac{precision\cdot recall}{precision+recall}$ +## Logistische Regression zur binären Klassifikation + +- **Klassifikationsverfahren** +- **Wahrscheinlichkeit** für die Klassenzugehörigkeit zwischen 0 und 1 +- **Nominalskalierte** Kriterien vorhersagen +- **Prädiktor** (Merkmal) +- **Kriterium** (Wahrscheinlichkeit) +- Je steiler die Kurve, desto besser die Vorhersage +- **Siehe auch** + - Multinomiale logistische Regression für mehrere Kriterien + +**Formel** +$$ +p(y = 1) \;=\; \beta_0 \,\cdot\, +\frac{e^{\beta_0 + \beta_1 \cdot x_1}} + {1 \;+\; e^{\beta_0 + \beta_1 \cdot x_1}}\;=\; +\\~\\p\!\bigl(\underbrace{y = 1}_{\text{(Zielvariable)}}\bigr) +\;=\; +\underbrace{\beta_0}_{\text{(Achsenabschnitt)}} +\,\cdot\, +\frac{ + e^{ + \overbrace{\beta_0}^{\text{(Achsenabschnitt)}} + \;+\; + \overbrace{\beta_1}^{\text{(Regressionskoeffizient)}} + \cdot + \overbrace{x_1}^{\text{(Prädiktor)}} + } +}{ + 1 + e^{\beta_0 + \beta_1 \cdot x_1} +} +$$ + +**Variablen:** +- $y$: Zielvariable (nimmt Werte 0 oder 1) +- $\beta_0$: Achsenabschnitt (Intercept) +- $\beta_1$: Regressionskoeffizient (Gewicht für den Prädiktor) +- $x_1$: Prädiktor (Eingangsvariable) +- $e^{\beta_0 + \beta_1 \cdot x_1}$: Exponentialterm, hier als Basis der logistischen Funktion +- $p(y=1)$: Wahrscheinlichkeit, dass $y=1$ eintritt + + +# 10. Unsupervised Learning +## Clustering +### K-Means +![[Pasted image 20250129121823.png]] +### Euclidean Distance +$d\left( x,y \right)=\sqrt{\sum_{i=1}^{n}\left( x_i-y_i \right)^2}$ + +### Elbow Method +![[Pasted image 20250129121853.png]] +- **Optimale Anzahl an Clustern** finden +>Wenn die **Distanz** zwischen den **Clustern** nicht mehr **signifikant** sinkt, ist die **optimale Anzahl** an Clustern erreicht. +### Silhouette Coefficient +$$ +\begin{align*} +a=\textrm{cohesion}=\textrm{intra cluster distance}\\ +b=\textrm{separation}=\textrm{nearest cluster distance}\\ +s\left( i \right)=\frac{b\left( i\right)-a\left( i\right)}{max\{a\left( i\right),b\left( i\right)\}} +\end{align*} +$$ diff --git a/CDS1011-Introduction-to-Data-Science/tasks/Arbeitsauftrag - Finanzen.pdf b/CDS1011-Introduction-to-Data-Science/tasks/Arbeitsauftrag - Finanzen.pdf new file mode 100644 index 0000000000000000000000000000000000000000..02f423bb1891437dcfd17dfa45c698efd9b7a2e1 GIT binary patch literal 76811 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zW#!6s(>gpjXPV5fH)u;Ih92%qJj|Wb)f}BOLnou%(PPqQH5X60)YlhLHK(s=WR@-4 z>G8sVIJED6cr5wHbdB3OeDj67oB9vsOjk5$YYNgs%wN2^cHQe#|Es>Nar357Cge}) ze2HUjNJQ$%$AjsvHvK~McDJ9(>3E5rU-CHly@wma%WJQFml`FtNj_ukc)!(tX`a9I zfSykRnm7HXWy4lId4kK<1_gEAT7j2(ZIV_L-Z2vYma#W(eo2IIZqnptKTL1q>R9h} z3%CWlcQ>3byqPrf=q1&t#AkcT*vlNiZ2!?d`Sc|Jw^Uk`4FRRANt1$eHX(3>IW&>YB2ymAQ~Al3rhX`F zB24{gSy=7CM#CH*{B@AbYa?M>W3@+82vbvd>jY)RUUD91r89}36tf&$EdV#w>%wCJ*4m<{E_05PAPZ9i5+_-64c-$9 VEF Graph + +## Eulersche Formel +$$ +n_V - n_E + n_F = 2 +$$ + +Gleiches für Dreiecke + + +### Speicherung als Matrix +- Einscannen der Vertecies durch z.B. LiDar + +Vertex: +- Koordinaten + +Kanten: +- V1 und v2\ + +Fläche: +- e1, e2, e3 + +## Darstellungsformen des Volumenmodells +Primitive: +- Würfel, Tetraeder ... + + +## Spacetrees +- Rekursives runterbrechen von Voxels + - Die die Auf dem Rand liegen werden fortgeführt + +### Reihenfolge +links unten nach rechts unten, dann nach oben links dann oben rechts + + diff --git a/CDS1012-Introduction-to-Computational-Science/Notes/20.09.24.md b/CDS1012-Introduction-to-Computational-Science/Notes/20.09.24.md new file mode 100644 index 0000000..7069f7d --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/Notes/20.09.24.md @@ -0,0 +1,6 @@ +# Organisatorisches +## Projekte +- Verkehrssimulation + +## Leistungsnachweis +- 100% Prüfung diff --git a/CDS1012-Introduction-to-Computational-Science/Notes/25.10.24.md b/CDS1012-Introduction-to-Computational-Science/Notes/25.10.24.md new file mode 100644 index 0000000..d765caa --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/Notes/25.10.24.md @@ -0,0 +1,6 @@ +## Codierung von Spacetrees +1 für Elternknoten +0 für Blätter + +0 für Außen +1 für auf Geometrie diff --git a/CDS1012-Introduction-to-Computational-Science/Notes/27.09.2024.md b/CDS1012-Introduction-to-Computational-Science/Notes/27.09.2024.md new file mode 100644 index 0000000..38ccebc --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/Notes/27.09.2024.md @@ -0,0 +1,88 @@ +# Simulationspipeline +## 1. Modellierung + +- Model erstellen, das auf eine niedrige Abstrakte Ebene heruntergebrochen wurde. +- Verienfachung / Formale Abstraktion der Realität -> Dimension wegnehmen +### Model Erstellen +Eine Annahme z.B. eine Formel + +3D - Modell + +### Mathematische Modellierung +Eine Szenario auf einer Karte -> Graphen + +#### Graphen +Problem: Jeder Pfad auf einem Graphen einmal verwenden. Funktioniert nur bei gerader Anzahl der Pfade an einem Knoten. Bei ungerade Anzahl -> Keine Weg zurück + + +#### Herleitung +- Welche Größen sind wichtig +- Wie beeinflussen sie sich? +- Was ist die Aufgabe ? + +#### Analyse +- Existens und Eindeutigkeit: Gibt es eine Lösung? Welche Lösung ist richtig? +- Wie beinflussen Ausgangswerte das Ergebnis? +- Wie genau ist das Modell? Mehrere Modelle sind möglich. + + + +## 2. Numerische Behandlung + +### Diskretisierung +Aus Unendlich wird endlich also z.B. 10 Werte. + +h : (Gitterabstand) Abstand zwischen den Punkten. + +![[diskretion.png]] + +### Algorithmen Entwicklung +- Genauer Ablauf entwickeln + + +### Klassifikation von Modellen + +- Diskrete und kontinuierliche Modelle + +#### diskrete Modelle +- Basieren auf diskrete oder kombinatorischen Beschreibungen +- z.B. Graphen, (endliche) Automaten + +#### kontinuierliche Modelle +- basieren auf stetigen Beschreibung +- in der Regel Gleichungen + +#### Deterministische und Stochastische Modelle +- Beispiel 1: Würfel + - Stochastisches Modell + +- Beispiel 2: Crashtest + - deterministisches Phänomen + +#### Skalen und Hierarchien +- Wahl der Skala in Abhängigkeit von + - Ergebnis + - kosten der Berechnung und Zeit + +- Auflösung der Abstraction: Mensch oder Atom + +#### Inverse Probleme +- Forward Problem + - Wetter + +- Inverse Problem + - Lösung kennt man: Wie kommt man da hin? + + +## 3. Implementierung +- Softwareentwicklung, Parallelisierung + +## 4. Visualisierung +- An Zielgruppe anpassen +- Ergenisse Zeigen + +## 5. Validierung +- Fehlerbetrachtung +- Was kann es? +## 6. Einbettung +- Verkauf und Vertrieb diff --git a/CDS1012-Introduction-to-Computational-Science/Notes/4.10.24 b/CDS1012-Introduction-to-Computational-Science/Notes/4.10.24 new file mode 100644 index 0000000..364f24d --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/Notes/4.10.24 @@ -0,0 +1,23 @@ +## Numerische Behandlung: Diskretisierung +> Auflösung des Problems + +Lösung nur an diskreten Stellen +- Differenzenquotient + +### Zeitaufwand +$$ +Rechenoperationen / Taktfrequenz = Zeit in sek +$$ + +# Implementierung +- Sprache egal +- Bibliotheken verwenden wo geht +- Parallelisierung + +# Visualisierung +- + +# Geometrische Modellierung +- Keine Offene Fläche (nicht verbundene Vertecies) +- Keine Durchdringende Geometrie +- diff --git a/CDS1012-Introduction-to-Computational-Science/Notes/Canvas.canvas b/CDS1012-Introduction-to-Computational-Science/Notes/Canvas.canvas new file mode 100644 index 0000000..2289854 --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/Notes/Canvas.canvas @@ -0,0 +1,4 @@ +{ + "nodes":[], + "edges":[] +} \ No newline at end of file diff --git a/CDS1012-Introduction-to-Computational-Science/code/fld.py b/CDS1012-Introduction-to-Computational-Science/code/fld.py new file mode 100644 index 0000000..9e4686d --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/code/fld.py @@ -0,0 +1,226 @@ +import sys +import numpy as np +import cupy as cp +import pygame + +# Minimaler, GPU-beschleunigter 2D-Fluid-Solver mit persistenter Flüssigkeit und Gravitation. +# Installation: +# pip install cupy-cudaXX pygame numpy +# XX = passende CUDA-Version, z.B. cupy-cuda112 für CUDA 11.2 + +# ============================== +# EINSTELLUNGEN +# ============================== +N = 64 # Gittergröße (N x N) +ITER = 2 # Iterationen; verringern für Geschwindigkeit +DT = 0.1 # Zeitschritt +DIFF = 0.0001 # Diffusionskoeffizient (klein, aber nicht null für Stabilität) +VISC = 0.0001 # Viskosität (klein, aber nicht null für Stabilität) +ZOOM = 8 # Pixel pro Gitterzelle in der Anzeige +GRAVITY = 0.1 # Gravitationskraft (anpassbar) + +WIDTH, HEIGHT = N * ZOOM, N * ZOOM + +# ============================== +# HILFSFUNKTIONEN +# ============================== +def set_bnd(b, x): + """ + Randbehandlung für die Simulation. + b = 0: Dichte + b = 1: u-Geschwindigkeit + b = 2: v-Geschwindigkeit + """ + if b == 1: + # u-Geschwindigkeit: links und rechts auf 0 setzen (Wände) + x[0, 1:-1] = 0 + x[-1, 1:-1] = 0 + elif b == 2: + # v-Geschwindigkeit: oben und unten auf 0 setzen (Wände) + x[1:-1, 0] = 0 + x[1:-1, -1] = 0 + else: + # Dichte: Spiegeln an den Rändern + x[1:-1, 0] = x[1:-1, 1] + x[1:-1, -1] = x[1:-1, -2] + x[0, 1:-1] = x[1, 1:-1] + x[-1, 1:-1] = x[-2, 1:-1] + + # Ecken setzen + x[0, 0] = 0.5 * (x[1, 0] + x[0, 1]) + x[0, -1] = 0.5 * (x[1, -1] + x[0, -2]) + x[-1, 0] = 0.5 * (x[-2, 0] + x[-1, 1]) + x[-1, -1] = 0.5 * (x[-2, -1] + x[-1, -2]) + +def add_source(x, s): + """ + Fügt eine Quelle zur Simulation hinzu. + """ + x += DT * s + +def diffuse(b, x, x0, diff): + """ + Diffusionsschritt der Simulation. + """ + a = DT * diff * (N - 2) * (N - 2) + for _ in range(ITER): + x[1:-1, 1:-1] = (x0[1:-1, 1:-1] + a * ( + x[0:-2, 1:-1] + x[2:, 1:-1] + + x[1:-1, 0:-2] + x[1:-1, 2:] + )) / (1 + 4*a) + set_bnd(b, x) + +def advect(b, d, d0, u, v): + """ + Advektionsschritt der Simulation. + """ + dt0 = DT * (N - 2) + + # Rückverfolgte Position berechnen + X, Y = cp.meshgrid(cp.arange(N), cp.arange(N), indexing='ij') + X = X[1:-1, 1:-1] - dt0 * u[1:-1, 1:-1] + Y = Y[1:-1, 1:-1] - dt0 * v[1:-1, 1:-1] + + X = cp.clip(X, 0.5, N - 1.5) + Y = cp.clip(Y, 0.5, N - 1.5) + + i0 = X.astype(int) + i1 = i0 + 1 + j0 = Y.astype(int) + j1 = j0 + 1 + + s1 = X - i0 + s0 = 1 - s1 + t1 = Y - j0 + t0 = 1 - t1 + + d_new = (s0 * (t0 * d0[i0, j0] + t1 * d0[i0, j1]) + + s1 * (t0 * d0[i1, j0] + t1 * d0[i1, j1])) + + d[1:-1, 1:-1] = d_new + set_bnd(b, d) + +def project(u, v, p, div): + """ + Projektionsschritt zur Inkompressibilität der Flüssigkeit. + """ + # Divergenz berechnen + div[1:-1,1:-1] = -0.5 * ( + (u[2:,1:-1] - u[0:-2,1:-1]) + + (v[1:-1,2:] - v[1:-1,0:-2]) + ) / N + p.fill(0) + set_bnd(0, div) + set_bnd(0, p) + + # Poisson-Gleichung lösen + for _ in range(ITER): + p[1:-1,1:-1] = (div[1:-1,1:-1] + p[0:-2,1:-1] + p[2:,1:-1] + + p[1:-1,0:-2] + p[1:-1,2:]) / 4 + set_bnd(0, p) + + # Geschwindigkeiten aktualisieren + u[1:-1,1:-1] -= 0.5 * (p[2:,1:-1] - p[0:-2,1:-1]) * N + v[1:-1,1:-1] -= 0.5 * (p[1:-1,2:] - p[1:-1,0:-2]) * N + set_bnd(1, u) + set_bnd(2, v) + +# ============================== +# FLUID-KLASSE +# ============================== +class Fluid: + def __init__(self): + self.dens = cp.zeros((N, N), dtype=cp.float32) + self.dens_prev = cp.zeros((N, N), dtype=cp.float32) + self.u = cp.zeros((N, N), dtype=cp.float32) + self.u_prev = cp.zeros((N, N), dtype=cp.float32) + self.v = cp.zeros((N, N), dtype=cp.float32) + self.v_prev = cp.zeros((N, N), dtype=cp.float32) + + def step(self, gravity): + """ + Führt einen Simulationsschritt durch, einschließlich der Anwendung von Gravitation. + """ + # Gravitation hinzufügen (nur zur v-Komponente) + self.v_prev += gravity * DT + + # Dichte + add_source(self.dens, self.dens_prev) + diffuse(0, self.dens, self.dens_prev, DIFF) + advect(0, self.dens, self.dens_prev, self.u, self.v) + + # Geschwindigkeit + add_source(self.u, self.u_prev) + add_source(self.v, self.v_prev) + diffuse(1, self.u, self.u_prev, VISC) + diffuse(2, self.v, self.v_prev, VISC) + project(self.u, self.v, self.u_prev, self.v_prev) + + advect(1, self.u, self.u_prev, self.u, self.v) + advect(2, self.v, self.v_prev, self.u, self.v) + project(self.u, self.v, self.u_prev, self.v_prev) + + # Quellen zurücksetzen + self.dens_prev.fill(0) + self.u_prev.fill(0) + self.v_prev.fill(0) + +# ============================== +# HAUPTPROGRAMM +# ============================== +def main(): + pygame.init() + screen = pygame.display.set_mode((WIDTH, HEIGHT)) + pygame.display.set_caption("GPU Fluid Simulation mit Gravitation") + clock = pygame.time.Clock() + + fluid = Fluid() + + # Zentrum für kontinuierliche Dichtequelle + center = (N//2, N//2) + + running = True + while running: + clock.tick(60) # 60 FPS anvisiert + + for event in pygame.event.get(): + if event.type == pygame.QUIT: + running = False + + # Mausinteraktion: Linke Maustaste fügt Dichte und zufällige Velocity hinzu + if pygame.mouse.get_pressed()[0]: + mx, my = pygame.mouse.get_pos() + i = int(mx // ZOOM) + j = int(my // ZOOM) + if 1 <= i < N-1 and 1 <= j < N-1: + fluid.dens_prev[j, i] += 100 + fluid.u_prev[j, i] += cp.random.uniform(-2, 2) + fluid.v_prev[j, i] += cp.random.uniform(-2, 2) + + # Immer ein klein wenig Dichte in der Mitte + fluid.dens_prev[center[1], center[0]] += 10 # Hinweis: [j, i] entspricht [y, x] + + # Ein Schritt Fluid-Simulation mit Gravitation + fluid.step(GRAVITY) + + # Dichte von GPU holen für Pygame + dens_host = cp.asnumpy(fluid.dens) + # In Graustufen umwandeln und skalieren + dens_normalized = np.clip(dens_host * 255, 0, 255).astype(np.uint8) + color_img = np.stack((dens_normalized, dens_normalized, dens_normalized), axis=-1) + + # Skalieren des Bildes auf die Bildschirmgröße + color_img_scaled = np.repeat(np.repeat(color_img, ZOOM, axis=0), ZOOM, axis=1) + + # Erstellen eines Pygame-Surfaces aus dem Array + surface = pygame.surfarray.make_surface(color_img_scaled.swapaxes(0, 1)) + + # Blitten der Surface auf den Bildschirm + screen.blit(surface, (0, 0)) + pygame.display.flip() + + pygame.quit() + sys.exit() + +if __name__ == "__main__": + main() diff --git a/CDS1012-Introduction-to-Computational-Science/formulas/Zusammenfassung.md b/CDS1012-Introduction-to-Computational-Science/formulas/Zusammenfassung.md new file mode 100644 index 0000000..4c2c6ec --- /dev/null +++ b/CDS1012-Introduction-to-Computational-Science/formulas/Zusammenfassung.md @@ -0,0 +1,279 @@ +# 1. Einführung +## Modellbildung +- **Modell**: Vereinfachte Darstellung eines Systems +- **Simulation**: Virtuelles Experiment + +## Mathematische Modellierung +- Features, Größen ( Objekte z.B. Staub, Wandposter ; Parameter z.B. Temperatur, Druck) +- Wechselwirkungen (Temperatur auf Personen im Raum; Druck und Staub) + +- Eindeutige Lösung: Existiert eine Lösung und ist sie eindeutig? +- Abhängigkeit von Initialwerten: Je nachdem wo man in einem Shooter startet, sind die Überlebenschancen unterschiedlich gut. +- Genauigkeit / Auflösung: Personenströme als Flüssigkeit modellieren, kann nicht eine einzelne Person darstellen. + +## Modellarten +- **Diskret**: In Schritten Simuliert +- **Kontinuierlich**: In Echtzeit / stetig +Simuliert +- **Deterministisch**: Immer das gleiche Ergebnis, mit den gleichen Anfangsbedingungen +- **Stochastisch**: Zufällige Ergebnisse, auch mit den gleichen Anfangsbedingungen (z.B. Würfel) +- **Inverse**: Ausgangsbedingungen aus Ergebnissen bestimmen oder Ergebnisse aus Ausgangsbedingungen bestimmen + +### Skalen +- 1D, 2D, 3D, so einfach wie möglich, so komplex wie nötig +- Abstraktionsebene: Menschen, Zellen oder Moleküle, oder gar Atome + +## Diskretitierung +- Gitterbreite h (delta zwischen punkten) + +# 2. Geometrische Modellierung +## Rigid Body Anforderungen +- keine losen Teile (Zusammenhängend) +- keine 1D-Teile (z.B Linie, Punkt) +- Es muss ein Innen und Außen geben (Volumen) +- keine Löcher +- keine Überlappungen +- Orientierbar (Außen und Innen unterscheidbar) +- Manigfaltig möglich (Zwei körper teilen sich eine Kante) + +![alt text](image.png) + +## Darstellungsformen (Speicherung) +- Vertecies (Ecken), Edges (Kanten), Faces (Flächen) +- $n_{v} - n_{e} + n_{f} = 2$ (Euler-Formel) -> Gilt für Rechtecke und Dreiecke + +### vef-Graph +> Es werden nur die Koordinaten der **Eckpunkte** gespeichert, Kanten und Flächen speichern referenzen. Dadurch werden änderungen an den Eckpunkten automatisch in den Kanten und Flächen übernommen. \ +> Eine **Kante** wird durch zwei Eckpunkte definiert. \ +> Eine **Fläche** durch drei Kanten. \ + +![alt text](image-1.png) + +## Volumenmodell +![alt text](image-3.png) + +### Spacetree +- **Balanzierung**: Jeder Knoten entweder **0** oder **4** Kinder (Eine Fläche 2 hat Angrezende Flächen) +![alt text](image-7.png) + +![alt text](image-4.png) + +#### Codierung +![alt text](image-10.png) + +#### Linearisierung +![alt text](image-11.png) + +# 3. Spieltheorie +## Definition +- **Gewissheit**: Konsequenzen zu Aktionen bekannt +- **Risiko**: Wahrscheinlichkeit von Konsequenz zu Aktion teilweise bekannt +- **Unsicherheit**:keine Konsequenz zu Aktion bekannt + +**Modellierung** +- Numbers of Players $N=\{1,2, \ldots, n\}$ +- Anzahl Aktionen (Strategien)$A_i\qquad i\in \mathbb{N}$ +- Menge Aktionsprofile (Strategieprofile) $A=\{\left( a_i \right)_{i\in \mathbb{N}}, a_i\in A_i, i\in\mathbb{N} \}$ +- Auszahlungsfunktion $u_i: A\rightarrow\mathbb{R}$ + +## Unterteilung +- **Kooperation** + - **kooperative Spiele**: Spieler kooperieren müssen um zu gewinnen. + - **nicht-kooperative Spiele**: Spieler konkurrieren um zu gewinnen / spielen gegeneinander. (z.B. nur einer kann gewinnen) +- **Serie** + - **simultane Spiele**: In einem Durchlauf treffen alle für sich gleichzeitig die Wahl. + - **sequentielle Spiele**: Regeln wer wann Entscheiden darf.Sequenzielle Abfolge. + +## Darstellung +- Bimatrixform / Normalform + - Gewinn wird angezeigt ($u_i$, $u_i$) + ![bimatrix](image-12.png) + +## Reine Strategien +### Gefangenen-Dilema +- Zwei Subjects $A_1\textrm{ und }A_2$ +- Mögliche Aktionen + - **Leugnen L**: Niedrigere Strafe (2 Jahre) + - **Gestehen G**: Volle Strafe 5 (Kronzeuge) oder 8 Jahre +### Nash-Gleichgewicht +> Ist das Optimale Ergebnis für **alle** Spieler +- Wenn ein Spieler durch Ändern seiner Strategie, seinen Gewinn erhöhen kann, ist der aktuelle Zustand nicht im Nash-Gleichgewicht. + +**Vorsicht!**: +- Jede Zelle muss einzeln betrachtet werden +- WICHTIG: Nicht mit der gleichen Strategie vergleichen +- Nash-Gleichgewicht muss für BEIDE Spieler gelten + +![nash](image-14.png) + +#### Dominante Strategien +> Eine Strategie die besser oder gleich gut ist wie alle anderen Strategien (>=) +**Beispiel:** +Sicht von S1: +- S2 nimmt Leugnen: Nimmt S1 Gestehen = 8, nimmt S1 Leugnen = 6. 8 >= 6 +- S2 nimmt Gestehen: Nimmt S1 Gestehen = 3, nimmt S1 Leugnen = 0. 3 >= 0 + +- Kreuzen sich Dominante Strategien, ist dort das Nash-Gleichgewicht (aber nicht umgekehrt) +#### Stark Dominante Strategien +> Eine Strategie die echt besser ist als alle anderen Strategien (>) +#### Falke-Taube-Spiel +#### Zwei-Personen-Nullsummenspiel +- Spieler 2 will gewinn maximieren, Spieler 1 will verlust minimieren +- nicht jedes 2PNS in reinen Strategien hat ein Nash-Gleichgewicht. +- **Gewinnfunktion**: $u_2 = -u_1$ + +- Matrixform + ![bimatrix_und_matrix](image-13.png) + +##### Konservative Strategien +- Wenn die Konservate Strategie beider Spieler gleich ist, ist dort ein Nash-Gleichgewicht. + +**MinMax-Strategie** +Was sollte ich höchstens verlieren? + +> 1. Mache eine liste mit dem jeweils größten Verlust +> 2. Daraus wähle den kleinsten Verlust + +$$U_{-} = \min_{i}(\max_{j}(a_{ij}) )$$ + +**MaxMin-Strategie** +Was sollte ich mindestens gewinnen? + +> 1. Mache eine liste mit dem minimalen gewinn +> 2. Daraus wähle den größten Gewinn + +$$U_{+} = \max_{j}(\min_{i}(a_{ij}) )$$ + +**Sattelpunkt** +> Nash-Gleichgewicht, wenn die Untergrenze und Obergrenze übereinstimmen. Dort ist auch der Sattelpunkt. +$$U_{-} = a_{i^* j^*} = U_{+}$$ + +![alt text](image-15.png) +## Gemischte Strategien + +> Spieler entscheidet zufällig zwischen Strategien +- Jede Strategie hat eine Wahrscheinlichkeit. $p_i$ +- Alle Wahrscheinlichkeiten addieren sich zu 1: $\sum_{i} p_i = 1$ +- Minimum 2 Strategien + +**Beispiel Spieler 1:** +$$p_1 = 70\%$$ + +$$p_2 = 1-p_1 = 30\%$$ + +**Beispiel Spieler 2:** +$$q_1 = 60\%$$ + +$$q_2 = 1-q_1 = 40\%$$ + +>Summe aus allen Zellen, eine Zelle wird mit der Wahrscheinlichkeit von Zeile und Spalte Multipliziert. + +**ACHTUNG KANN MAN NUR GRAPHISCH LÖSEN** +![alt text](image-19.png) + +### Lösen von Gemischten Strategien +> **Tip**: Beim Vereinfachen nicht die Klammern ausmultiplizeren, weil man einfach kürzen kann. +1. Setze Erwartungswerte gleich von Beiden Strategien für Beide Spieler + +**Berechnen von q aus Sicht von Spieler 1** +$$\mu_{S_{11}} = \mu_{S_{12}}$$ + +**Berechnung von p aus Sicht von Spieler 2** +$$\mu_{S_{21}} = \mu_{S_{22}}$$ + +2. Einsetzen der Erwartungswerte und U- und U+ ausrechnen +3. Wenn U- und U+ gleich sind, ist das Nash-Gleichgewicht + +> **INFO**: Wenn eine Freie Variable in der Gleichung ist, die gleichung in abhängigkeit von der freien Variable umstellen, nash-gleichgewicht ist dann von z.B. s abhängig. + +![alt text](image-20.png) +![alt text](image-22.png) + +## Gruppenentscheidungen +- Menge an Auswahlmöglichkeiten +- Jeder Möglichkeit wird eine Zahl ( $k \isin \N$ ) zugeordnet ( Rang ) +- Rangabbildung ist Surjetiv (Jede Möglichkeit hat einen Rang. Ein Rang kann aber mehrmals auftreten ) + +### Relationen +$$r(x) < r(y): \text{x ist besser als y}$$ +$$x \ R \ y: \text{x steht in Relation zu y}$$ + +>reflexiv schließt asymetrisch aus und umgekehrt + +- **transitiv**: $x \ R \ y \ und \ y \ R \ z \Rightarrow x \ R \ z$ + - $\text{Wenn 2 < 3 und 3 < 4, dann auch 2 < 4}$ +- **reflexiv**: $x \ R \ x$ + - $\text{2 <= 2}$ +- **asymetrisch**: $x \ R \ y \And y \ R \ x \Rightarrow False$ + - $\text{2 < 3, aber nicht 3 < 2}$ + +### Präferenzen +![](image-23.png) +![alt text](image-24.png) +>*Lesen*: Von links starten dann relation dann rechts +![](image-26.png) + +### Extern/Intern Diktator +![alt text](image-27.png) + +### Rangaddition +>Jede Wahlmöglichkeit wird mit der Summe der Ränge bewertet. Die Wahlmöglichkeit mit der niedrigsten Summe ist die beste. +![alt text](image-29.png) +![](image-30.png) + +### Condorcet-Verfahren (BEI MINUTE 40:00 bis 1:10:00) +![alt text](image-32.png) +![](image-33.png) + +### Einstimmigkeit +![alt text](image-34.png) + +![alt text](image-37.png) +- I.e. Irrelevante Änderungen hinten in der Reihenfolge, dürfen nicht vorne in der Reihenfolge etwas ändern. + +![](image-38.png) +![alt text](image-39.png) + +# 4. Automaten +- **Forderung**: Simulation schneller als Echtzeit + +## Bestandteile +- **Zellraum**: Spielfeld, i.d.R. eine Matrix in der jedes Element einer Zelle entspricht +- **Zustandsmenge**: Menge aller möglichen ( diskrete ) Zustände einer Zelle -> State Machine +- **Nachbarschaftsbeziehung**: Zustände der Nachbarzellen werden berücksichtigt +- **diskrete Zeit**: Zustandsänderung eines ZA erfolgt in diskreten Zeitschritten $\delta t$ (z.B. 1s) +- **lokale übergangsfunktion**: Berechnet den Zustand $Z_{t+\delta t}$ einer Zelle aus dem aktuellen Zustand $Z_t$ und den Zuständen der Nachbarzellen + +## Nachbarzellen (im 2D und Schachbrett ) +- **Moore-Nachbarschaft**: 8 Zellen um die Zelle +- **Von-Neumann-Nachbarschaft**: 4 Zellen um die Zelle + +## Übergangsfunktion ( Beispiel Conways Game of Life) +```python +def transition_function(cell, neighbors): + if cell == 0 and sum(neighbors) == 3: # Wenn Zelle tot und 3 Nachbarn lebendig + return 1 # Dann wird die Zelle wiederbelebt + elif cell == 1 and sum(neighbors) in [2, 3]: # Wenn Zelle lebendig und 2 oder 3 Nachbarn lebendig + return 1 + else: + return 0 +``` + +## Game of Life +Start mit 2D-Gitter. +Jede Zelle ist entweder lebendig oder tot und ändert ihren Zustand gemäß der Nachbarzellen: \ +Eine lebende Zelle mit 2 oder 3 lebenden Nachbarn bleibt am Leben, sonst stirbt sie. +Eine tote Zelle mit genau 3 lebenden Nachbarn wird lebendig. +Es werden ALLE Zellen mit dem aktuellen Zustand und den Nachbarzellen betrachtet, daraus wird der nächste Zustand berechnet, damit die Zellen parallel aktualisiert werden. + +**Kurzfassung**: +- Grid anlegen (2D-Liste) +- Jede Runde Nachbarn zählen +- Nach obigen Regeln neue Zustände berechnen +- Endlos aktualisieren und ausgeben. +# Mathe schriftlich rechnen +## Dividieren +![alt text](image-16.png) + +## Multiplizieren +![alt text](image-18.png) diff --git a/CDS1012-Introduction-to-Computational-Science/formulas/Zusammenfassung.pdf 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+![[Pasted image 20250131123257.png]] + +![[Pasted image 20250131123144.png]] +![[Pasted image 20250131123158.png]] +![[Pasted image 20250131123224.png]] +![[Pasted image 20250131123251.png]] + +

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